Special functions of mathematical (Geo-)physics [without chapter 1] 9783034805636, 3034805632

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Applied and Numerical Harmonic Analysis Series Editor John J. Benedetto University of Maryland College Park, MD, USA

Editorial Advisory Board Akram Aldroubi Vanderbilt University Nashville, TN, USA

Jelena Kovaˇcevi´c Carnegie Mellon University Pittsburgh, PA, USA

Andrea Bertozzi University of California Los Angeles, CA, USA

Gitta Kutyniok Technische Universit¨at Berlin Berlin, Germany

Douglas Cochran Arizona State University Phoenix, AZ, USA

Mauro Maggioni Duke University Durham, NC, USA

Hans G. Feichtinger University of Vienna Vienna, Austria

Zuowei Shen National University of Singapore Singapore, Singapore

Christopher Heil Georgia Institute of Technology Atlanta, GA, USA

Thomas Strohmer University of California Davis, CA, USA

St´ephane Jaffard University of Paris XII Paris, France

Yang Wang Michigan State University East Lansing, MI, USA

For further volumes: www.birkhauser-science.com/series/4968

Willi Freeden



Martin Gutting

Special Functions of Mathematical (Geo-)Physics

Willi Freeden Martin Gutting Geomathematics Group University of Kaiserslautern Germany

ISBN 978-3-0348-0562-9 ISBN 978-3-0348-0563-6 (eBook) DOI 10.1007/978-3-0348-0563-6 Springer Basel Heidelberg New York Dordrecht London Library of Congress Control Number: 2013931107 Mathematical Subject Classification (2010): 33-01, 42C10, 35Q86, 42B37, 42C05, 43A90, 33C10, 33C45, 33C55, 11P21, 42A16, 42B05, 42B05, 65B15, 86A10, 86A25, 86A30 c Springer Basel 2013  This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer Basel is part of Springer Science+Business Media (www.birkhauser-science.com)

Preface

An essential aim of geomathematics is the investigation of qualitative and quantitative structures of the Earth’s system to deepen our understanding of its complexity. In this respect, special functions comprise the essential instruments for mathematical interaction of abstraction and concretization. Special functions enable the formulation of a geoscientific problem by reduction such that a new, more concrete problem can be attacked within a well-structured framework, usually in the context of differential equations. A good understanding of special functions provides the capacity to recognize causality between the abstractness of the geomathematical concept and the impact on, as well as cross-sectional importance to, the geoscientific reality. Our purpose in this work is to present a textbook that allows the reader to concentrate on special fields such as the geosphere, hydrosphere, or atmosphere. In other words, the special functions to be discussed vary widely, depending on the chosen measurement parameters (gravitation, electric and magnetic fields, deformation, climate observables, fluid flow, etc.) and on the field characteristic (potential field, diffusion field, wave field). The differential equation under consideration determines the type of special functions that are needed in the desired reduction process. The diversity of geomathematical problems involves such a large number of scientific manifestations that our approach to any of them has to be selective. In consequence, since greater weight has to be given to some topics than to others, we have chosen to restrict ourselves to gravitation, geomagnetism, elasticity, and fluid flow theory. Gravitational field theory defines a canonical need to generate special function systems for the Laplace equation. Geomagnetism and electric current systems are closely related to the (pre-)Maxwell equation; the deformation of the solid Earth leads to function systems solving the Cauchy–Navier equation (at least when linear material behavior is assumed). Oceanic circulation and wind motion have to be handled in terms of vectorial function systems involving the Navier–Stokes equation or modifications of it. Unfortunately, we are confronted with the difficult challenge to characterize special function systems under adequate consistency in terms of less mathematically structured geometric features of a reference model (such as the geoid or the real Earth’s surface) as well as the intrinsic structure of v

vi

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underlying differential equations involving the laws of physics. Thus, at the present stage of geoscience, no compendium can be expected that is both geometrically consistent with modern navigation results and geophysically reflected by advanced mathematical settings. The complexity of a real “potato-like” Earth model is a striking obstacle that can only be overcome to some extent in today’s mathematics. Accordingly, the principles lie in the suitable transition to a regularly structured geometry for the Earth, namely, the ball in first approximation. This leads us to a prestructured framework, namely, spherically oriented special function systems. Looking at the special functions available in the geophysical literature today, we find that a spherical shape of the Earth is used in almost all publications. Indeed, by modern satellite positioning methods, the maximum deviation of the actual Earth’s surface from the average Earth’s radius (6,371 km) can be determined to be less than 0.4 %. Although a spheriodization, i.e., a mathematical formulation simply in spherical reference geometry, amounts to a strong restriction, it is at least acceptable for a large number of problems. Standard special functions since the time of C.F. Gauß are polynomial trial functions, conventionally called spherical harmonics. Spherical harmonics represent the analogs of trigonometric functions for orthogonal (Fourier) expansions on the sphere. In consequence, the use of spherical harmonics in diverse areas of geosciences is a well-established method, particularly for the purpose of decomposing scalar potentials. Nowadays, reference models for the Earth’s gravitational and magnetic potential, e.g., are widely known by tables of expansion coefficients of the frequency constituents of their potentials. However, it should be mentioned that vectorial potentials—even in a spherical Earth’s reference model— have their own nature. Concerning the mathematical modeling of vector fields, one is usually not interested in their separation into scalar Cartesian component functions. Instead, inherent physical properties should be observed. For example, the external gravitational field is curl-free, the magnetic field is divergence-free, the equations for incompressible flow, i.e., the Navier–Stokes equations, imply divergence-free vector solutions. In a spherical nomenclature as intended in our approach, all these physical constraints result in a formulation by certain operators, such as the surface gradient, surface curl gradient, surface divergence, surface curl. Our types of vector spherical harmonics satisfy these requirements by splitting the tangential part into a curl-free and a divergence-free field, thereby avoiding artificial singularities arising from the use of local coordinates. Basically, two transitions are undertaken in our approach to harmonics: first, the extension from the scalar to the vectorial case is strictly realized under physical constraints and, second, the definition of Legendre functions is canonically described under the phenomenon of rotational invariance on the sphere. The Legendre functions act as constituting elements for zonal functions, i.e., one-dimensional functions only depending on the polar distance of their two arguments. Altogether, the concept of spherical harmonics plays the central role in a geomathematical presentation of special functions, reflecting the significance of a polynomial nature in a spherically shaped Earth. In addition, spherical harmonics comprise the canonical candidates to represent the angular part in a radial/angular decomposition of solution systems for Laplace, Helmholtz, Cauchy–Navier, (pre-)Maxwell, and Navier–Stokes equations.

Preface

vii

It is surprising that, besides the geometrically implied spheriodization, the methodologically oriented periodization should take some space in a modern collection of special function systems of geomathematical importance. The reasons are twofold. First, the periodization leads back to the Fourier transform in Euclidean spaces that has been well understood for a long time and is extremely efficient in numerical computation. Second, the procedure of periodization leads to the Euler summation formula and the Poisson summation formula which show a close relationship to each other. The Green (lattice) functions forming the essential basis of these summation formulas indeed enable us to express key volume integrals in geophysics, such as the Newton integral, Mie potentials, elastic potentials, by mass lattice point conglomerates that discretely fill out the integration domain under consideration in an equidistributed way. A variety of examples for combined periodization and spheriodization occur in the theory of Earth-satellite relations (cf., e.g., Kaula 1966), mixing time-wise obligations on periodic orbits with space-wise approaches on torus and/or sphere. Satellite gravimetry (see, e.g., Pail and Plank (2002), Sneeuw (2000), Xu et al. (2008), and the references therein) is a particularly interesting area of spaceborne technology, where one-dimensional periodization in time is adequately involved in three-dimensional periodization and/or spheriodization in space. This textbook presents material used by the Geomathematics Group, University of Kaiserslautern, during the last several years to set up a contemporary theory of special functions of mathematical (geo-)physics. Our work canonically shows a threefold subdivision. Part I provides preparatory material concerning auxiliary functions such as the Gamma function and important classes of orthogonal polynomials. The general concept of orthogonal polynomials is introduced before we start to consider the classical polynomials, in particular the Jacobi polynomials and— as a special and very important case of them—the ultraspherical or Gegenbauer polynomials. Several basic mathematical and physical applications are included, such as quadrature rules, modeling of the electrostatic potential, and the quantummechanical description of oscillations. Part II deals with spherically structured function systems. It starts with the scalar theory of spherical harmonics in the Euclidean space R3 including the addition theorem, the Funk–Hecke formula, as well as the closure and completeness of spherical harmonics in the space of squareintegrable functions, i.e., the space of functions with finite signal energy. It follows the physically based theory of vector spherical harmonics. The basic tool to establish divergence-free and curl-free tangential fields is the Helmholtz decomposition theorem. An alternative system of vector spherical harmonics is also constructed in such a way that they can be identified as eigenfunctions of the Beltrami operator. This eigenfunction system plays a particular role in geomagnetism to separate, e.g., the crustal field from other magnetic sources. Both vector spherical harmonic systems are shown to be closed and complete in the space of square-integrable vector fields on the sphere. All properties characterize vector spherical harmonics as suitable trial functions to constitute the angular ingredients in a radial/angular decomposition of solutions of the Cauchy–Navier as well as the Navier–Stokes equation. Part III is devoted to the lattice function as the multi-dimensional,

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2.1

3.0

3.1

3.4

4

5

Fig. 1 Selected sections and chapters for a basic one-term course (note that some parts of Sect. 3.3 have to be included to complete Sect. 3.4. Sections 5.5, 5.6, 5.9, and 5.10 can be skipped in the oneterm course)

periodic analog to the well-known Bernoulli function. From a physical point of view, the lattice function is interpreted to be the Green function for the Laplace operator corresponding to the boundary conditions of periodicity. It turns out to be the most essential tool for the process of periodization in the context of Euler and Poisson summation formulas. Lattice point sums such as the Zeta and Theta functions, generated by the interaction of point potentials to each other, conclude our multi-periodic theory. It should be remarked that the whole palette of multi-periodic functions is provided in relation to the Laplace operator and arbitrary lattices so that this approach serves as a prototype for further formulations of more general (elliptic) partial differential equations. Essential ingredients of the textbook are the work of M¨uller (1952, 1969, 1998), Freeden et al. (1998), Freeden and Schreiner (2009), and Freeden (2011). Each chapter of the book is followed by exercises related to the presented material. The exercises reflect significant topics, mostly in computational geoapplications. In doing so, they not only confront the reader directly with the contents of the chapter, but also with additional knowledge in geomathematical fields of research, where special functions play a decisive role in applications. Students who wish to continue further studies should consult the literature given as supplements for each topic worked out by exercises. All in all, the content of the book is equally suitable for an education in geomathematics and a study in applied and harmonic analysis. The book is primarily meant to be a self-consistent introductory text for an advanced undergraduate or graduate course in special functions. The schedule of topics allows a selected subdivision into a one-term course (see Fig. 1) as well as a two-term course. In addition to the proposed sections and chapters in Fig. 1, further contents can be selected from Chap. 3, such as Sect. 3.2 and all details of Sects. 3.3 or 3.5–3.7, if the schedule allows it. The examples of Chap. 1 can be presented at any appropriate time. A two-term course with special emphasis on particular research fields should include additional material from Chaps. 5 and 7 documenting the special interest of a graduate student in gravitation, geomagnetism, deformation, atmospheric/oceanic flow, respectively. Chapters 6 and 8 give multi-dimensional radial/angular decompositions of harmonic and metaharmonic functions as a reference tool, thereby assuming as preparatory material the whole theory of the Gamma function as presented in Chap. 2. Another separate route going exclusively into the field of lattice functions includes Chaps. 9 and 10 while also requiring all the material of Chap. 2.

Preface

ix

In a book of this type, special precautions have been taken to ensure the accuracy of formulas and examples. It is a pleasure to acknowledge with thanks the valuable reading of the manuscript by Dipl.-Math. C. Blick, Dr. C. Gerhards, and Dr. I. Ostermann. We thank our student cand.-phys. Hanna Haug for pointing out some errors and slips in an early version of the manuscript. Finally, we want to thank Prof. Dr. J.J. Benedetto, Series Editor of Applied and Numerical Harmonic Analysis, and Dr. T. Hempfling, Managing Director of Springer Basel and Executive Editor of Birkh¨auser Basel, for valuable remarks and hints. Additionally, it is a pleasure to acknowledge the courtesy and ready cooperation of Dr. B. Hellriegel and all staff members of Birkh¨auser/Springer, Basel. Kaiserslautern, Germany

Willi Freeden, Martin Gutting

About the Authors

Willi Freeden: Studies in mathematics, geography, and philosophy at the RWTH Aachen; 1971, diploma in mathematics; 1972, Staatsexamen in mathematics and geography; 1975, Ph.D. in mathematics; 1979, habilitation in mathematics; 1981/1982, visiting research professor at The Ohio State University, Columbus (Department of Geodetic Science and Surveying); 1984, professor of mathematics at the RWTH Aachen (Institute of Pure and Applied Mathematics); 1989, professor of technomathematics (industrial mathematics); 1994, head of the Geomathematics Group; 2002–2006, vice-president for research and technology at the University of Kaiserslautern; 2009, editor in chief of the International Journal on Geomathematics (GEM); 2010, editor of the Handbook of Geomathematics, member of the editorial board of eight international journals. Martin Gutting: Studies in mathematics at the University of Kaiserslautern; 2003, diploma in mathematics, focus on geomathematics; 2007, Ph.D. in mathematics, postdoc researcher at the University of Kaiserslautern, lecturer in the course of geomathematics (in particular for constructive approximation, special functions, and inverse problems); 2011, lecturer for engineering mathematics at the University of Kaiserslautern and DHBW Mannheim.

xi

Contents

1

Introduction: Geomathematical Motivation .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Example: Gravitation (Laplace and Poisson Equation) . . . . . . . . . . . . 1.2 Example: Geomagnetism (Maxwell’s Equations) . . . . . . . . . . . . . . . . . . 1.3 Example: Fluid Flow (Navier–Stokes Equation) . . . . . . . . . . . . . . . . . . . 1.4 Example: Elastic Field (Cauchy–Navier Equation).. . . . . . . . . . . . . . . .

Part I 2

3

Auxiliary Functions

The Gamma Function .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Definition and Functional Equation . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Euler’s Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Stirling’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Pochhammer’s Factorial . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Exercises (Incomplete Gamma and Beta Function, Applications in Statistics) . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

25 25 29 33 38 43

Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 47 3.1 Properties of Orthogonal Polynomials . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 56 3.2 Quadrature Rules and Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . 66 3.3 The Jacobi Polynomials .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 70 3.4 Ultraspherical Polynomials . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 80 3.5 Application of the Legendre Polynomials in Electrostatics . . . . . . . . 90 3.6 Hermite Polynomials and Applications . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95 3.7 Laguerre Polynomials and Applications . . . . . . . .. . . . . . . . . . . . . . . . . . . . 98 3.8 Exercises (Gaussian Integration, Legendre Series, Kernel Expansions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 101

Part II 4

1 1 9 13 16

Spherically Oriented Functions

Scalar Spherical Harmonics in R3 . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 113 4.1 Basic Notation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 114 4.2 Orthogonal Invariance.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 119 xiii

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Contents

Homogeneous Polynomials on the Unit Sphere in R3 .. . . . . . . . . . . . . Closure and Completeness of Spherical Harmonics .. . . . . . . . . . . . . . . The Funk–Hecke Formula and the Irreducibility of Scalar Harmonics.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Green’s Function with Respect to the Beltrami Operator . . . . . . . . . . The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Exercises (Low Discrepancy Method, Locally Supported Wavelets, Up Function, Anharmonic Functions for the Ball, Fast Multipole Method, Wigner Matrices, Quaternionic Generation of Spherical Harmonics) .. . . . .

125 144

Vectorial Spherical Harmonics in R3 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Basic Notation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Definition of Vector Spherical Harmonics . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 The Helmholtz Decomposition Theorem . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Closure and Completeness of Vector Spherical Harmonics .. . . . . . . 5.5 Homogeneous Harmonic Vector Polynomials . .. . . . . . . . . . . . . . . . . . . . 5.6 Vectorial Beltrami Operator . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.7 Vectorial Addition Theorem . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.8 Vectorial Funk–Hecke Formulas . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.9 Vectorial Counterparts of the Legendre Polynomial .. . . . . . . . . . . . . . . 5.10 Application to Elastic Fields . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.11 Exercises (Uncertainty Principle, Classification of Zonal Functions, Coupling Integrals and Navier–Stokes Equation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

211 212 218 220 223 231 236 239 246 251 254

4.3 4.4 4.5 4.6 4.7 4.8

5

6

7

8

q

154 159 165

169

264

Spherical Harmonics in R . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Nomenclature and Basics . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Integral Theorems for the Laplace Operator . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Integral Theorems for the Laplace–Beltrami Operator .. . . . . . . . . . . . 6.4 Homogeneous Harmonic Polynomials . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Spherical Harmonics of Dimension q . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Integral Theorems for the Helmholtz–Beltrami Operator . . . . . . . . . . 6.7 Exercises (Cartesian Generation of Spherical Harmonics, Best Approximations) . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

285 286 290 297 302 312 330

Classical Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Derivation and Definition of Bessel Functions .. . . . . . . . . . . . . . . . . . . . 7.2 Orthogonality Relations .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 Bessel Functions with Integer Index . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Exercises (Bessel Function Expansions, Hankel Transform and Discontinuous Integrals) . . . . . . . .. . . . . . . . . . . . . . . . . . . .

347 347 354 355

q

Bessel Functions in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Regular Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Modified Bessel Functions .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 Hankel Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

337

359 363 363 370 372

Contents

xv

8.4 8.5 8.6

Kelvin Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 378 Expansion Theorems .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 385 Exercises (Helmholtz Equation, Entire Solutions, Bessel Function Like Asymptotics) . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 390

Part III 9

Periodically Oriented Functions

Lattice Functions in R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Bernoulli Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Periodic Polynomials.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Lattice Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 Euler Summation Formula .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.5 Riemann Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.6 Poisson Summation Formula for the Laplace Operator . . . . . . . . . . . . 9.7 Theta Function.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.8 Exercises (Trapezoidal Rule, Periodic Sobolev Spaces, Projection Method) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . q

10 Lattice Functions in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1 Lattices in Euclidean Spaces. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 Periodic Polynomials.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3 Lattice Function for the Laplace Operator . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4 Euler Summation Formula for the Laplace Operator .. . . . . . . . . . . . . . 10.5 Zeta Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.6 Integral Asymptotics for Lattice Functions . . . . .. . . . . . . . . . . . . . . . . . . . 10.7 Poisson Summation Formula . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.8 Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.9 Exercises (Algebraic, Periodic, and Spherical Splines, Lattice Point Sums, Lattice Point Distributions) .. . . . . . . . . . . . . . . . . . .

395 395 398 400 404 411 415 419 420 427 427 430 432 436 439 443 448 455 460

11 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 483 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 485 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 495

Chapter 2

The Gamma Function

In what follows, we introduce the classical Gamma function in Sect. 2.1. It is essentially understood to be a generalized factorial. However, there are many further applications, e.g., as part of probability distributions (see, e.g., Evans et al. 2000). The main properties of the Gamma function are explained in this chapter (for a more detailed discussion the reader is referred to, e.g., Artin (1964), Lebedev (1973), M¨uller (1998), Nielsen (1906), and Whittaker and Watson (1948) and the references therein). We briefly consider Euler’s Beta function in Sect. 2.2 and use it to recursively compute the volume of the .q  1/-dimensional unit sphere Sq1  Rq . As outstanding property of the Gamma function the Stirling formula is verified in Sect. 2.3. It leads us to the so-called duplication formula (Lemma 2.3.3) which will simplify a lot of calculations in later chapters. The extension of the Gamma function to complex values is studied in Sect. 2.4. In doing so, we introduce Pochhammer’s factorial and Euler’s constant  . Moreover, we establish product representations for the Gamma function as well as for trigonometric functions. In Sect. 2.5 the incomplete Gamma and Beta functions are briefly presented in form of some exercises and their relation to probability distributions is indicated.

2.1 Definition and Functional Equation For real values x > 0, we consider the integrals Z

1 0

and

Z

et t x1 dt;

1 1

et t x1 dt:

W. Freeden and M. Gutting, Special Functions of Mathematical (Geo-)Physics, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-0348-0563-6 2, © Springer Basel 2013

(2.1.1)

(2.1.2)

25

26

2 The Gamma Function

In order to show the convergence of (2.1.1) we observe that 0 < et t x1  t x1 holds true for all t 2 .0; 1. Therefore, for " > 0 sufficiently small, we have Z

1

t x1

e t

"

Z

1

dt 

t

"

x1

ˇ t x ˇˇ1 1 "x dt D D  : x ˇ" x x

(2.1.3)

Consequently, for all x > 0, the integral (2.1.1) is convergent. To assure the convergence of (2.1.2) we observe that 1 et t x1 D P1

tk kD0 kŠ

t x1 

1

tn nŠ

t x1 D

nŠ t nxC1

(2.1.4)

for all n 2 N and t  1. This shows us that Z

A 1

Z

et t x1 dt  nŠ

A 1

1 t nxC1

dt D nŠ

ˇA   1 t nCx ˇˇ nŠ D  1 x  n ˇ1 x  n Anx (2.1.5)

provided that A is sufficiently large and n is chosen such that n  x C 1. Thus, the integral (2.1.2) is convergent which we summarize in the following lemma. Lemma 2.1.1. For all x > 0, the integral Z

1 0

et t x1 dt

(2.1.6)

is convergent. Definition 2.1.2. The function x 7!  .x/; x > 0, defined by Z  .x/ D

1 0

et t x1 dt

(2.1.7)

is called the Gamma function (see Fig. 2.3 (right) for an illustration). Obviously, we have the following properties: (i)  is positive R 1 for all x > 0, (ii)  .1/ D 0 et dt D 1. We can use integration by parts to obtain for x > 0: Z  .x C 1/ D

1 0

Z

Dx

ˇ1 et t x dt D et t x ˇ0 

1 0

et t x1 dt D x  .x/:

Z 0

1

.et /xt x1 dt

(2.1.8)

2.1 Definition and Functional Equation

27

Lemma 2.1.3. The Gamma function  satisfies the functional equation  .x C 1/ D x  .x/; x > 0:

(2.1.9)

Moreover, by iteration for x > 0 and n 2 N;  .x C n/ D .x C n  1/  : : :  .x C 1/x  .x/ D  .n C 1/ D

.x C i  1/ .x/; (2.1.10)

i D1

Y n  n Y i  .1/ D i D nŠ: i D1

n Y

(2.1.11)

i D1

In other words, the Gamma function can be interpreted as an extension of factorials. Lemma 2.1.4. The Gamma function  is differentiable for all x > 0 and we have Z 1  0 .x/ D et ln.t/t x1 dt: (2.1.12) 0

Proof. For x > jhj > 0, we use the formula t y D eln.t /y , y > 0, t > 0: Z 1 Z 1 t xCh1 e t dt D et Cln.t /.xCh1/ dt:  .x C h/ D

(2.1.13)

By Taylor’s formula we find 0 < # < 1 such that Z 1    .x C h/   .x/ D et e ln.t / eln.t /.xCh/  eln.t /x dt

(2.1.14)

0

Z

0

0

1

D 0

Z

Dh

  et e ln.t / h ln.t/t x C 12 h2 .ln.t//2 t xC#h dt

1

e

t

ln.t/t

x1

dt C

0

1 2 2h

Z

1 0

et .ln.t//2 t xC#h1 dt:

This gives us the differentiability of  if the second integral is bounded. Consider the following estimate (we employ that .ln.t//2  t 2 for t  1 and that et  1 for t 2 Œ0; 1/ Z 0

1

et .ln.t//2 t xC#h1 dt D

Z Z



1

0 1 0

et .ln.t//2 t xC#h1 dt C

.ln.t//2 t xC#h1 dt C

  .2 C x C #h/ C

This provides us with the desired result.

Z 1

Z

1

1

1

et .ln.t//2 t xC#h1 dt

et t 2 t xC#h1 dt

2 < 1: .x C #h/3

(2.1.15) t u

28

2 The Gamma Function

An analogous proof can be given to show that  is infinitely often differentiable for all x > 0 and Z 1  .k/ .x/ D

et .ln.t//k t x1 dt; k 2 N:

0

(2.1.16)

Lemma 2.1.5 (Gauß’ Expression of the Second Logarithmic Derivative). For x > 0,  0 2 (2.1.17)  .x/ <  .x/ 00 .x/: Equivalently, we have 

d dx

2

 00 .x/  ln. .x// D  .x/



 0 .x/  .x/

2

> 0;

(2.1.18)

i.e., x 7! ln. .x//, x > 0, is a convex function or  is logarithmic convex. Proof. We start with 

 0 .x/

2

Z

1

D Z

et ln.t/t x1 dt

2

0 1

D

e

 2t

0

t

x1 2

ln.t/e

 2t

t

x1 2

2 dt

:

(2.1.19)

The Cauchy–Schwarz inequality yields (note that equality cannot occur since the two functions are linearly independent): 

 0 .x/

2

Z
0; dx 2 dx  .x/ . .x//2

(2.1.21) t u

which yields (2.1.18). Note that ln. .// is convex, i.e., for t 2 Œ0; 1 ln . .tx C .1  t/y//  t ln. .x// C .1  t/ ln. .y//     D ln  t .x/ C ln  1t .y/   D ln  t .x/   1t .y/

(2.1.22)

which is equivalent to  .tx C .1  t/y/   t .x/   1t .y/ with x; y > 0.

2.2 Euler’s Beta Function

29

Fig. 2.1 The illustration of the coordinate transformation relating the Beta and the Gamma functions

2.2 Euler’s Beta Function Next, we notice that for x; y > 0; the integral Z

1 0

t x1 .1  t/y1 dt

(2.2.1)

is convergent. Definition 2.2.1. The function .x; y/ 7! B.x; y/, x; y > 0, defined by Z B.x; y/ D

1 0

t x1 .1  t/y1 dt

(2.2.2)

is called the Euler Beta function. For x; y > 0, we see that Z  .x/ .y/ D

1 0

et t x1 dt

Z

1

0

es s y1 ds D

Z

1

Z

0

0

1

e.t Cs/ t x1 s y1 dt ds: (2.2.3)

Note that the transition from one-dimensional to two-dimensional integrals is permitted by Fubini’s theorem. We make a coordinate transformation (see Fig. 2.1) as follows: t D u.1  v/;

0  u < 1;

(2.2.4)

s D uv;

0  v  1:

(2.2.5)

It is not difficult to verify that the functional determinant of the coordinate transformation is given by ˇ ˇ ˇ1  v uˇ @.t; s/ ˇ ˇ D u.1  v/ C uv D u  0: Dˇ (2.2.6) v uˇ @.u; v/

30

2 The Gamma Function

Thus, we find Z

1

Z

0

1 0

e.t Cs/ t x1 s y1 dt ds D

Z Z

Z

1

Z

0

D Z

1

0

0 1 0

1

D

1

eu .u.1  v//x1 .uv/y1 u du dv eu uxCy2 .1  v/x1 vy1 u du dv

eu uxCy1 du

Z

0

0

1

vy1 .1  v/x1 dv: (2.2.7)

This leads us to the following theorem: Theorem 2.2.2. For x; y > 0, B.x; y/ D

 .x/ .y/ :  .x C y/

(2.2.8)

In particular,  B

1 1 ; 2 2



Z

1

D 0

1

1

t  2 .1  t/ 2 dt D 2

D 2 arcsin.1/ D 2 Therefore, we have

This shows that

Z 0

1

1

.1  u2 / 2 du

(2.2.9)

 D : 2

   2 12 D :  .1/

(2.2.10)

  Z 1 p 1 1 D D  et t  2 dt: 2 0

(2.2.11)

Other types of integrals can be derived from Z

1

e 0

t ˛

1 dt D ˛ uDt ˛

Z

1

u

e u 0

1 ˛ 1

1 du D  ˛

  1 ; ˛

˛ > 0:

(2.2.12)

Lemma 2.2.3. For ˛ > 0, Z

1

e 0

t ˛

 dt D 

 ˛C1 : ˛

(2.2.13)

2.2 Euler’s Beta Function

31

In particular, Lemma 2.2.3 yields for ˛ D 2 Z

1 0

2

et dt D 

p     1 1  3 D  D : 2 2 2 2

(2.2.14)

Moreover, we have Z

1 0

and

Z

1 0

˛

t x1 et dt D

2

t x1 e˛t dt D

1 x   ; ˛ ˛

x; ˛ > 0;

1 x x  ˛ 2 ; 2 2

x; ˛ > 0:

(2.2.15)

(2.2.16)

Within the notational framework of polar coordinates (see (6.1.17) and (6.1.18) for details) we are now prepared to give the well-known calculation of the area kSq1 k of the unit sphere Sq1 in Rq : By definition, we set kS0 k D 2. Clearly, S1 is the unit circle in R2 , i.e., S1 D fx 2 R2 W jxj D 1g. Hence, its area is equal to kS1 k D 2:

(2.2.17)

Furthermore, S2 D fx 2 R3 W jxj D 1g is the unit sphere in R3 . Thus, its area is known to be equal to kS2 k D 4:

(2.2.18)

We are interested in deriving the area of the sphere Sq1 in Rq .q > 3/: q1

kS

Z kD

Sq1

dS.q1/ ..q/ /:

(2.2.19)

In terms of spherical coordinates (6.1.17) and (6.1.18) in Rq the surface element dS.q1/ ./ of the sphere Sq1 in Rq admits the representation p    dS.q1/ .q/ D dS.q2/ 1  t 2 .q1/ dt p  C .1/q1 t dV.q1/ 1  t 2 .q1/ :

(2.2.20)

Now, we notice that dV.q1/

p    q3 1  t 2 .q1/ D t.1  t 2 / 2 dt dS.q2/ .q1/ D .1/q1 t.1  t 2 /

q3 2

(2.2.21)

  dS.q2/ .q1/ dt:

32

2 The Gamma Function

In addition, it is not difficult to see that p    q1 1  t 2 .q1/ D .1  t 2 / 2 dS.q2/ .q1/ : dS.q2/

(2.2.22)

Combining our results we are led to the identity   p dS.q1/ t "q C 1  t 2 .q1/ D .1  t 2 /

q3 2



(2.2.23)

   1  t 2 C t 2 dS.q2/ .q1/ dt;

p where we have used the decomposition .q/ D t "q 1  t 2 .q1/ . Note that "1 ; : : : ; "q is the canonical orthonormal system in Rq . In brief, we obtain     q3 dS.q1/ .q/ D .1  t 2 / 2 dS.q2/ .q1/ dt;

(2.2.24)

such that Z

kSq1 k D

1 1

Z Sq2

Z

D kSq2 k

.1  t 2 / 1

1

q3 2

.1  t 2 /

dS.q2/ ..q1/ / dt

q3 2

(2.2.25)

dt:

For the computation of the remaining integral it is helpful to use some facts known from the Gamma function as well as Euler’s Beta function. More explicitly, Z

1 1

2

.1  t /

q3 2

Z dt D 2

0

 DB

1

2

.1  t /

1 q1 ; 2 2

q3 2

t 2 Dv

Z

1

dt D

 D



0

1

v 2 .1  v/

q3 2

dv

(2.2.26)

    p q1 q1   2 2 2     D :  q2  q2

1

By recursion we get the following lemma from (2.2.26): Lemma 2.2.4. For q  2, q

q1

kS

2 k D 2 q :  2

(2.2.27)

q1

The area of the sphere SR .y/ with center y 2 Rq and radius R > 0 is given by q

q1

kSR .y/k D kSq1 k Rq1 D 2

2   Rq1 :  q2

(2.2.28)

2.3 Stirling’s Formula

33

Furthermore, using .q/ D radius R > 0 is given by q kBR .y/k

x , jxj

q

the volume of the ball BR .y/ with center y 2 Rq and

Z

Z

D

q

BR .y/

dV.q/ .x/ D

Z

R

 q1

Sr

rD0 q

2 D 2 q   2

Z 0

.y/

R

dS.q1/ ..q/ /

dr

(2.2.29)

q

r

q1

2  Rq : dr D  q  2 C1

2.3 Stirling’s Formula Next, we are interested in the behavior of the Gamma function  for large positive values x. This provides us with the so-called Stirling’s formula, a result which we apply to verify the helpful duplication formula and to extend the Gamma function in Sect. 2.4. Theorem 2.3.1 (Stirling’s Formula). For x > 0, ˇ ˇ r ˇ ˇ 2 ˇ p  .x/ ˇ ˇ 2x x1=2 ex  1ˇ  x :

(2.3.1)

Proof. Regard x as fixed and substitute t D x.1 C s/;

1  s < 1;

dt Dx ds

(2.3.2)

in the defining integral of the Gamma function. We obtain Z  .x/ D

1 0

et t x1 dt D

x x

Dx e

Z

1 1

Z

1 1

exxs x x1 .1 C s/x1 x ds

(2.3.3)

.1 C s/x1 exs ds D x x ex I.x/:

Our aim is to verify that I.x/ satisfies ˇ r ˇˇ ˇ 2 ˇ 2 ˇ ˇI.x/  ˇ : ˇ x ˇ x such that ˇ ˇ ˇ  .x/ r 2 ˇ 2 ˇ ˇ ˇ x x  ˇ ; ˇx e x ˇ x

ˇ ˇ r ˇ ˇ 2  .x/ ˇ i:e:; ˇ : p  1ˇˇ  x1=2 x x x e 2

(2.3.4)

(2.3.5)

34

2 The Gamma Function 5 4 3 2 1 0 −1 −2 −3 −1

0

1

2

3

4

5

6

Fig. 2.2 The functions s 7! u2 .s/ D s  ln.1 C s/ (blue) and u.s/ defined by (2.3.7) (red)

For that purpose we write 2 .s/

.1 C s/x exs D exp .x.s  ln.1 C s/// D exu

;

(2.3.6)

where (cf. Fig. 2.2) ( u.s/ D

1

js  ln.1 C s/j 2 js  ln.1 C s/j

1 2

;

s 2 Œ0; 1/;

;

s 2 .1; 0/:

(2.3.7)

We set up the Taylor expansion of u2 for s 2 .1; 1/ at 0: d2 u2 du2 s2 .0/s C .#s/ ds ds 2 2   1 1 s2 sC ; D0C 1 1C0 .1 C s#/2 2

u2 .s/ D u2 .0/ C

(2.3.8)

where # 2 .0; 1/. Therefore, u2 .s/ D

1 s2 2 .1 C s#/2

(2.3.9)

with 0 < # < 1. We interpret # as a uniquely defined function of s, i.e., # W s 7! #.s/; such that

2.3 Stirling’s Formula

35

1 u.s/ 1 D p s .1 C s#.s// 2

(2.3.10)

is a positive continuous function for s 2 .1; 1/ with the property ˇ ˇ ˇ ˇ ˇ  ˇ ˇ u.s/ ˇ 1 ˇˇ s#.s/ ˇˇ 1 1 ˇˇ ˇˇ 1 ˇ ˇ ˇ s  p2 ˇ D ˇ p2 1 C s#.s/  1 ˇ D p2 ˇ 1 C s#.s/ ˇ ˇ ˇ ˇ s#.s/u.s/ ˇ ˇ ˇ D j#.s/j  ju.s/j  ju.s/j: Dˇ ˇ s

(2.3.11)

From u2 .s/ D s  ln.1 C s/ follows that 2u du D

s ds: 1Cs

(2.3.12)

Obviously, s W u 7! s.u/; u 2 R, is of class C.1/ .R/ and thus, Z I.x/ D

1 1

.1 C s/x1 exs ds D 2

Z

C1

2

exu

1

u du: s.u/

(2.3.13)

We are able to deduce that ˇ ˇ ˇ ˇ p Z 1 xu2 ˇ ˇ Z 1 xu2 u p Z C1 xu2 ˇ ˇ ˇI.x/  2 2 ˇ D ˇ2 ˇ du  e du e 2 e du ˇ ˇ ˇ ˇ s.u/ 0 1 1 ˇ Z 1   ˇ ˇ ˇ 1 u 2 D ˇˇ2 p duˇˇ exu s.u/ 2 1 ˇ ˇ Z 1 1 ˇ 2 ˇ u  p ˇˇ du exu ˇˇ 2 s.u/ 2 1 Z 1 Z 1 2 xu2 e exu u du: (2.3.14) 2 juj du D 4 1

0

Note that we can use the integral (2.2.16) with ˛ D x and x D 1; Z

1

2

u11 exu du D

Z

0

1

2

exu du D

0

1 2

x 1=2 

1 2

p  D p ; 2 x

(2.3.15)

as well as with ˛ D x and x D 2 in (2.2.16) Z

1

u 0

21 xu2

e

Z

1

du D 0

2

exu u du D

1 2

x 1  .1/ D

1 : 2x

(2.3.16)

36

2 The Gamma Function

This yields: r ˇˇ r ˇ ˇ ˇ p 1  ˇ ˇˇ ˇ ˇ D ˇI.x/  2 ˇˇ  4 1 D 2 : ˇI.x/  2 2 ˇ 2 xˇ ˇ x ˇ 2x x

(2.3.17) t u

This leads to the desired result. Remark 2.3.2. Stirling’s formula can be rewritten in the form lim p

x!1

 .x/ 2x x1=2 ex

D 1:

(2.3.18)

a > 0:

(2.3.19)

An immediate application is the limit relation lim

x!1

 .x C a/ D 1; x a  .x/

This can be seen from Stirling’s formula by lim p

x!1

 .x C a/ 1

2.x C a/xCa 2 exa

D1

(2.3.20)

due to the relation 1

1

1

.x C a/xCa 2 D x xCa 2 .1 C xa /xCa 2

(2.3.21)

and the limits .1 C xa /x D 1; x!1 ea lim

1

lim .1 C xa /a 2 D 1:

x!1

(2.3.22)

Next, we prove the so-called Legendre relation or duplication formula. Lemma 2.3.3 (Duplication Formula). For x > 0 we have 2

x1



x  2

 

xC1 2

 D

p   .x/:

(2.3.23)

Proof. We consider the function x 7! ˚.x/; x > 0; defined by ˚.x/ D

2x1  . x2 / . xC1 2 /  .x/

(2.3.24)

for x > 0. Setting x C 1 instead of x we find the following functional equation for the numerator     x  x C 1 x xC1 2x   C 1 D 2x1 x  ; (2.3.25) 2 2 2 2

2.3 Stirling’s Formula

37

such that the numerator satisfies the same functional equation as the denominator. This means ˚.x C 1/ D ˚.x/; x > 0. By repetition we get for all n 2 N and x fixed ˚.x C n/ D ˚.x/. We let n tend to 1. For the numerator of ˚.x C n/ we then find by use of the result in Remark 2.3.2, i.e., by using (2.3.19) twice, that xCnC1 2xCn1  . xCn / 2 / . 2 x    n 2 D 1: n!1 xCn1 n 2 n xC1 2 .2/ 2  .2/ 2

lim

(2.3.26)

For the denominator we consider lim

n!1 xCn1 2

 .x C n/  n  x2 n xC1  n 2 .2/ 2  .2/ 2

D lim

n!1

D lim

n!1

2xCn1

n!1

 .x C n/

2

2

. . n / / . n2 /n1 en 2 . n /n12en 2 2

en 

2 !1  . n2 / . n2 /n1 en 2

p 1 2nnCx 2 en 

 n 2 !1  .2/ n n1 n . 2 / e 2

2xCn

D lim p

 n xC 12

 .x C n/  n nCx 12 2



 .x C n/

1 Dp ; 

(2.3.27)

since Stirling’s formula yields that lim n!1 p i.e.,

 . n2 / n

1

n

2. n2 / 2  2 e 2

D 1;

(2.3.28)

 n 2 .2/ D 1; lim n!1 2. n /n1 en 2

(2.3.29)

and by the same arguments as in Remark 2.3.2 (set a in (2.3.20) to x and x in (2.3.20) to n) we find that  .x C n/ lim p D 1: 1 2nnCx 2 en

(2.3.30)

n!1

Therefore, we get for every x > 0 and all n 2 N; ˚.x/ D ˚.x C n/ D lim ˚.x C n/ D n!1

p :

(2.3.31)

38

2 The Gamma Function

A periodic function with this property must be constant. This proves the lemma. u t A generalization of the Legendre relation (“duplication formula”) is the Gauß multiplication formula that can be verified by analogous arguments. Lemma 2.3.4. For x > 0 and n  2,



x n

 

xC1 n



  ::: 

 xCn1 n1 p nx D .2/ 2 n  .x/ : (2.3.32) n

2.4 Pochhammer’s Factorial Thus far, the Gamma function  is defined for positive values, i.e., x 2 R>0 . We are interested in an extension of  to the real line R (or even to the complex plane C) if possible. Definition 2.4.1. The so-called Pochhammer factorial .x/n with x 2 R and n 2 N is defined by .x/n D x.x C 1/ : : : .x C n  1/ D

n Y

.x C i  1/:

(2.4.1)

i D1

For x > 0, it is clear that .x/n D or

 .x C n/  .x/

.x/n 1 D :  .x C n/  .x/

(2.4.2)

(2.4.3)

The left-hand side is defined for x > n and gives the same value for all n 2 N 1 with n > x. We may use this relation to define  .x/ for all x 2 R, and we see that this function vanishes for x D 0; 1; 2; : : : (see Fig. 2.3 (left)). We know that the Gamma integral is absolutely convergent for x 2 C with Re.x/ > 0 and represents a holomorphic function for all x 2 C with Re.x/ > 0. Moreover, the Pochhammer factorial .x/n can be defined for all complex x. Because of (2.4.3) with n chosen 1 sufficiently large, we have a definition of  .x/ for all x 2 C. This is summarized in the following lemma. Lemma 2.4.2. The  -function is a meromorphic function that has simple poles in 1 is an entire analytic 0; 1; 2; : : : (see Fig. 2.3 (right)). The reciprocal x 7!  .x/ function (see Fig. 2.3 (left) for an illustration).

2.4 Pochhammer’s Factorial

39

5

10

4

8 6

3

4

2

2

1

0

0

−2 −4

−1

−6

−2

−8

−3 −4

−3

−2

−1

0

1

2

3

4

−10 5 −5

−4

−3

−2

−1

0

1

2

3

4

5

Fig. 2.3 The reciprocal of the Gamma function (left) and the Gamma function (right) on the real line R

Lemma 2.4.3. For x 2 C, n1 Y  1 D lim nx x 1 C xk : n!1  .x/

(2.4.4)

kD1

Proof. Because of (2.4.3) the identity .x/n  .n/ 1 D  .n/  .x C n/  .x/

(2.4.5)

is valid for all x 2 C with Re.x/ > n. Furthermore it is easy to see that n1 Y .x/n x .x C 1/.x C 2/ : : : .x C n  1/ 1C Dx Dx :  .n/ 1  2 : : : .n  1/ k

(2.4.6)

kD1

Combining the two and multiplying with nx we find that n1 Y  .x C n/ x x 1C D n : x x n  .n/ .x/ k

(2.4.7)

kD1

Now, we use (2.3.19) with x D n and a D x, i.e., lim

n!1

 .x C n/ D 1;  .n/nx

x > 0;

(2.4.8)

on the left-hand side and obtain for x > 0; n1 Y x 1  .x C n/ 1 x 1 C D lim D lim ; n x n!1 nx  .n/  .x/ n!1  .x/ k kD1

(2.4.9)

40

2 The Gamma Function

which proves Lemma 2.4.3 for x > 0. To determine if this limit is also defined for x  0 we consider once again s  ln.1 C s/ (cf. Fig. 2.2) with 1 < s < 1 from (2.3.9) in the proof of Theorem 2.3.1: 0  s  ln.1 C s/ D Therefore, we can put s D i.e., with # D 0:

1 s2 2 .1 C #s/2 1 k

   ln 1 C k1 

1 k

This immediately proves that lim

n  P 1

n!1 kD1

lim

n!1

n X  kD1

1 k

# D #.s/ 2 .0; 1/:

(2.4.10)

and estimate the right-hand side with its maximum,

0

Moreover,

;

k

1 : 2k 2

(2.4.11)

   ln 1 C k1 exists and is positive.

n X  1   1  ln 1 C k D lim k  ln.k C 1/ C ln.k/ n!1

D lim

n!1

kD1

X n kD1

1 k

  ln.n C 1/ D ;

(2.4.12)

where  denotes Euler’s constant  D lim

m!1

 m1 X kD1

 1  ln m  0:577215665 : : : : k

(2.4.13)

Assume now that x 2 R. If k  2jxj, then 0

x k

   ln 1 C xk
e k 2 ;

(2.4.16)

2.4 Pochhammer’s Factorial

41

which shows that n1 Y

lim

n!1

kD1

1 Y   x   x x k 1C k e 1 C xk e k D

(2.4.17)

kD1

exists for all x. Furthermore, n1 Y

x ej

j D1

 X  X  n1  n1 1 1 D exp x D exp x j j  x ln.n/ C x ln.n/ j D1

j D1

 X  n1 x 1  x ln.n/ ; D n exp x j

(2.4.18)

j D1

where

 X  n1 1 lim exp x  x ln.n/ D ex : j

n!1

Therefore, lim nx x n!1

n1 Q kD1

1C

lim nx x

n!1

(2.4.19)

j D1

x k



exists for all x 2 R and it holds that

n 1 Y Y     x 1 C xk D xex 1 C xk e k ;

kD1

(2.4.20)

kD1

where the infinite product is also convergent for all x 2 R. By similar arguments these results can be extended for all x 2 C. t u The proof of Lemma 2.4.3 also shows us the following lemma. Lemma 2.4.4. For x 2 C, 1  Y 1 x  x x 1C D xe e k:  .x/ k

(2.4.21)

kD1

Let us consider the expression Q.x/ D

1  .x/ .1  x/ sin.x/; 

(2.4.22)

which has no singularities and is holomorphic for all x 2 C. It is not difficult to show that

42

2 The Gamma Function

1  .1 C x/ .1  x/ sin.x/ x 1 D  .x/ .2  x/ sin.x/: .1  x/

Q.x/ D

(2.4.23) (2.4.24)

Obviously, by using (2.4.23) for x D 0 and (2.4.24) for x D 1, we obtain sin.x/ D 1; x!0 x sin.x/ D 1: Q.1/ D  .1/ .1/ lim x!1 .1  x/

Q.0/ D  .1/ .1/ lim

(2.4.25) (2.4.26)

In the interval Œ0; 1 the function Q is positive and twice continuously differentiable. With the duplication formula (Lemma 2.3.3) we get x 



xC1 Q Q 2 2

 D Q.x/;

(2.4.27)

which is easily verified. Setting R.x/ D ln.Q.x//, we see that x 



xC1 R CR 2 2

 D R.x/:

(2.4.28)

By differentiation we obtain 1 00  x  1 00 R C R 4 2 4



xC1 2



D R00 .x/:

(2.4.29)

As the second order derivative R00 is continuous on the compact interval Œ0; 1, there is a value  2 Œ0; 1 such that jR00 ./j  jR00 .x/j; x 2 Œ0; 1: Therefore, we obtain from (2.4.29) ˇ  ˇ ˇ  ˇ 1 ˇˇ 00  ˇˇ 1 ˇˇ 00  C 1 ˇˇ 1 00 00 jR ./j  ˇR C R ˇ  2 jR ./j; 4 2 ˇ 4ˇ 2

(2.4.30)

(2.4.31)

which implies jR00 ./j D 0, i.e., R00 .x/ D 0. From R.1/ D R.0/ D 0 we then deduce R.x/ D 0. Therefore, Q.x/ D 1. This result can be written in the form  .x/ .1  x/ D

 : sin.x/

(2.4.32)

2.5 Exercises

43

It establishes an identity between the meromorphic functions  ./,  .1  /, and .sin /1 . Altogether, we have  1  Y 1 x2 1 1 2 : Dx  .x/  .1  x/ k

(2.4.33)

kD1

In connection with (2.4.32) we obtain Lemma 2.4.5. For x 2 C, sin.x/ D x

 1  Y x2 1 2 : k

(2.4.34)

kD1

2.5 Exercises (Incomplete Gamma and Beta Function, Applications in Statistics) In this section the discussion of the so-called incomplete Gamma functions and their relation to the error functions erf and erfc as well as the incomplete Beta function is left to the reader in the form of some exercises. The results demonstrate an immediate transition to probability distributions in statistics.

Incomplete Gamma Function Definition 2.5.1. By definition, we let for x; a > 0; Z  .a; x/ D

1 x

et t a1 dt Z

and .a; x/ D  .a/   .a; x/ D

x 0

et t a1 dt:

(2.5.1)

(2.5.2)

The functions  .; x/, .; x/ are called the incomplete Gamma functions related to x. Exercise 2.5.2. Prove that .a; x/ D x a

Z

1 0

a x

 .a; x/ D x e

ext t a1 dt; Z

1 0

ext dt:

(2.5.3) (2.5.4)

44

2 The Gamma Function

Exercise 2.5.3. Show that the so-called error functions 2 erf.x/ D p 

Z

x

2

et dt;

0

2 erfc.x/ D 1  erf.x/ D p 

(2.5.5) Z

1

2

et dt

(2.5.6)

x

admit the representations 1 erf.x/ D p . 21 ; x 2 /; 

(2.5.7)

1 erfc.x/ D p  . 12 ; x 2 /: 

(2.5.8)

Exercise 2.5.4. Verify that  .n C 1; x/ D nŠex

n X xm ; mŠ mD0

.n C 1; x/ D nŠ 1  e

x

(2.5.9)

n X xm mŠ mD0

! (2.5.10)

hold true for all n 2 N0 . Exercise 2.5.5. Prove the following recurrence relations .a C 1; x/ D a.a; x/  x a ex ; a x

 .a C 1; x/ D a .a; x/ C x e

(2.5.11) :

(2.5.12)

Incomplete Beta Function Definition 2.5.6. The function .x; y/ 7! B.x; y; ˛/ defined by Z ˛ B.x; y; ˛/ D t x1 .1  t/y1 dt

(2.5.13)

0

is called incomplete Beta function relative to ˛ 2 .0; 1. Exercise 2.5.7. Show that I.x; y; ˛/ D satisfies

B.x; y; ˛/ B.x; y/

(2.5.14)

2.5 Exercises

45

I.x; y; 1/ D 1;

(2.5.15)

I.x; y; ˛/ D 1  I.y; x; 1  ˛/:

(2.5.16)

Exercise 2.5.8. Prove the recurrence relation .x C y/I.x; y; ˛/ D xI.x C 1; y; ˛/ C yI.x; y C 1; ˛/:

(2.5.17)

Exercise 2.5.9. Verify the binomial expansion ! n X n l ˛ .1  ˛/nl ; I.m; n C 1  m; ˛/ D l

n; m 2 N:

(2.5.18)

lDm

As already announced, the incomplete Gamma and Beta functions possess a variety of applications in probability theory and statistics, from which we mention only two examples. We restrict ourselves to continuous random variables with nonnegative realizations. Definition 2.5.10 (Gamma Distribution). A random variable X with density distribution ( 0 ; if x  0; F .x/ D ˛p p1 ˛x (2.5.19) e ; if x > 0;  .p/ x is called a Gamma distribution with p > 0 and ˛ > 0. Clearly, F .x/  0 for all x 2 R. An easy calculation gives Z R

F .x/ dx D 1:

(2.5.20)

The probability density function of the Gamma distribution reads as follows: Z x 7!

x 0

F .t/ dt D

˛p  .p/

Z 0

x

t p1 e˛t dt D

1  .p/

Z

˛x 0

t p1 et dt D .p; ˛x/: (2.5.21)

The Gamma distribution is widely used as a conjugate prior in Bayesian statistics (for more details see, e.g., Papoulis and Pillai 2002).

Beta Distribution The Beta distribution is a family of continuous probability distributions defined on the interval .0; 1/ and parameterized by two positive values p and q.

46

2 The Gamma Function

Definition 2.5.11. A random variable X with density distribution ( F .x/ D

0

; if x … .0; 1/;

1 p1 .1 B.p;q/ x

 x/

q1

; if x 2 .0; 1/;

(2.5.22)

is called a Beta distribution with p; q > 0. The probability density function of the Beta distribution reads as follows: Z x 7!

x 0

1 F .t/ dt D B.p; q/

Z

x 0

t p1 .1  t/q1 dt D

B.p; q; x/ B.p; q/

(2.5.23)

for x 2 .0; 1/. The expectation value and the variance of a Beta distributed random variable X corresponding to the parameters p and q are given by  D E.X / D

p ; pCq

Var.X / D E.X  2 / D

pq : .p C q/2 .p C q C 1/

(2.5.24) (2.5.25)

Beta distributions are often used in Bayesian inference, since Beta distributions provide a family of conjugate prior distributions for binomial distributions (more details can be found, e.g., in van der Waerden 1969).

Chapter 3

Orthogonal Polynomials

In this chapter, we introduce polynomial function systems that are orthogonal with respect to a scalar product characterized by a measure d. We start with some very general results from Fourier analysis (see, e.g., Davis 1963; Reed and Simon 1972; Rudin 1991; Yoshida 1980), before we begin to specifically consider polynomials. As means of normalization we deal with monic polynomials, i.e., polynomials whose leading coefficient is equal to 1, and consider their properties (symmetry, zeros, best approximation) in Sect. 3.1. This enables us to find a three-term recurrence whose coefficients yield matrices that possess the zeros of the monic orthogonal polynomials as eigenvalues. Finally, the Christoffel–Darboux formula is included. In Sect. 3.2 we deal with one highly important application of orthogonal polynomials, namely quadrature rules, in particular Gauß quadratures. For more on the general orthogonal polynomials, in particular on computational techniques, we refer to Gautschi (2004) and the references therein. Starting with the Jacobi polynomials in Sect. 3.3, we concentrate on classical orthogonal polynomials which possess additional properties such as a defining differential equation and Rodrigues’ representation. We also investigate their relation to the hypergeometric function. In Sect. 3.4 we restrict ourselves to the ultraspherical (or Gegenbauer) polynomials which are a particular type of Jacobi polynomials. Later on, they are of special interest in geomathematics. They form the coefficients of the power series of their generating function and we establish a set of recurrence relations for the ultraspherical polynomials and their derivatives. Moreover, we provide their connection with the Legendre polynomials of dimension q (see, e.g., Freeden 2011; M¨uller 1998) which will be of great importance in Chap. 6. In Sect. 3.5, we develop the application of Legendre polynomials in electrostatics, briefly introduce Hermite polynomials in Sect. 3.6 as well as Laguerre polynomials in Sect. 3.7. We also give examples for their application and significance in physics. Finally, some exercises deal with error estimates for Gaussian integration, stable evaluation algorithms for Legendre series, and error estimates for kernel expansions in Sect. 3.8. For further details on classical orthogonal polynomials the reader is

W. Freeden and M. Gutting, Special Functions of Mathematical (Geo-)Physics, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-0348-0563-6 3, © Springer Basel 2013

47

48

3 Orthogonal Polynomials

referred to, e.g., Abramowitz and Stegun (1972), Lebedev (1973), Magnus et al. (1966), Sneddon (1980), and Szeg¨o (1967). We consider weighted Hilbert spaces of square-integrable functions on intervals in R. In general, these spaces are denoted by L2 .Œa; b/, where a; b can also be 1; 1, respectively. They possess the scalar product, for F; G 2 L2 .Œa; b/, Z

b

hF; Gid D

F .x/G.x/ d.x/:

(3.0.1)

a

 is a nondecreasing function on R which has finite limits as x ! ˙1 and whose induced positive measure d has finite moments of all orders, i.e., Z r D r .d/ D

x r d.x/ < 1

(3.0.2)

R

for r 2 N0 with 0 > 0. If  is absolutely continuous, the scalar product becomes Z

b

hF; Gid D

F .x/G.x/w.x/ dx:

(3.0.3)

a

Here w is a non-negative function which is Lebesgue-measurable and for which Rb a w.x/ dx > 0. It is called a weight function. Definition 3.0.1. Two functions F; G 2 L2 .Œa; b/ are orthogonal if hF; Gid D 0. If additionally kF kd D kGkd D 1, they are called orthonormal. Example 3.0.2. The trigonometric functions Fn .x/ D cos.nx/, n 2 N0 , form a system of orthogonal functions on the interval .0; / with the weight function w.x/ D 1 since Z 

cos.mx/ cos.nx/ dx D 0

(3.0.4)

0

for n ¤ m. Definition 3.0.3. Let fxk gk2N0 be a given orthonormal set (either finite or infinite). To an arbitrary real-valued function F let there correspond the formal Fourier expansion 1 X Fk xk .x/ (3.0.5) F .x/  kD0

with the coefficients Fk defined by Z

b

Fk D hF; xk id D

F .x/xk .x/ d.x/;

k 2 N0 :

(3.0.6)

a

These coefficients are called Fourier coefficients of F with respect to the given orthonormal system.

3 Orthogonal Polynomials

49

Theorem 3.0.4. Let xk , Fk be as before with F 2 L2 .Œa; b/. Let l  0 be a fixed integer and ak 2 R, k D 0; : : : ; l. The integral Z

b

 2 l X F .x/  ak xk d.x/

a

(3.0.7)

kD0

becomes minimal if and only if ak D Fk for k D 0; : : : ; l. P Proof. We consider the integral in question with G D lkD0 ak xk : kF  Gk2d D hF; F id  2Re.hF; Gid / C hG; Gid D kF k2d  2Re

X l kD0

D kF k2d C

l X

 l X ak hF; xk id C ak aj hxk ; xj id „ ƒ‚ … k;j D0

 X l jak j2  2Re ak Fk

kD0

D kF k2d 

l X

(3.0.8)

ık;j

kD0

jFk j2 C

kD0

l X

jak  Fk j2

kD0

which is minimal if and only if ak D Fk for all k D 0; : : : ; l. Therefore, the truncated Fourier series is the best approximating element. t u Remark 3.0.5. We also find Bessel’s equality for F , Fk , and xk as in Theorem 3.0.4: l l  2 X X   Fk xk  D kF k2d  jFk j2 F  d

kD0

(3.0.9)

kD0

and Bessel’s inequality (the left-hand side in (3.0.9) is non-negative): l X

jFk j2  kF k2d :

(3.0.10)

kD0

Lemma 3.0.6. For F , Fk , and xk as in Theorem 3.0.4 the Fourier series X Fk xk (3.0.11) k2N0

converges in the L2 -norm (in the L2 -sense) to an L2 -function S and .F  S /?xk for all k 2 N0 .

50

3 Orthogonal Polynomials

Proof. For n  m we have n n n 2 X X X   Fk xk  D jFk j2 kFk xk k2d D  kDm

d

kDm

(3.0.12)

kDm

due to the Theorem of Pythagoras and the orthonormality of the functions P xk . Bessel’s inequality (3.0.10) gives us the convergence of the series k2N0 jFk j2 . Thus, for all " > 0 there exists an m 2 N, such that for n  m; n X

jFk j2 < ":

(3.0.13)

kDm

Therefore, the sequence fSn gn with Sn D

n X

Fk xk

(3.0.14)

kD0

is a Cauchy sequence with respect to the L2 -norm. Since L2 .Œa; b/ is complete, there exists S 2 L2 .Œa; b/ which is the limit of this sequence, i.e., S D lim Sn D n!1

X

Fk xk

(3.0.15)

k2N0

in the sense of L2 .Œa; b/. It remains to show the orthogonality of S  F : If j 2 f0; : : : ; lg, then hS  F; xj id D

DX k2N0

E Fk xk ; xj

d

 hF; xj id D

X k2N0

Fk hxk ; xj id Fj D 0; „ ƒ‚ … ık;j

(3.0.16) since for Fn ! F , Gn ! G we have hFn ; Gn id ! hF; Gid .

t u

Now, we only have to show completeness of the system fxk gk . Then, F  S D 0 (in the L2 -sense), i.e., the limit of the Fourier series is the function F . Therefore, we consider properties of Fourier series in a general inner product space. Theorem 3.0.7. Let fxk gk2N  X be a sequence of orthonormal elements of the inner product space X . Consider the following statements: (A) The elements xk are closed in X , i.e., for any element x 2 X and for all " > 0 there exist an n 2 N and coefficients a1 ; : : : ; an 2 K such that n   X   ak xk   ": x  kD1

X

(3.0.17)

3 Orthogonal Polynomials

51

(B) The Fourier series of any y 2 X converges to y (in the norm of X ), i.e., n   X   hy; xk iX xk  D 0: lim y 

n!1

X

kD1

(3.0.18)

(C) Parseval’s identity holds, i.e., for all y 2 X , kyk2X D hy; yiX D

1 X

jhy; xk iX j2 :

(3.0.19)

kD1

(D) The extended Parseval identity holds, i.e., for all x; y 2 X , hx; yiX D

1 X hx; xk iX hxk ; yiX :

(3.0.20)

kD1

(E) There is no strictly larger orthonormal system containing the set fxk gk2N . (F) fxk gk2N is complete, i.e., if y 2 X and hy; xk iX D 0 for all k 2 N, then y D 0. (G) An element of X is determined uniquely by its Fourier coefficients, i.e., if hx; xk iX D hy; xk iX for all k 2 N, then x D y. Then, it holds that .A/ ” .B/ ” .C / ” .D/ H) .E/ ” .F / ” .G/:

(3.0.21)

If X is a complete inner product space, then also .E/ H) .D/ and all statements are equivalent. Proof. Assume (A). Due to the minimizing property of the truncated Fourier series (see Theorem 3.0.4 which holds for any orthonormal system fxk gk ) n n     X X     hy; xk iX xk   y  ak xk   "; y  X

kD1

kD1

X

(3.0.22)

where the last estimate is provided by .A/. If on the other hand .B/ holds, we can approximate any y by its truncated Fourier series which shows closure. Thus, .A/ ” .B/. By orthogonality, D

x

n X kD1

hx; xk iX xk ; y 

n X kD1

hy; xk iX xk

E X

D hx; yiX 

n X hx; xk iX hxk ; yiX : kD1

(3.0.23)

52

3 Orthogonal Polynomials

Using the Schwarz inequality, n n n ˇ ˇ     X X X ˇ ˇ     hx; xk iX hxk ; yiX ˇ  x  hx; xk iX xk   y  hy; xk iX xk  : ˇhx; yiX  kD1

X

kD1

kD1

X

(3.0.24) If .B/ holds, the right-hand members both tend to 0, hence .B/ H) .D/. Selecting x D y in .D/ shows .C /, i.e., .D/ H) .C /. Since n n  2 X X   2 0  y  hy; xk iX xk  D kykX  jhy; xk iX j2 ; kD1

X

(3.0.25)

kD1

we see .C / H) .B/ and thus, .A/ ” .B/ ” .C / ” .D/. Now, assume .A/ and suppose that fxk gk2N [fwg, w ¤ xk , is also an orthonormal system. This system is also closed in X . Since .A/ H) .C / kwk2X D

1 X

jhw; xk iX j2 C jhw; wiX j2 ; and kwk2X D

kD1

1 X

jhw; xk iX j2 :

kD1

(3.0.26) Thus, hw; wiX D 0 which contradicts kwkX D 1. This means that .A/ H) .E/. Suppose there is a 0 ¤ y 2 X , such that hy; xk iX D 0 for all k. Then, the set fxk gk2N [ fy= kykX g would be a strictly larger orthonormal system thanfxk gk2N . Therefore, .E/ ” .F /. Suppose that hw; xk iX D hy; xk iX , k 2 N. Then, hw  y; xk iX D 0, k 2 N. Assuming .F /, w  y D 0 and .F / H) .G/. If .F / were false, we could find z ¤ 0 with hz; xk iX D 0, k 2 N. For any y, hy; xk iX D hy Cz; xk iX , k 2 N. So y and y Cz would be two distinct elements with the same Fourier coefficients. Thus, .G/ would be false and we obtain .G/ H) .F /. This completes the chain of implications (3.0.21). Assume now that additionally X is complete. We want to show .G/ H) .B/ which is the missing implication. Let w 2 X and consider Sn D

n X hw; xk iX xk :

(3.0.27)

kD1

For n > m, we find that kSn  Sm k2X D

n X kDmC1

jhw; xk iX j2 :

(3.0.28)

3 Orthogonal Polynomials

53

Bessel’s inequality gives us 1 X

jhw; xk iX j2 < 1:

(3.0.29)

kD1

Thus, for a given " > 0, we can find an N."/ such that the right-hand side of (3.0.28) is less than " for all m; n  N."/. Thus, fSn gn is a Cauchy sequence and since X is complete, it converges to an element S 2 X . If l be fixed and n  l, then hS  Sn ; xl iX D hS; xl iX  hSn; xl iX D hS; xl iX  hw; xl iX

(3.0.30)

and by the Schwarz inequality jhS; xl iX  hw; xl iX j D jhS  Sn ; xl iX j  kS  Sn kX kxl kX D kS  Sn kX : (3.0.31) Together with the convergence of Sn to S this gives us hS; xl iX D hw; xl iX

for l 2 N:

(3.0.32)

By .G/, this implies that S D w, such that the convergence reads n   X   hw; xk iX xk  D 0: lim w 

n!1

kD1

X

(3.0.33) t u

This is precisely .B/.

From here on, we consider polynomials. At first, we have to investigate the positive definiteness of the inner product on the space of polynomials. Definition 3.0.8. The space of real polynomials up to degree n is denoted by ˘n . The space of real polynomials of all degrees is ˘ , ˘n  ˘ for all n 2 N0 . For P; Q 2 ˘ we use the scalar product Z hP; Qid D

P .x/Q.x/ d.x/

(3.0.34)

R

and the induced norm. Note that these integrals exist by definition of  (finite moments). Definition 3.0.9. The scalar product (3.0.34) is called positive definite on the space of all polynomials ˘ if kP kd > 0 for all P 2 ˘ , P 6 0. It is called positive definite on the space ˘n (polynomials of degree less than or equal to n) if kP kd > 0 for all P 2 ˘n , P 6 0.

54

3 Orthogonal Polynomials

Theorem 3.0.10. The scalar product (3.0.34) is positive definite on the space ˘ if and only if for the corresponding moments r (see (3.0.2)), ˇ ˇ ˇ 0 1    k1 ˇ ˇ ˇ ˇ 1 2    k ˇ ˇ ˇ det Mk D ˇ : :: :: ˇˇ > 0; ˇ :: : : ˇ ˇ ˇ ˇ k1 k    2k2

k 2 N:

(3.0.35)

It is positive definite on the space ˘n if and only if det Mk > 0 for k D 1; 2; : : : ; n C 1. n P ck x k . Proof. At first, we treat the case of ˘n . Let P 2 ˘n , i.e., P .x/ D kD0

kP k2d

D

n X k;lD0

Z ck cl

n X

x kCl d.x/ D R

ck cl kCl D c T MnC1 c;

(3.0.36)

k;lD0

where c D Œc0 ; c1 ; : : : ; cn T 2 RnC1 . Therefore, positive definiteness on ˘n is equivalent to positive definiteness of the matrix MnC1 , i.e., det Mk > 0 for k D 1; 2; : : : ; n C 1:

(3.0.37) t u

Taking n to infinity yields the result for ˘ .

Definition 3.0.11. Polynomials whose leading coefficient is 1, i.e., Pk .x/ D x k C : : : with k 2 N0 , are called monic polynomials. The set of monic polynomials of degree n is denoted by ˘nı . Monic real polynomials Pk , k 2 N0 , are called monic orthogonal polynomials with respect to the measure d if hPk ; Pl id D 0 for k ¤ l; k; l 2 N0 and kPk kd > 0 for k 2 N0 :

(3.0.38)

Normalization yields the orthonormal polynomials PQk .x/ D Pk .x/= kPk kd . Lemma 3.0.12. Let fPk gkD0;1;:::;n be monic orthogonal polynomials. If Q 2 ˘n satisfies hQ; Pk id D 0 for k D 0; 1; : : : ; n, then Q  0. Proof. We write Q as Q.x/ D

n P i D0

ai x i . Then,

0 D hQ; Pn id D

n X i D0

ai hx i ; Pn id

(3.0.39)

3 Orthogonal Polynomials

55

and x i can be written as Pi .x/ C Ri 1 .x/ with Ri 1 2 ˘i 1 : Thus hx i ; Pn id D hPi ; Pn id C hRi 1 ; Pn id D ıi;n hPn ; Pn id ;

(3.0.40)

since Ri 1 can be written as a linear combination of monic orthogonal polynomials whose degree is i 1. Thus, they are orthogonal to Pn and so is Ri 1 . This gives us 0 D hQ; Pn id D an hPn ; Pn id . Since hPn ; Pn id > 0, we obtain an D 0. Similarly, we can show that an1 D 0, an2 D 0; : : : ; a0 D 0. t u Lemma 3.0.13. A set P0 ; : : : ; Pn of monic orthogonal polynomials is linearly independent. Moreover, any polynomial P 2 ˘n can be uniquely represented in P the form P D nkD0 ck Pk for some real coefficients ck , i.e., P0 ; : : : ; Pn is a basis of ˘n . P Proof. If nkD0 k Pk  0, taking the scalar product on both sides of the equation with Pj , j D 0; : : : ; n, yields that j D 0. This gives us linear P independence. If we take the scalar product with Pj on both sides of P D nkD0 ck Pk , we find that hP; Pj id D

n X

ck hPk ; Pj id D cj hPj ; Pj id ;

j D 0; : : : ; n:

(3.0.41)

kD0

P The difference P  nkD0 ck Pk is orthogonal to P0 ; : : : ; Pn and by Lemma 3.0.12 has to be the zero polynomial. t u Theorem 3.0.14. If the scalar product h; id is positive definite on ˘ , there exists a unique infinite sequence fPk gk2N0 of monic orthogonal polynomials. Proof. The polynomials Pk can be constructed by applying Gram-Schmidt orthogonalization to the sequence of powers fEk gk2N0 , where Ek .x/ D x k . Therefore, we choose P0 D 1 and for k 2 N we use the recursion Pk D Ek 

k1 X lD0

cl Pl ;

cl D

hEk ; Pl id : hPl ; Pl id

(3.0.42)

Since the scalar product is positive definite, hPl ; Pl id > 0. Thus, the monic polynomial Pk is uniquely defined and by construction orthogonal to all Pj , j < k. t u The prerequisites of Theorem 3.0.14 are fulfilled if  has many points of increase, i.e., points t0 such that .t0 C "/ > .t0  "/ for all " > 0. The set of all points of increase of  is called support of the measure d, its convex hull is the support interval of d. Theorem 3.0.15. If the scalar product h; id is positive definite on ˘n , but not on ˘m for all m > n, there exist only n C 1 orthogonal polynomials P0 ; : : : ; Pn .

56

3 Orthogonal Polynomials

Proof. We apply Gram–Schmidt orthogonalization, i.e., (with Ek .x/ D x k ) Pk D Ek 

k1 X hEk ; Pl id lD0

hPl ; Pl id

k 2 N0 ;

Pl ;

(3.0.43)

as long as hPl ; Pl id > 0, i.e., for k  n C 1. In the end we obtain PnC1 which is orthogonal to Pj for all j  n and all Pj , j D 0; : : : ; n, are orthogonal polynomials with positive norm. We now show that kPnC1 kd D 0. Since by assumption h; id is not positive definite on ˘nC1 , there exists a monic polynomial w 2 ˘nC1 with kwkd D 0. Moreover, deg.w  PnC1 /  n and thus, w D PnC1 C

n X

aj Pj

(3.0.44)

j D0

with coefficients aj 2 R. Since 0 D kwk2d D kPnC1 k2d C

n X j D0

 2 aj2 Pj d

(3.0.45)

  and Pj d > 0, we find that aj D 0, j D 0; : : : ; n, and kPnC1 kd D 0. Hence, PnC1 is not an orthogonal polynomial. t u Theorem 3.0.16. If the moments of d exist only for r D 0; 1; : : : ; r0 , there exist only n C 1 orthogonal polynomials P0 ; : : : ; Pn , where n D br0 =2c. Proof. The Gram–Schmidt procedure can be performed as long as the scalar products in (3.0.42), in particular hPk ; Pk id , exist. This is the case as long as 2k  r0 , i.e., k  n D br0 =2c. t u

3.1 Properties of Orthogonal Polynomials For this section we assume that the measure d is a positive measure on R which possesses an infinite number of points of increase and finite moments of all orders. Definition 3.1.1. An absolutely continuous measure d.x/ D w.x/dx is symmetric (with respect to the origin) if its support interval is Œa; a with 0 < a  1 and w.x/ D w.x/ for all x 2 R. Remark 3.1.2. Note that if no confusion is likely to arise, we skip the subscript d for norms and scalar products and we just use the notation L2 .Œa; b/ for the spaces. Theorem 3.1.3. If d is symmetric, then Pk .x/ D .1/k Pk .x/;

k 2 N0 :

(3.1.1)

3.1 Properties of Orthogonal Polynomials

57

Thus, Pk is either an even or an odd polynomial depending on the degree k. Proof. Set POk .x/ D .1/k Pk .x/. We compute for k ¤ l; hPOk ; POl id D .1/kCl

Z R

Pk .x/Pl .x/ d.x/ D .1/kCl hPk ; Pl id D 0:

(3.1.2) Since all POk are monic, POk .x/ D Pk .x/ by the uniqueness of monic orthogonal polynomials. t u Theorem 3.1.4. If d is symmetric on Œa; a, 0 < a  1, and the polynomials PkC and Pk are related to the monic orthogonal polynomials with respect to d such that P2k .x/ D PkC .x 2 /; P2kC1 .x/ D xPk .x 2 /; (3.1.3) then fPk˙ g are the monic orthogonal polynomials with respect to the measure 1

1

1

dC .x/ D x  2 w.x 2 / dx

1

or d .x/ D x C 2 w.x 2 / dx;

(3.1.4)

respectively, on Œ0; a2 . Proof. We restrict ourselves to the proof for PkC , the other case follows analogously. Obviously, the polynomials PkC are monic. By symmetry it holds that Z

a

0 D hP2k ; P2l id D 2

P2k .x/P2l .x/w.x/ dx

(3.1.5)

0

for k ¤ l. Therefore, we obtain for k ¤ l Z 0D2 0

a

PkC .x 2 /PlC .x 2 /w.x/

Z

a2

dx D 0

1

1

PkC .t/PlC .t/t  2 w.t 2 / dt;

where the substitution t D x 2 has been used.

(3.1.6) t u

Theorem 3.1.5. All zeros of Pk , k 2 N, are real, simple, and located in the interior of the support interval Œa; b of d. R Proof. Since R Pk .x/ d.x/ D 0 for k  1, there must exist at least one point in the interior of Œa; b at which Pk changes sign. Let x1 ; x2 ; : : : ; xn , n  k, be all these points. If we had n < k, then due to orthogonality Z R

Pk .x/

n Y

.x  xj / d.x/ D 0;

(3.1.7)

j D1

which is impossible since the integrand does no longer change sign. Therefore, we obtain that n D k. t u

58

3 Orthogonal Polynomials

Theorem 3.1.6. The zeros of PkC1 alternate with those of Pk , i.e., .kC1/

.k/

.kC1/

xkC1 < xk < xk .kC1/

where xj

.k/

, xi

.k/

.k/

.kC1/

< xk1 <    < x1 < x1

;

(3.1.8)

are the zeros of PkC1 and Pk , respectively, in descending order.

Proof. See Remark 3.1.21. The proof uses the Christoffel–Darboux formula, i.e., Theorem 3.1.19. t u Theorem 3.1.7. In any open interval .c; d / in which d  0 there can be at most one zero of Pk . .k/

Proof. We perform a proof by contradiction. Suppose there are two zeros xi ¤ .k/ .k/ xj in .c; d /. Let all the other zeros (within .c; d / or not) be denoted by xn . Then, Z R

Y

Pk .x/ Z

D R

.x  xn.k/ / d.x/

n¤i;j

Y

.k/

.k/

.x  xn.k/ /2 .x  xj /.x  xi / d.x/ > 0;

(3.1.9)

n¤i;j

since the integrand is non-negative outside of .c; d /. This is a contradiction to the Q .k/ orthogonality of Pk to polynomials of lower degree such as .x  xn /. t u n¤i;j

Theorem 3.1.8. For any monic polynomial P 2 ˘nı we have Z

Z P .x/ d.x/  2

R

R

Pn2 .x/ d.x/;

(3.1.10)

where equality is achieved only for P D Pn . In other words, Pn minimizes the integral on the left-hand side of (3.1.10) over all P 2 ˘nı , i.e., Z

Z P 2 .x/ d.x/ D

minı

P 2˘n

R

R

Pn2 .x/ d.x/:

(3.1.11)

Proof. Due to Lemma 3.0.13 the polynomial P can be represented as P .x/ D Pn .x/ C

n1 X

ck Pk .x/:

(3.1.12)

kD0

Therefore, Z

Z P 2 .x/ d.x/ D

R

R

Pn2 .x/ d.x/ C

n1 X kD0

Z ck2

R

Pk2 .x/ d.x/:

(3.1.13)

3.1 Properties of Orthogonal Polynomials

59

This shows the desired inequality. Equality in (3.1.10) holds if and only if the coefficients c0 D c1 D : : : D cn1 D 0, i.e., for P D Pn . u t Remark 3.1.9. If we consider the function Z  n1 2 X ˚.a0 ; a1 ; : : : ; an1 / D xn C ak x k d.x/; R

(3.1.14)

kD0

we can compute the partial derivatives and set them equal to zero which yields Z P .x/x k d.x/ D 0; k D 0; 1; : : : ; n  1: (3.1.15) R

These are exactly the conditions of orthogonality that Pn has to satisfy. Furthermore, the Hessian of ˚ is 2Mn , where Mn is the matrix of Theorem 3.0.10 which is positive definite. This confirms that Pn gives us a minimum. Theorem 3.1.10. Let 1 < p < 1. Then, the extremal problem of determining Z minı

P 2˘n

jP .x/jp d.x/

(3.1.16)

R

possesses the unique solution Pn . Proof. The search for the desired minimum is equivalent to the problem of best approximation of x n by polynomials of degree n  1 in the Lp -norm. The problem of best approximation in normed spaces is uniquely solvable if the space is strictly convex, i.e., if from kxk D kyk D 1 and x ¤ y we can conclude that kx C yk < 2. A normed space X is strictly convex if and only if kx C yk D kxk C kyk yields x D ˛y or y D ˛x with an ˛  0. The Minkowski inequality guarantees this for the Lp -norm. For further details see, e.g., Davis (1963), Reed and Simon (1972), Rudin (1991), and Yoshida (1980) or other books on functional analysis. t u Orthogonal polynomials fulfill a three-term recurrence relation which can be used for: • Generating values of the polynomials and their derivatives, • Computation of the zeros as eigenvalues of a symmetric tridiagonal matrix via the recursion coefficients, • Normalization of the orthogonal polynomials, • Efficient evaluation of expansions in orthogonal polynomials. The reason for the existence of these three-term recurrences is the shift property of the scalar product, i.e., Z hxU; V i D

xU.x/V .x/ d.x/ D hU; xV i R

for all U; V 2 ˘:

(3.1.17)

60

3 Orthogonal Polynomials

Note that there are other scalar products that do not possess this property (even though they are positive definite). Theorem 3.1.11. Let Pk , k 2 N0 , be the monic orthogonal polynomials with respect to the measure d (see Definition 3.0.11). Then, P1 .x/ D 0; P0 .x/ D 1; PkC1 .x/ D .x  ˛k /Pk .x/  ˇk Pk1 .x/; k 2 N0 ; (3.1.18) where ˛k D

hxPk ; Pk i ; hPk ; Pk i

k 2 N0 ;

ˇk D

hPk ; Pk i ; hPk1 ; Pk1 i

k 2 N:

(3.1.19)

The index range for k is infinite if the scalar product is positive definite on ˘ . The range is finite (k  d  1) if the scalar product is positive definite on ˘d , but not on ˘n with n > d . Remark 3.1.12. Although ˇ0 in (3.1.18) can be arbitrary since it is multiplied with P1  0, we define it for later purposes as Z ˇ0 D hP0 ; P0 i D

d.x/:

(3.1.20)

R

Proof (of Theorem 3.1.11). Since PkC1  xPk is a polynomial of degree k, we can write PkC1 .x/  xPk .x/ D ˛k Pk .x/  ˇk Pk1 .x/ C

k2 X

k;j Pj .x/

(3.1.21)

j D0

for certain constants ˛k , ˇk , k;j , where P1 .x/ D 0 and empty sums are also zero. We take the scalar product with Pk on both sides which gives us (using orthogonality): hPkC1 ; Pk i  hxPk ; Pk i D ˛k hPk ; Pk i  ˇk hPk1 ; Pk i C

k2 X

k;j hPj ; Pk i:

j D0

(3.1.22) Therefore, we find that  hxPk ; Pk i D ˛k hPk ; Pk i; or ˛k D

hxPk ; Pk i : hPk ; Pk i

(3.1.23)

(3.1.24)

3.1 Properties of Orthogonal Polynomials

61

This proves the relation for ˛k . For ˇk we need to take the scalar product with Pk1 , where k  1: hPkC1 ; Pk1 i  hxPk ; Pk1 i D ˛k hPk ; Pk1 i  ˇk hPk1 ; Pk1 i C

k2 X

k;j hPj ; Pk1 i:

j D0

(3.1.25) Thus,  hxPk ; Pk1 i D ˇk hPk1 ; Pk1 i:

(3.1.26)

Since hxPk ; Pk1 i D hPk ; xPk1 i D hPk ; Pk C Rk1 i with Rk1 2 ˘k1 , we find that hxPk ; Pk1 i D hPk ; Pk i which provides us with ˇk D

hPk ; Pk i : hPk1 ; Pk1 i

(3.1.27)

As a last step we take the scalar product on both sides with Pi , i < k  1, and obtain hPkC1 ; Pi i  hxPk ; Pi i D ˛k hPk ; Pi i  ˇk hPk1 ; Pi i C

k2 X

k;j hPj ; Pi i:

j D0

(3.1.28) This immediately leads to  hxPk ; Pi i D k;i hPi ; Pi i:

(3.1.29)

Here we make use of the shift property of the scalar product, i.e., hxPk ; Pi i D hPk ; xPi i D 0 since xPi 2 ˘k1 . Therefore, k;i D 0 for i < k  1 which finally proves (3.1.18). t u Remark 3.1.13. If the index range in Theorem 3.1.11 is finite (k  d  1), the relations for ˛d and ˇd still make sense, ˇd > 0, but the polynomial Pd C1 that results from (3.1.18) has norm kPd C1 k D 0 (see also Theorem 3.0.15). Remark 3.1.14. Note that ˇk > 0 for all k 2 N0 and for n 2 N0 ; kPn k2 D ˇn  ˇn1  : : :  ˇ1  ˇ0 :

(3.1.30)

There is a converse result to Theorem 3.1.11 saying that any sequence of polynomials is orthogonal with respect to a positive measure with infinite support if the polynomials satisfy a three-term recurrence relation of the form (3.1.18) and all ˇk are positive. Theorem 3.1.15. Let the support interval Œa; b of d be finite. Then, a  ˛k  b;

k 2 N0 ;

0 < ˇk  maxfa2 ; b 2 g;

k 2 N;

(3.1.31)

where the index range is k  1 or k  d (d as in Theorem 3.1.11), respectively.

62

3 Orthogonal Polynomials

Proof. Since for x in the support of d we know that a  x  b, the definition of ˛k in Theorem 3.1.11 immediately yields the desired estimates. By definition 0 < ˇk and it remains to show the upper bound. We notice that kPk k2 D hPk ; Pk i D jhxPk1 ; Pk ij

(3.1.32)

and apply the Cauchy–Schwarz inequality to obtain kPk k2  maxfjaj; jbjg kPk1 k kPk k :

(3.1.33)

Therefore, kPk k2 kPk k  maxfa2 ; b 2 g:  maxfjaj; jbjg and ˇk D 2 kPk1 k kPk1 k

(3.1.34) t u

Definition 3.1.16. If the index range in Theorem 3.1.11 is infinite, the Jacobi matrix associated with the measure d is the infinite, symmetric, tridiagonal matrix

J1

p 3 2 ˛ ˇ1 p 0 0 p 7 6 ˇ1 ˛1 ˇ2 p 7 6 p 7 6 ˇ2 ˛2 ˇ3 D6 7: 6 :: :: :: 7 4 : : :5 0

(3.1.35)

Its leading principal minor matrix of size n n is denoted by 2

Jn D ŒJ1 Œ1Wn;1Wn

3 p ˇ1 p 0 ˛0 p 6 ˇ ˛ 7 ˇ2 p 1 p1 6 7 6 7 ˇ2 ˛2 ˇ3 6 7 6 7: D6 :: :: :: 7 : : : 6 7 p p 6 7 4 ˇn2 p˛n2 ˇn1 5 0 ˇn1 ˛n1

(3.1.36)

If the index range in Theorem 3.1.11 is finite (k  d  1), then Jn is well-defined for 0  n  d . Remark 3.1.17. Based on the monic orthogonal polynomials Pk , k 2 N0 , we can define orthonormal polynomials by PQk .x/ D Pk .x/= kPk k ;

such that hPQk ; PQj i D ık;j :

(3.1.37)

3.1 Properties of Orthogonal Polynomials

63

.n/ Theorem 3.1.18. The zeros xi of Pn (or the orthonormal version PQn ) are the eigenvalues of the Jacobi matrix Jn of order n. The corresponding eigenvectors are .n/ given by p.x Q i /, where

 T p.x/ Q D PQ0 .x/; PQ1 .x/; : : : ; PQn1 .x/ :

(3.1.38)

Proof. A simple calculation transforms the three-term recurrence relation (3.1.18) for the Pk into a relation for the PQk , i.e., p p ˇkC1 PQkC1 .x/ D .x  ˛k /PQk .x/  ˇk PQk1 .x/

(3.1.39)

for k 2 N0 , where we use the ˛k , ˇk of Theorem 3.1.11. Together with the usual convention PQ1 D 0 we obtain the following system of equations x p.x/ Q D Jn p.x/ Q C

p ˇn PQn .x/"n ;

(3.1.40)

where "n D Œ0; 0; : : : ; 1T is the n-th unit vector in Rn . .n/ If we put x D xi in (3.1.40), the second summand on the right-hand side drops p .n/ out. Since the first component of the vector p.x Q i / is 1= ˇ0 , the vector cannot be .n/ 0 and is indeed the eigenvector to the eigenvalue xi . t u Theorem 3.1.19 (Christoffel–Darboux Formula). Let PQk denote the orthonormal polynomials with respect to the measure d. Then, n X

PQk .x/PQk .t/ D

kD0

p PQnC1 .x/PQn .t/  PQn .x/PQnC1 .t/ ˇnC1 xt

(3.1.41)

and n X

PQk .x/

2

D

p 0

ˇnC1 PQnC1 .x/PQn .x/  PQn0 .x/PQnC1 .x/ :

(3.1.42)

kD0

Proof. We use the recurrence relation for the orthonormal polynomials (3.1.39), where PQ1 .x/ D 0 and PQ0 .x/ D p1ˇ . It is multiplied by PQk .t/ to obtain: 0

p p ˇkC1 PQkC1 .x/PQk .t/ D .x  ˛k /PQk .x/PQk .t/  ˇk PQk1 .x/PQk .t/:

(3.1.43)

Now, we interchange the roles of t and x and subtract that from the first relation, i.e.,

64

3 Orthogonal Polynomials

p

ˇkC1 PQkC1 .x/PQk .t/ 

p

ˇkC1 PQkC1 .t/PQk .x/

(3.1.44) p

D .x  ˛k /PQk .x/PQk .t/  ˇk PQk1 .x/PQk .t/   p  .t  ˛k /PQk .t/PQk .x/  ˇk PQk1 .t/PQk .x/ D .x  t/PQk .x/PQk .t/ 

p

ˇk PQk1 .x/PQk .t/  PQk1 .t/PQk .x/ :

Therefore, .x  t/PQk .x/PQk .t/ D

p

ˇkC1 PQkC1 .x/PQk .t/  PQkC1 .t/PQk .x/ p

 ˇk PQk1 .t/PQk .x/  PQk1 .x/PQk .t/ :

(3.1.45)

We sum up both sides from k D 0 to k D n and make use of the telescoping sum on the right-hand side (note that PQ1 D 0): n X

PQk .x/PQk .t/ D

kD0

p

ˇnC1

PQnC1 .x/PQn .t/  PQn .x/PQnC1 .t/ : xt

(3.1.46)

For the second part we take the limit t ! x on both sides. Thus, on the right-hand side we find PQnC1 .x/PQn .t /  PQn .x/PQnC1 .t / PQn .t /  PQn .x/ PQn .x/ D PQnC1 .x/ C PQnC1 .x/ xt xt xt  PQn .x/ D PQnC1 .x/

(3.1.47)

PQnC1 .t /  PQnC1 .x/ PQnC1 .x/  PQn .x/ xt xt PQn .t /  PQn .x/ PQnC1 .t /  PQnC1 .x/  PQn .x/ xt xt

Letting t tend to x we obtain the limit



0 0 PQnC1 .x/ PQn0 .x/  PQn .x/ PQnC1 .x/ D PQnC1 .x/PQn .x/  PQn0 .x/PQnC1 .x/; (3.1.48) t u

which is the desired result.

Corollary 3.1.20. Let Pk denote the monic orthogonal polynomials with respect to the measure d. Then, for x ¤ t, n  Y n X kD0

 n X ˇi Pk .x/Pk .t/ D ˇn  ˇn1  : : :  ˇkC1 Pk .x/Pk .t/

i DkC1

kD0

D

PnC1 .x/Pn .t/  Pn .x/PnC1 .t/ : xt

(3.1.49)

3.1 Properties of Orthogonal Polynomials

65

Proof. Put PQk D Pkp = kPk k in the first formula of Theorem 3.1.19 and use from Theorem 3.1.11 that ˇnC1 D kPnC1 k = kPn k: n X

PQk .x/PQk .t/ D

kD0

D

n X

1

kD0

kPk k2

Pk .x/Pk .t/

(3.1.50)

kPnC1 k PnC1 .x/Pn .t/  Pn .x/PnC1 .t/ : kPn k kPnC1 k kPn k .x  t/

This yields n X kPn k2 2

kD0

kPk k

Pk .x/Pk .t/ D

PnC1 .x/Pn .t/  Pn .x/PnC1 .t/ : xt

Together with kPn k2 D ˇn  ˇn1      ˇ1  ˇ0 D

n Y

ˇi

(3.1.51)

(3.1.52)

i D0

this provides us with n  Y n X kD0

i DkC1

 PnC1 .x/Pn .t/  Pn .x/PnC1 .t/ : ˇi Pk .x/Pk .t/ D xt

(3.1.53) t u

Remark 3.1.21. From the second part of Theorem 3.1.19 we obtain the inequality 0 PQnC1 .x/PQn .x/  PQn0 .x/PQnC1 .x/ > 0:

(3.1.54)

This can be used to prove Theorem 3.1.6 in the following way: Let  and  be consecutive zeros of PQn , such that PQn0 ./PQn0 ./ < 0 (which holds since all zeros are simple). Then, (3.1.54) tells us that  PQn0 ./PQnC1 ./ > 0

and

 PQn0 ./PQnC1 ./ > 0:

(3.1.55)

This implies that PQnC1 has opposite signs at  and . Therefore, there is at least one zero of PQnC1 between  and . In this way we find at least n  1 zeros of PQnC1 . .n/ .n/ For the largest zero of PQn , i.e., for 1 , it holds that PQn0 .1 / > 0 and by (3.1.55) .n/ .n/ we have PQnC1 .1 / < 0. The polynomial PQnC1 has another zero at the right of 1 Q since PnC1 .x/ > 0 for x sufficiently large. .n/ A similar argument holds for the smallest zero of PQn , i.e., for n . This proves Theorem 3.1.6.

66

3 Orthogonal Polynomials

3.2 Quadrature Rules and Orthogonal Polynomials In this section, let d be a measure with bounded or unbounded support, positive definite or not. An n-point quadrature rule for d is a formula of the type Z F .x/ d.x/ D R

n X

wj F .xj / C Rn .F /:

(3.2.1)

j D1

The mutually distinct points xj are called nodes, the coefficients wj are the weights of the quadrature rule. Rn .F / is the remainder or error term. Definition 3.2.1. The quadrature rule (3.2.1) possesses degree of exactness d if Rn .P / D 0 for all P 2 ˘d . It has precise degree of exactness d if it has degree of exactness d , but not d C 1. A quadrature rule (3.2.1) with degree of exactness d D n  1 is called interpolatory.

Interpolatory Quadrature Rules A quadrature rule is interpolatory if and only if it is obtained by integration of the Lagrange interpolation, i.e., by integrating F .x/ D

n X

F .xj /Lj .x/ C rn1 .F I x/;

(3.2.2)

j D1

where Lj .x/ D

n Y x  xi : x  xi i D1 j

(3.2.3)

i ¤j

Thus, we obtain Z wj D

Z R

Lj .x/ d.x/; j D 1; 2; : : : ; n; and Rn .F / D

R

rn1 .F I x/ d.x/: (3.2.4)

It is well-known that for P 2 ˘n1 the interpolation error rn1 .P I /  0, i.e., the remainder term Rn .P / D 0 and d D n  1. Given any n distinct nodes an interpolatory quadrature can always be constructed.

3.2 Quadrature Rules and Orthogonal Polynomials

67

Theorem 3.2.2. Let 0  k  n be an integer. The quadrature rule (3.2.1) has degree of exactness d D n  1 C k if and only if both of the following two conditions are fulfilled: (i) (3.2.1) is interpolatory, (ii) The node polynomial !n .x/ D

n Q

.x  xj / satisfies

j D1

Z R

!n .x/P .x/ d.x/ D 0 for all P 2 ˘k1 :

(3.2.5)

Proof. First, let the degree of exactness be d D n1Ck. Obviously, the quadrature rule is interpolatory. For P 2 ˘k1 we know that !n P 2 ˘nCk1 . Therefore, Z R

!n .x/P .x/ d.x/ D

n X

wj !n .xj /P .xj / D 0

(3.2.6)

j D1

by definition of !n . Let now the conditions .i / and .i i / be fulfilled. Let P 2 ˘nCk1 . We have to prove that the remainder term Rn .P / D 0. Polynomial long division by the node polynomial !n yields that P D Q!n C R, where Q 2 ˘k1 and R 2 ˘n1 . Then, Z

Z

Z

P .x/ d.x/ D R

R

Q.x/!n .x/ d.x/ C

R.x/ d.x/ D R

n X

wj R.xj /;

j D1

(3.2.7) since the first integral vanishes due to .i i / and the second integral is evaluated exactly by the quadrature rule due to .i /. However, R.xj / D P .xj /  Q.xj /!n .xj / D P .xj /: Therefore,

Z P .x/ d.x/ D R

i.e., Rn .P / D 0.

n X

wj P .xj /;

(3.2.8)

(3.2.9)

j D1

t u

Remark 3.2.3. If d is positive definite on ˘n , then k D n is optimal. k D n C 1 requires that !n is orthogonal to all elements of ˘n , i.e., also to itself, which is not possible.

Gauß Quadratures Definition 3.2.4. The quadrature rule (3.2.1) with k D n is called the Gauß quadrature rule with respect to the measure d. Its degree of exactness is d D 2n  1.

68

3 Orthogonal Polynomials

Remark 3.2.5. The second condition in Theorem 3.2.2 shows that for a Gauß quadrature (i.e., k D n) we have !n D Pn . Therefore, the nodes xjG are the zeros of the n-th orthogonal polynomial with respect to d. The weights wG j can be found by interpolation. Note that for the Gauß quadratures we assume that d is positive definite on ˘n . Theorem 3.2.6. All nodes xjG of the Gauß quadrature rule are mutually distinct and contained in the interior of the support interval Œa; b of d. All weights wG j are positive. Proof. Since the nodes xjG are the zeros of Pn , the first part follows directly from Theorem 3.1.5. Now, we consider the weights: Z 0< R

L2i .x/

d.x/ D

n X

2 G G wG j Li .xj / D wi

for i D 1; 2; : : : ; n, since L2i ./ 2 ˘2n2  ˘2n1 . Theorem 3.2.7. Let rule. Then,

wG j n X

(3.2.10)

j D1

be the weights and

xjG

t u

be the nodes of a Gauß quadrature

Z G wG j F .xj /

j D1

D R

P2n1 .F I x/ d.x/;

(3.2.11)

where P2n1 .F I / is the Hermite interpolation polynomial of degree 2n  1 which satisfies the equations 0 P2n1 .F I xjG / D F 0 .xjG /

P2n1 .F I xjG / D F .xjG /;

for j D 1; 2; : : : ; n: (3.2.12)

Proof. The Hermite interpolation polynomial is given by P2n1 .F I x/ D

n  X

 Hj .x/F .xjG / C Kj .x/F 0 .xjG / ;

(3.2.13)

j D1

where   Hj .x/ D 1  2.x  xjG /L0j .xjG / L2j .x/;

(3.2.14)

Kj .x/ D.x  xjG /L2j .x/;

(3.2.15)

Lj .x/ D

n Y i D1 i ¤j

x  xiG : xjG  xiG

(3.2.16)

3.2 Quadrature Rules and Orthogonal Polynomials

69

Note that Hj .xiG / D ıj;i , Hj0 .xiG / D 0, as well as Kj .xiG / D 0, Kj0 .xiG / D ıj;i . Obviously, Hj ; Kj 2 ˘2n1 . Hence, Z R

P2n1 .F I x/ d.x/ D

n Z X j D1

R

Z Hj .x/ d.x/F .xjG / C

R

 Kj .x/ d.x/F 0 .xjG / :

(3.2.17) Moreover, Z n   X Hj .x/ d.x/ D wi 1  2.xiG  xjG /L0j .xjG / L2j .xiG / R

i D1

D

n X

  wi 1  2.xiG  xjG /L0j .xjG / ıj;i D wj ;

(3.2.18)

i D1

and Z R

Kj .x/ d.x/ D

n X

wi .xiG xjG /L2j .xiG / D

i D1

n X

wi .xiG xjG /ıj;i D 0: (3.2.19)

i D1

t u

This finally gives us the desired result, i.e., (3.2.11).

Corollary 3.2.8. If F 2 C .Œa; b/ and Œa; b is the support interval of d, then the remainder term of the Gauß quadrature rule can be expressed as Z F .2n/ ./ Rn .F / D .Pn .x//2 d.x/ (3.2.20) .2n/Š R .2n/

with  2 .a; b/. Proof. We know from the theory of interpolation (see, e.g., Davis 1963) that F .x/ D P2n1 .F I x/ C r2n1 .F I x/ D P2n1 .F I x/ C

n F .2n/ ..x// Y .x  xj /2 .2n/Š j D1

(3.2.21) with .x/ 2 .a; b/. This yields for the remainder of the numerical integration that Z Rn .F / D

Z R

r2n1 .F I x/ d.x/ D

R

n F .2n/ ..x// Y .x  xj /2 d.x/: .2n/Š j D1

The mean value theorem of integration completes the proof.

(3.2.22) t u

Theorem 3.2.9. The first n orthogonal polynomials Pk , k D 0; : : : ; n  1, are discretely orthogonal in the sense that n X j D1

2 G G wG j Pk .xj /Pl .xj / D ık;l kPk k ;

k; l D 0; 1; : : : ; n  1;

(3.2.23)

70

3 Orthogonal Polynomials

where xjG , wG j are the nodes and weights of the n-point Gauß quadrature rule. Proof. This follows directly since the degree of exactness of the n-point Gauß quadrature is 2n  1 and the product of Pk and Pl with k; l 2 f0; 1; : : : ; n  1g is a polynomial of degree 2n  2. t u Remark 3.2.10. If a > 1, b  1, it can be desirable to have x0 D a. To achieve this, we replace n by n C 1 in (3.2.1) and write !nC1 .x/ D .x  a/!n .x/. The optimal formula (see Theorem 3.2.2) requires !n to satisfy Z R

!n .x/P .x/.x  a/ d.x/ D 0 for all P 2 ˘n1 :

(3.2.24)

Therefore, !n .x/ D Pn .xI da / with da .x/ D .x  a/ d.x/. The remaining n nodes have to be zeros of Pn .I da /. The resulting rule is called the Gauß–Radau rule. If also b < 1 and a as well as b are both desired as nodes, we find the .n C 2/-point Gauß–Lobatto rule similarly with the measure da;b .x/ D .x  a/.b  x/ d.x/:

(3.2.25)

These rules have degrees of exactness equal to 2n and 2n C 1, respectively. For further details, the reader is referred to, e.g., Davis and Rabinowitz (1967), Gautschi (2004), H¨ammerlin and Hoffmann (1992).

3.3 The Jacobi Polynomials In this section we consider the Jacobi polynomials which are classical orthogonal polynomials depending on two parameters ˛; ˇ. For special choices of these parameters we obtain other important classes such as the ultraspherical polynomials which are treated in more detail in Sect. 3.4. The exercises in Sect. 4.8 come back to the general Jacobi polynomials and show their importance for geomathematical applications. For further details on Jacobi polynomials we refer to e.g., Abramowitz and Stegun (1972), Lebedev (1973), Magnus et al. (1966), Sneddon (1980), and Szeg¨o (1967). Definition 3.3.1. Let d.x/ D w.x/ dx with w.x/ D .1  x/˛ .1 C x/ˇ , ˛; ˇ > 1, and the support interval Œ1; 1. The corresponding orthogonal polynomials are the .˛;ˇ/ .˛;ˇ/ .˛;ˇ/ Jacobi polynomials Pn 2 ˘n which fulfill hPn ; Pm id D 0 if n ¤ m and the normalization condition ! nC˛ .n C ˛ C 1/ .˛;ˇ/ D Pn .1/ D : (3.3.1) n .n C 1/ .˛ C 1/ Note that we generalize the notation of the binomial coefficients naturally via the Gamma function.

3.3 The Jacobi Polynomials

71

Remark 3.3.2. Similar to Theorem 3.1.3 we find the identity Pn.˛;ˇ/ .x/ D .1/n Pn.ˇ;˛/ .x/ and so we obtain Pn.˛;ˇ/ .1/

D .1/

n

! nCˇ : n

(3.3.2)

(3.3.3)

Theorem 3.3.3. For ˛ > 1, we have .˛;˛/

P2n

.x/ D

.2n C ˛ C 1/ .n C 1/ .˛;1=2/ P .2x 2  1/ .n C ˛ C 1/ .2n C 1/ n

D .1/n .˛;˛/

P2nC1 .x/ D

.2n C ˛ C 1/ .n C 1/ .1=2;˛/ P .1  2x 2 /; .n C ˛ C 1/ .2n C 1/ n

.2n C ˛ C 2/ .n C 1/ x Pn.˛;1=2/ .2x 2  1/ .n C ˛ C 1/ .2n C 2/

D .1/n

(3.3.4)

(3.3.5)

.2n C ˛ C 2/ .n C 1/ x Pn.1=2;˛/ .1  2x 2 /: .n C ˛ C 1/ .2n C 2/

Proof. At first, we show (3.3.4). For that we use the notation d1 .x/ D .1  x/˛ .1 C x/˛ dx D .1  x 2 /˛ dx;

(3.3.6)

d2 .x/ D .1  x/˛ .1 C x/1=2 dx:

(3.3.7)

It suffices to prove that Z I D Z

R

Pn.˛;1=2/ .2x 2  1/P .x/ d1 .x/

1

D 1

Pn.˛;1=2/ .2x 2  1/P .x/.1  x 2 /˛ dx D 0;

(3.3.8)

where P 2 ˘2n1 . If P is an odd polynomial, this is fulfilled. Therefore, let P be even, i.e., P .x/ D R.x 2 / with R 2 ˘n1 . Then, Z

1

I D 1

Z

D2 0

Z

1

D 0

Pn.˛;1=2/ .2x 2  1/R.x 2 /.1  x 2 /˛ dx 1

Pn.˛;1=2/ .2x 2  1/R.x 2 /.1  x 2 /˛ dx Pn.˛;1=2/ .2u  1/R.u/.1  u/˛ u1=2 du;

(3.3.9)

72

3 Orthogonal Polynomials

where we have substituted x 2 by u. Applying the substitution rule another time with u D 1Cx gives us: 2 I D 2˛1=2 D2

˛1=2

Z Z

1 1

R

˛ 1=2 Pn.˛;1=2/ .x/R. xC1 dx 2 /.1  x/ .1 C x/

Pn.˛;1=2/ .x/R. xC1 / d2 .x/ D 0: 2

(3.3.10) t u

A similar argument can be used to prove (3.3.5). Remark 3.3.4 (Special Cases).

(i) For ˛ D ˇ, we are dealing with the special case of ultraspherical polynomials (or Gegenbauer polynomials) which due to Theorem 3.3.3 are even or odd polynomials (depending on n being even or odd). Cn./ .x/ D D

.˛ C 1/ .n C 2˛ C 1/ .˛;˛/ P .x/ .2˛ C 1/ .n C ˛ C 1/ n . C 12 / .n C 2/ .2/ .n C  C 12 /

Pn.1=2;1=2/ .x/;

(3.3.11)

where ˛ D ˇ D   12 ,  >  12 since ˛ > 1. .0/ If ˛ D  12 (or  D 0), the Gegenbauer polynomial Cn vanishes identically for n  1 (see Lemma 3.4.1). Another consequence of Theorem 3.3.3 is that Jacobi polynomials with ˛ D ˙ 12 or ˇ D ˙ 12 can be expressed by ultraspherical polynomials. We will treat these classical orthogonal polynomials in detail in Sect. 3.4. (ii) For ˛ D ˇ D  12 , we obtain the Chebyshev polynomials of first kind Tn , i.e., Qn

i D1 .2i  2n nŠ

Pn.1=2;1=2/ .x/ D

1/

Qn

i D1 .2i  2n nŠ

Tn .x/ D

1/

cos.n#/; (3.3.12)

where x D cos.#/. (iii) For ˛ D ˇ D 12 , we obtain the Chebyshev polynomials of second kind Un , i.e., Qn Pn.1=2;1=2/ .x/ D

i D0 .2i C 1/ Un .x/ 2n .n C 1/Š

Qn D

i D0 .2i C 1/ sin..n C 1/#/ ; 2n .n C 1/Š sin.#/ (3.3.13)

where x D cos.#/. (iv) The mixed variants ˛ D  12 , ˇ D 12 and ˛ D 12 , ˇ D  12 yield the Chebyshev polynomials of third kind Vn and of fourth kind Wn , respectively, i.e.,

3.3 The Jacobi Polynomials

73

Qn Pn.1=2;1=2/ .x/

i D1 .2i  2n nŠ

D

1/

Qn Vn .x/ D

Qn Pn.1=2;1=2/ .x/ D

i D1 .2i  2n nŠ

i D1 .2i  1/ Wn .x/ D 2n .n C 1/Š

1/ cos

Qn

.2nC1/#

cos

#2

;

2

(3.3.14) .2nC1/#

i D1 .2i  1/ sin 2 ; 2n .n C 1/Š sin #2 (3.3.15)

where x D cos.#/. For details on Chebyshev polynomials see, e.g., Rivlin (1990). (v) For ˛ D ˇ D 0, we find the Legendre polynomials Pn , i.e., Pn.0;0/ .x/ D Cn.1=2/ .x/ D Pn .x/:

(3.3.16)

Note that in this case the weight function is w.x/ D 1, x 2 Œ1; 1. .˛;ˇ/

Theorem 3.3.5. The Jacobi polynomials y D Pn .x/ satisfy the following linear, homogeneous differential equation of the second order: .1  x 2 /y 00 C .ˇ  ˛  .˛ C ˇ C 2/x/y 0 C n.n C ˛ C ˇ C 1/y D 0; (3.3.17) or equivalently

d .1  x/˛C1 .1 C x/ˇC1 y 0 C n.n C ˛ C ˇ C 1/.1  x/˛ .1 C x/ˇ y D 0: dx (3.3.18) Proof. We note that since y 2 ˘n we have that

d .1  x/˛C1 .1 C x/ˇC1 y 0 D .1  x/˛ .1 C x/ˇ z; dx

(3.3.19)

where z 2 ˘n . To show that z is a constant multiple of y, i.e., z D Cy, we have to prove the orthogonality to any P 2 ˘n1 , i.e., Z

1 1

d .1  x/˛C1 .1 C x/ˇC1 y 0 P .x/ dx D 0: dx

(3.3.20)

We use integration by parts on the left-hand side which yields (since ˛ C 1 > 0 and ˇ C 1 > 0): Z

d .1  x/˛C1 .1 C x/ˇC1 y 0 P .x/ dx dx 1 Z 1 D .1  x/˛C1 .1 C x/ˇC1 y 0 P 0 .x/ dx 1

1

74

3 Orthogonal Polynomials

Z

1

D

y 1

d .1  x/˛C1 .1 C x/ˇC1 P 0 .x/ dx; dx „ ƒ‚ …

(3.3.21)

CQ .1x/˛ .1Cx/ˇ R.x/

where we performed another integration by parts and R 2 ˘n1 . Therefore, the integral vanishes and we find that z D Cy. The constant factor C can be calculated by comparing the highest terms, i.e., y D kn x n C : : : ;

y 0 D nkn x n1 C : : : ;

y 00 D n.n  1/kn x n2 C : : : ; (3.3.22)

and 0D

d .1  x/˛C1 .1 C x/ˇC1 y 0 C C.1  x/˛ .1 C x/ˇ y dx

(3.3.23)

D .˛ C 1/.1  x/˛ .1 C x/ˇC1 y 0 C .ˇ C 1/.1  x/˛C1 .1 C x/ˇ y 0 C.1  x/˛C1 .1 C x/ˇC1 y 00 C C.1  x/˛ .1 C x/ˇ y D .˛ C 1/.1  x/˛ .1 C x/ˇ .1 C x/y 0  .ˇ C 1/.1  x/˛ .1 C x/ˇ .x  1/y 0 .1  x/˛ .1 C x/ˇ .x 2  1/y 00 C C.1  x/˛ .1 C x/ˇ y: Thus, C D .n.˛ C 1/  n.ˇ C 1/  n.n  1// D n.n C ˛ C ˇ C 1/;

(3.3.24) t u

which proves the differential equation (3.3.18). Theorem 3.3.6. Let ˛; ˇ > 1. The differential equation .1  x 2 /y 00 C .ˇ  ˛  .˛ C ˇ C 2/x/y 0 C y D 0;

(3.3.25)

where  2 R is a parameter, has a polynomial solution not identically zero if and .˛;ˇ/ only if  D n.n C ˛ C ˇ C 1/, n 2 N0 . This solution is A  Pn .x/, A ¤ 0, and .˛;ˇ/ no solution which is linearly independent of Pn can be a polynomial. P1 k Proof. Substitute y D kD0 ak .x  1/ in the differential equation. This gives us 0 D  .x C 1/

1 X

k.k  1/ak .x  1/k1  .2.˛ C 1/ C .˛ C ˇ C 2/.x  1//

kD2



1 X kD1

kak .x  1/k1 C 

1 X kD0

ak .x  1/k

(3.3.26)

3.3 The Jacobi Polynomials

D  .x  1 C 2/

1 X

75

k.k  1/ak .x  1/k1  2.˛ C 1/

kD2

 .˛ C ˇ C 2/

1 X

D

kak .x  1/k C 



1 X

ak .x  1/k

kD0

k.k  1/ak .x  1/k  2

kD2 1 X

kak .x  1/k1

kD1

kD1 1 X

1 X

1 X

.k C 1/kakC1 .x  1/k  2.˛ C 1/

kD1

.k C 1/akC1 .x  1/k  .˛ C ˇ C 2/

kD0

1 X

kak .x  1/k C 

kD1

1 X

ak .x  1/k :

kD0

Thus, the coefficients have to fulfill the relation .  k.k C ˛ C ˇ C 1// ak  2.k C 1/.k C ˛ C 1/akC1 D 0

(3.3.27)

for k 2 N0 . If we assume that y is a polynomial, we can suppose that an denotes the last nonzero coefficient, i.e., anC1 D 0. Therefore, the factor in front of an in the recurrence relation above has to vanish, i.e.,  D n.n C ˛ C ˇ C 1/:

(3.3.28)

On the other hand, if (3.3.28) holds for  , we find that ai D 0 for i  n C 1 since the factor of akC1 in (3.3.28) differs from zero. Let  D n.n C ˛ C ˇ C 1/ and let z be a second solution of the differential equation (3.3.18), i.e.,

d .1  x/˛C1 .1 C x/ˇC1 y 0 C n.n C ˛ C ˇ C 1/.1  x/˛ .1 C x/ˇ y D 0; dx (3.3.29)

d .1  x/˛C1 .1 C x/ˇC1 z0 C n.n C ˛ C ˇ C 1/.1  x/˛ .1 C x/ˇ z D 0: dx (3.3.30) Multiply the first equation by z and the second by y and subtract them: d dx d D dx

0D





d .1  x/˛C1 .1 C x/ˇC1 y 0 z  .1  x/˛C1 .1 C x/ˇC1 z0 y dx

(3.3.31) .1  x/˛C1 .1 C x/ˇC1 .y 0 z  z0 y/ :

Thus, for all x 2 Œ1; 1, .1  x/˛C1 .1 C x/ˇC1 .y 0 z  yz0 / D c D const:

(3.3.32)

76

3 Orthogonal Polynomials

If we let x ! ˙1, we see that y and z cannot both be polynomials unless c D 0. Therefore, y 0 z D z0 y for all x 2 .1; 1/, i.e., the two solutions y and z are linearly .˛;ˇ/ dependent. Therefore, z.x/ D cP Q n .x/. t u Remark 3.3.7. The Jacobi polynomials can also be defined as the polynomial solutions of the corresponding differential equation that additionally take the value ! nC˛ .˛;ˇ/ Pn .1/ D : (3.3.33) n Definition 3.3.8. The hypergeometric function F (sometimes denoted by 2 F1 ) is defined by 1 1 X .c/ X .a C k/ .b C k/ x k .a/k .b/k x k D F .a; bI cI x/ D .c/k kŠ .a/ .b/ .c C k/ kŠ kD0 kD0 (3.3.34)

or its analytic continuation with a; b 2 R, c 2 R n fN0 g, x 2 .1; 1/. If a D n or b D n, n 2 N0 , the hypergeometric function reduces to a polynomial in x whose degree is n. Theorem 3.3.9. For the Jacobi polynomials the following representations hold: !

nC˛ .˛;ˇ/ Pn .x/ D (3.3.35) F n; n C ˛ C ˇ C 1I ˛ C 1I 1x 2 n !  n X .n C ˛ C 1/ x1 k .n C ˛ C ˇ C 1 C k/ n D : nŠ .n C ˛ C ˇ C 1/ .˛ C 1 C k/ k 2 kD0

(3.3.36) This can be reformulated as ! n 1 X n .n C ˛ C 1/ .n C ˇ C 1/ .˛;ˇ/ .x  1/nk .x C 1/k Pn .x/ D n k .n C ˛ C 1  k/ .ˇ C 1 C k/ 2 nŠ kD0 ! ! n 1 X nC˛ nCˇ .x  1/nk .x C 1/k : D n (3.3.37) k nk 2 kD0

Proof. This can be shown via the properties of the hypergeometric function, in particular its differential equation for which we refer to Abramowitz and Stegun (1972), Magnus et al. (1966), and Szeg¨o (1967). Another way uses Rodrigues’ representation of the Jacobi polynomials which can be found in Theorem 3.3.12. t u .˛;ˇ/

Corollary 3.3.10. The leading coefficient of the Jacobi polynomial Pn n is given by

of degree

3.3 The Jacobi Polynomials

77

kn.˛;ˇ/

D2

n

! 2n C ˛ C ˇ : n

(3.3.38)

Proof. Consider the summand for k D n in representation (3.3.36) of Theorem 3.3.9: kn.˛;ˇ/

! .n C ˛ C ˇ C 1 C n/ n 1 .n C ˛ C 1/ D nŠ .n C ˛ C ˇ C 1/ .˛ C 1 C n/ n 2n ! n 2n C ˛ C ˇ D2 : (3.3.39) n t u

Corollary 3.3.11. For the derivative of the Jacobi polynomials we have n C ˛ C ˇ C 1 .˛C1;ˇC1/ d .˛;ˇ/ Pn .x/ D Pn1 .x/: dx 2

(3.3.40)

Proof. This follows immediately if both sides are expanded according to Theorem 3.3.9. u t Theorem 3.3.12 (Rodrigues’ Formula). Let ˛; ˇ > 1. Then, .1  x/ .1 C x/ ˛

ˇ

Pn.˛;ˇ/ .x/

.1/n D n 2 nŠ



d dx

n



.1  x/nC˛ .1 C x/nCˇ : (3.3.41)

Proof. We use Leibniz’ rule for the differentiation of products to find 

d dx

n 

!   nk n  X n d k d .1  x/nC˛ .1 C x/nCˇ D .1  x/nC˛ .1 C x/nCˇ dx dx k kD0

! n X n D .1  x/˛ Rnk .x/.1 C x/ˇ Sk .x/ k kD0

D .1  x/˛ .1 C x/ˇ %.x/;

(3.3.42)

where Rj ; Sj 2 ˘j , deg.Rj / D deg.Sj / D j , j D 0; : : : ; n, and % 2 ˘n , deg.%/ D n. In detail (we will need that later): (3.3.43) .1  x/˛ .1 C x/ˇ %.x/ ! n X n .n C ˛ C 1/ .n C ˇ C 1/ .1/k .1  x/nC˛k .1 C x/ˇCk : D k .n C ˛  k C 1/ .ˇ C k C 1/ kD0

78

3 Orthogonal Polynomials

Therefore, ! n X n .n C ˛ C 1/ .n C ˇ C 1/ .1/k .1  x/nk .1 C x/k : %.x/ D k .n C ˛  k C 1/ .ˇ C k C 1/ kD0 (3.3.44) .˛;ˇ/

We now have to show that % D C  Pn Z

1

with a constant C . It suffices to show that

.1  x/˛ .1  x/ˇ %.x/R.x/ dx D 0

(3.3.45)

1

for all R 2 ˘n1 . We use (3.3.42) and apply integration by parts n times: Z

Z

 d n   .1  x/nC˛ .1 C x/nCˇ R.x/ dx 1 1 dx   ˇ1  ˇ d n1  D .1  x/nC˛ .1 C x/nCˇ R.x/ ˇˇ dx 1 „ ƒ‚ … 1

.1  x/˛ .1  x/ˇ %.x/R.x/ dx D

Z

1

D0

 d n1   .1  x/nC˛ .1 C x/nCˇ R0 .x/ dx  1 dx Z 1  D : : : D .1/n .1  x/nC˛ .1 C x/nCˇ R.n/ .x/ dx D 0; 1

1

(3.3.46)

since deg.R/  n1, i.e., R.n/  0. Now, we just have to determine the constant C . Consider x D 1 in %, i.e., only the summand k D n remains, i.e., ! ! n .n C ˛ C 1/ .n C ˇ C 1/ nC˛ n n n n .1/ 2 D .1/ 2 nŠ %.1/ D : .˛ C 1/ .n C ˇ C 1/ n n (3.3.47) .˛;ˇ/

Thus, %.1/ D .1/n 2n nŠ Pn

.1/, i.e., C D .1/n 2n nŠ.

t u

Remark 3.3.13. The Theorem 3.3.12 also gives us the explicit representation (3.3.37) of the second part of Theorem 3.3.9. Theorem 3.3.14. Let ˛; ˇ > 1. Then, for n 2 N, 2  h.˛;ˇ/ D Pn.˛;ˇ/  D n D and

Z

1 1

.˛;ˇ/ 2 Pn .x/ .1  x/˛ .1 C x/ˇ dx

2 .n C ˛ C 1/ .n C ˇ C 1/ 2n C ˛ C ˇ C 1 .n C 1/ .n C ˛ C ˇ C 1/ ˛CˇC1

(3.3.48)

3.3 The Jacobi Polynomials

.˛;ˇ/

h0

79

Z    .˛;ˇ/ 2 D P0  D

1

1

D 2˛CˇC1

 2 .˛;ˇ/ P0 .x/ .1  x/˛ .1 C x/ˇ dx

.˛ C 1/ .ˇ C 1/ : .˛ C ˇ C 2/

(3.3.49)

Proof. Let n 2 N. We have, from the proof of Theorem 3.3.12, Z

1

1

D



.˛;ˇ/

Pn

.x/

2

Z

1

1

.˛;ˇ/

Pn

.x/x n .1  x/˛ .1 C x/ˇ dx

 d n   .1  x/nC˛ .1 C x/nCˇ x n dx 1 dx !Z 1 2n C ˛ C ˇ .1  x/nC˛ .1 C x/nCˇ dx; n 1

.1/n .˛;ˇ/ kn 2n nŠ

D 22n

Z

.˛;ˇ/

.1  x/˛ .1 C x/ˇ dx D kn 1

(3.3.50)

where we used integration by parts n times and Corollary 3.3.10. Z

1

1

.˛;ˇ/ 2 Pn .x/ .1  x/˛ .1 C x/ˇ dx

D2

D2 D

˛Cˇ

2n C ˛ C ˇ n

˛CˇC1

!Z

2n C ˛ C ˇ n



1 1

!Z

1x 2

nC˛ 

(3.3.51) 1Cx 2

nCˇ dx

1

y nCˇ .1  y/nC˛ dy 0

2˛CˇC1 .n C ˛ C 1/ .n C ˇ C 1/ : 2n C ˛ C ˇ C 1 .n C 1/ .n C ˛ C ˇ C 1/

The case n D 0 follows similarly.

t u

Theorem 3.3.15. The Jacobi polynomials fulfill the following three-term recurrence relation: .˛;ˇ/

(3.3.52) 2.n C 1/.n C ˛ C ˇ C 1/.2n C ˛ C ˇ/PnC1 .x/

D .2n C ˛ C ˇ C 1/ .2n C ˛ C ˇ C 2/.2n C ˛ C ˇ/x C ˛ 2  ˇ 2 Pn.˛;ˇ/ .x/ .˛;ˇ/

 2.n C ˛/.n C ˇ/.2n C ˛ C ˇ C 2/Pn1 .x/ .˛;ˇ/

for n 2 N with P0

.˛;ˇ/

.x/ D 1 and P1

.x/ D 12 .˛ C ˇ C 2/x C 12 .˛  ˇ/.

80

3 Orthogonal Polynomials

Proof. Division by the leading coefficient (see Corollary 3.3.10) gives us POn.˛;ˇ/ D

1 .˛;ˇ/ kn

Pn.˛;ˇ/ ;

n 2 N0 ;

(3.3.53)

which are monic orthogonal polynomials. Therefore, they fulfill (3.1.18) of Theorem 3.1.11 with

ˇn.˛;ˇ/

  2   O .˛;ˇ/ 2 .˛;ˇ/ .˛;ˇ/ kn1 hn Pn  D   D 2  O .˛;ˇ/ 2 .˛;ˇ/ .˛;ˇ/ kn hn1 Pn1  4n.n C ˛/.n C ˇ/.n C ˛ C ˇ/ ; n 2 N; (3.3.54) .2n C ˛ C ˇ/2 .2n C ˛ C ˇ C 1/.2n C ˛ C ˇ  1/ Z 1 2˛CˇC1 .˛ C 1/ .ˇ C 1/ ; (3.3.55) D .1  x/˛ .1 C x/ˇ dx D .˛ C ˇ C 2/ 1 D

.˛;ˇ/

ˇ0

.˛;ˇ/

where we use the values of hn from Theorem 3.3.14. A lengthy, but uneventful calculation yields the missing values of the recurrence relation, i.e., for n 2 N0 , .˛;ˇ/

˛n.˛;ˇ/ D

hxPn

.˛;ˇ/

; Pn

.˛;ˇ/

hn

i

D

ˇ2  ˛2 : .2n C ˛ C ˇ/.2n C ˛ C ˇ C 2/

(3.3.56)

.˛;ˇ/

Note that in the case of ˇ1 , i.e., n D 1, the last factors in the numerator and in the denominator cancel each other out. The same happens with the factor ˛ C ˇ in .˛;ˇ/ .˛;ˇ/ ˛0 . By multiplying (3.1.18) with knC1 we obtain .˛;ˇ/ PnC1 .x/



D x

˛n.˛;ˇ/

.˛;ˇ/

knC1

P .˛;ˇ/ .x/ .˛;ˇ/ n kn



.˛;ˇ/ .˛;ˇ/ .˛;ˇ/ knC1 ˇn P .x/; .˛;ˇ/ n1 kn1

which can easily be transformed into the desired expression.

(3.3.57) t u

3.4 Ultraspherical Polynomials In Remark 3.3.4 we have already presented the ultraspherical (or Gegenbauer) ./ polynomials Cn and have established their connection to the general Jacobi polynomials, i.e.,

3.4 Ultraspherical Polynomials

Cn./ .x/ D D

81

.˛ C 1/ .n C 2˛ C 1/ .˛;˛/ P .x/ .2˛ C 1/ .n C ˛ C 1/ n . C 12 / .n C 2/ .2/ .n C  C 12 /

1 1 . ; / 2 2

Pn

.x/;

(3.4.1)

where ˛ D ˇ D   12 ,  >  12 since ˛ > 1. This relation gives us the following properties for  ¤ 0:

./ (i) The value at 1 is Cn .1/ D nC21 , the symmetry relation is given by n Cn./ .x/ D .1/n Cn./ .x/;

(3.4.2)

the polynomials are either even or odd (depending on n) due to Theorem 3.3.3. ./ (ii) The differential equation (3.3.17) whose solution is y D Cn becomes .1  x 2 /y 00  .2 C 1/xy 0 C n.n C 2/y D 0:

(3.4.3)

(iii) We know the explicit representation from Theorem 3.3.9, namely !

n C 2  1 Cn./ .x/ D F n; n C 2I  C 12 I 1x 2 n !   n . C 12 / X n .n C 2 C k/ x  1 k D : k . C 12 C k/ .2/nŠ 2

(3.4.4)

(3.4.5)

kD0

(iv) Due to Corollary 3.3.10 the leading coefficient is kn./ D lim x n Cn./ .x/ D 2n x!1

! nC1 : n

(3.4.6)

(v) The derivative is again an ultraspherical polynomial, i.e., by Corollary 3.3.11: d ./ .C1/ C .x/ D 2Cn1 .x/: dx n

(3.4.7)

(vi) Rodrigues’ representation (see Theorem 3.3.12) for the Gegenbauer polynomials reads as follows: ./

Cn .x/ D D

.1/n . C 12 / .n C 2/ 2n nŠ .2/ .n C  C 12 /

1

.1  x 2 / 2 

1 .2/n .n C / .n C 2/ .1  x 2 / 2  nŠ ./ .2n C 2/

 

d dx d dx

n  n 

1



1



.1  x 2 /nC 2

.1  x 2 /nC 2

:

(3.4.8)

82

3 Orthogonal Polynomials

(vii) Their L2 -norm is given by Theorem 3.3.14, i.e., 12  ./ 2 .n C 2/ C  D 2 h./ D : n n nŠ.n C /. .//2

(3.4.9)

(viii) The corresponding three-term recurrence (see Theorem 3.3.15) is given by ./

./

.n C 1/CnC1.x/ D 2.n C /xCn./ .x/  .n C 2  1/Cn1 .x/; ./

(3.4.10)

./

for n 2 N, with C0 .x/ D 1 and C1 .x/ D 2x. .0/

Lemma 3.4.1. If n  1, Cn  0, but ./

2 Cn .x/ D Tn .x/ !0  n lim

(3.4.11)

with the Chebyshev polynomial of first kind Tn . Proof. We consider ./ Cn .x/ . C 12 / .n C 2/ D P .1=2;1=2/ .x/   .2/ .n C  C 12 / n

D

(3.4.12)

2 . C 12 / .n C 2/ P .1=2;1=2/ .x/: .2 C 1/ .n C  C 12 / n

Now, we take the limit  ! 0 and obtain: ./ Cn .x/ 2 . 12 / .n/ D P .1=2;1=2/ .x/ !0  .1/ .n C 12 / n

lim

D

2 nŠ . 12 / P .1=2;1=2/ .x/ n .n C 12 / n

nŠ 2n 2 2 P .1=2;1=2/ .x/ D Tn .x/; D Qn n i D1 .2i  1/ n n where we have used (3.3.12) in the final step.

(3.4.13) t u

./

Note that C0 D 1, even as  ! 0. Combining Theorems 3.3.3 and 3.3.9 for the ultraspherical polynomials we obtain the following representations. Lemma 3.4.2. For n 2 N0 , we have the following representations in terms of the hypergeometric function: ./ C2n .x/

!

2n C 2  1 D F n; n C I  C 12 I 1  x 2 ; 2n

(3.4.14)

3.4 Ultraspherical Polynomials

./ C2nC1 .x/

83

!

2n C 2 D xF n; n C  C 1I  C 12 I 1  x 2 : 2n C 1

(3.4.15)

Proof. Consider the even case (3.4.14) first. We start with the relation between Gegenbauer and Jacobi polynomials and apply Theorem 3.3.3. ./

C2n .x/ D D

. C 12 / .2n C 2/ .2/ .2n C  C 12 /

.1=2;1=2/

P2n

(3.4.16)

.x/

. C 12 / .2n C 2/ .2n C  C 12 / .n C 1/ .2/ .2n C  C 12 / .n C  C 12 / .2n C 1/

Pn.1=2;1=2/.2x 2  1/

Now, we use the representation (3.3.35) of Theorem 3.3.9: ./ C2n .x/

!   . C 12 / .2n C 2/ .n C 1/ n C   12 2 1 D I 1  x F n; n C I  C 2 n .2/ .n C  C 12 / .2n C 1/ !   2n C 2  1 F n; n C I  C 12 I 1  x 2 : D (3.4.17) 2n

t u

The odd case (3.4.15) can be proved analogously.

Lemma 3.4.3. For n 2 N0 , the ultraspherical polynomials possess the following representations in terms of the hypergeometric function F : ! nC1 D .1/ F .n; n C I 12 I x 2 /; n ! n C  ./ C2nC1 .x/ D .1/n 2 x F .n; n C  C 1I 32 I x 2 /: n ./ C2n .x/

n

(3.4.18)

(3.4.19)

Proof. Consider the even case. This time we apply the second variant of Theorem 3.3.3. ./

C2n .x/ D

. C 12 / .2n C 2/ .2/ .2n C  C 12 /

.1=2;1=2/

P2n

D

. C 12 / .2n C 2/

.2n C  C 12 / .n C 1/

D

. C 12 / .2n C 2/ .n C 1/

.1=2;1=2/

Pn .1  2x 2 / .n C  C 12 / .2n C 1/ ! 1   2/ . C 12 / .2n C 2/ .n C 1/ n n 2 .1/ F n; n C I 12 I 1.12x D 2 1 n .2/ .n C  C 2 / .2n C 1/ .2/ .2n C  C 12 /

.1/n

(3.4.20)

.x/

.2/ .n C  C 12 / .2n C 1/

.1/n

.n C 12 / .n C 1/ . 21 /

  F n; n C I 12 I x 2 :

84

3 Orthogonal Polynomials

Due to the duplication formula of Lemma 2.3.3, we find that .1/ . C 12 / D 21 2 ; .2/ 2 ./

(3.4.21)

.1/ .n C  C 12 / D 2nC21 2 ; .2n C 2/ 2 .n C / .n C 12 / . 12 /

D

.2n/ 22n1 .n/

(3.4.22)

:

(3.4.23)

Therefore, ./

C2n .x/ D D

. 12 / 22nC21 .n C / .2n/ 21 22n1 .n/ 2 ./ . 12 /  2 .n C /.1/n  F n; n C I 12 I x 2 2 n ./ .n/

D .1/n

  .1/n F n; n C I 12 I x 2 .2n C 1/

!   nC1 F n; n C I 12 I x 2 : n

(3.4.24) t u

The odd case can be verified analogously.

Corollary 3.4.4. For N 2 N0 , the ultraspherical polynomials possess the following explicit representation: bN=2c ./

CN .x/ D

X

.1/m

mD0

.N  m C / .2x/N 2m ; ./ .m C 1/ .N  2m C 1/

where b N2 c is the largest integer that is less than or equal to

N 2

(3.4.25)

.

Proof. Let N D 2n, i.e., bN=2c D n. We use the definition of the hypergeometric function F (Definition 3.3.8) in Lemma 3.4.3. ./

CN .x/ D .1/n



.n C / F n; n C I 12 I x 2 ./ .n C 1/ .n C / X .n/k .n C /k x 2k : ./ .n C 1/ kŠ . 12 /k n

D .1/n

(3.4.26)

kD0

We resolve the Pochhammer symbols as follows (note that the duplication formula of Lemma 2.3.3 is used in (3.4.29)): .n/k D .1/k

.n C 1/ ; .n C 1  k/

(3.4.27)

3.4 Ultraspherical Polynomials

.n C /k D . 12 /k D

85

.n C  C k/ ; .n C / .k C 12 / . 12 /

D

(3.4.28) .2k/

22k1 .k/

:

(3.4.29)

We obtain: ./

CN .x/ D

n X .1/nCk kD0

D

n X .1/nk kD0

n X D .1/nk kD0

.n C 1/ .n C  C k/ . 12 / x 2k .n C / ./ .n C 1/ .n C 1  k/ .n C / .k C 12 / kŠ 22k1 .k/ 2k .n C  C k/ x ./ .n  k C 1/ .k C 1/ .2k/ .2x/2k .n C  C k/ : ./ .n  k C 1/ .2k C 1/

(3.4.30)

Now, we shift the index using k D n  m or m D n  k: ./

CN .x/ D

n X

.1/m

mD0

.2x/2n2m .n C n  m C / ./ .m C 1/ .2n  2m C 1/

bN=2c

D

X

mD0

.1/m

.N  m C / .2x/N 2m : ./ .m C 1/ .N  2m C 1/

(3.4.31)

For N D 2n C 1, the proof can be performed analogously.

t u

Theorem 3.4.5. For  > 0 and n 2 N0 , we have that ! ˇ ./ ˇ n C 2  1 : max ˇCn .x/ˇ D Cn./ .1/ D 1x1 n

(3.4.32)

For  < 0 and n 2 N0 , it holds that ˇ ˇ ˇ ˇ max ˇCn./ .x/ˇ D ˇCn./ .x 0 /ˇ ;

(3.4.33)

1x1

where x 0 is one of the two maximum points nearest to 0 if n is odd, x 0 D 0 if n is even. Proof. Let n  1. We consider

2 n.n C 2/F .x/ D n.n C 2/ Cn./ .x/ C .1  x 2 /



d ./ C .x/ dx n

2 :

(3.4.34)

86

3 Orthogonal Polynomials

 2 ./ Then, F .x/ D Cn .x/ if

./ d C .x/ dx n

max

1x1

D 0 or x D ˙1, i.e.,

./ 2 Cn .x/  max F .x/: 1x1

(3.4.35)

Now, we differentiate equation (3.4.34) and use the differential equation (3.4.3):   2 d ./ d ./ Cn .x/  2x Cn .x/ dx dx !   d ./ d2 ./ C .1  x 2 /2 Cn .x/ Cn .x/ dx dx 2       d ./ d ./ d ./ D2 .2 C 1/x Cn .x/ Cn .x/  x Cn .x/ dx dx dx 2  d ./ (3.4.36) Cn .x/ : D 4x dx

n.n C 2/F 0 .x/ D n.n C 2/2Cn .x/ ./



Therefore, we find that, if  > 0, F is increasing in Œ0; 1. If  < 0, F is decreasing in Œ0; 1. This (together with the symmetry relation of Theorem 3.1.3) gives us the desired result. t u Remark 3.4.6. For the Legendre polynomials, i.e.,  D 12 , Theorem 3.4.5 gives us jPn .x/j  Pn .1/ D 1

(3.4.37)

and ! ! ˇ 0 ˇ ˇˇ .3=2/ ˇˇ n1C31 nC1 n.n C 1/ ˇP .x/ˇ D ˇC D D n n1 .x/ˇ  2 n1 n1

(3.4.38)

for x 2 Œ1; 1. Theorem 3.4.7 (Generating Function). For  > 0 and h 2 .1; 1/, the ultraspherical polynomials form the coefficients of the following power series: 1 X

hn Cn./ .x/ D .1  2hx C h2 / ;

(3.4.39)

nD0

where x 2 Œ1; 1. Proof. First, we check the convergence of the series. ! ˇX ˇ 1 1 X ˇ 1 n ./ ˇ X ˇ ˇ n C 2  1 ˇ : h Cn .x/ˇˇ  jhjn ˇCn./ .x/ˇ  jhjn ˇ n nD0

nD0

nD0

(3.4.40)

3.4 Ultraspherical Polynomials

87

! n C 2  1 .n C 2/  C n2 D .n C 1/ .2/ n

Since

with a constant C > 0 and

1 X

n2 jhjn < 1;

(3.4.41)

(3.4.42)

nD0

we have absolute and uniform convergence of the series. Now, we consider the recurrence relation (3.4.10): 1 X

nCn./ .x/hn1 D 2x

nD1

1 1 X X ./ ./ .nC1/Cn1 .x/hn1  .nC22/Cn2 .x/hn1 ; nD1

nD1

(3.4.43) where

./ C1

 0. For a fixed x 2 Œ1; 1, we introduce the notation ˚.h/ D

1 X

Cn./ .x/hn :

(3.4.44)

nD0

Then, we obtain by differentiating with respect to h: 0

˚ .h/ D

1 X

nCn./ .x/hn1 ;

(3.4.45)

nD1

and for the first term on the right-hand side of (3.4.43)

0 h1 h ˚.h/ D ˚.h/ C h˚ 0 .h/ D

1 X

Cn./ .x/hn C

nD0

D

1 X

1 X

nCn./ .x/hn

nD1 ./

.n C   1/Cn1 .x/hn1 :

(3.4.46)

nD1

The second term on the right-hand side can be computed as

0 h22 h2 ˚.h/ D 2h˚.h/ C h2 ˚ 0 .h/ D

1 X

2Cn./ .x/hnC1 C

nD0

D

1 X

nCn./ .x/hnC1

nD1

1 X

./

.n C 2  2/Cn2 .x/hn1 ;

nD1

(3.4.47)

88

3 Orthogonal Polynomials ./

where in each case C1  0. Therefore,

0

0 ˚ 0 .h/ D 2xh1 h ˚.h/  h22 h2 ˚.h/

D 2x h1 h1 ˚.h/ C h1 h ˚ 0 .h/

 2h21 h22 ˚.h/ C h22 h2 ˚ 0 .h/



D 2x ˚.h/ C h˚ 0 .h/  2h˚.h/ C h2 ˚ 0 .h/ :

(3.4.48)

.1  2hx C h2 /˚ 0 .h/ D .2x  2h/˚.h/

(3.4.49)

2h  2x ˚ 0 .h/ D  : ˚.h/ 1  2hx C h2

(3.4.50)

This yields or equivalently

We perform integration with respect to the variable h:

ln j˚.h/j D  ln.1  2hx C h2 / C C D ln .1  2hx C h2 / C C: (3.4.51) Finally,

˚.h/ D CQ .1  2hx C h2 / ;

where CQ D 1, since ˚.0/ D 1.

(3.4.52) t u

Remark 3.4.8. For  D 0, we can find the generating function of the Chebyshev polynomials of first kind, namely  ln.1  2hx C h2 / D

1 1 n X X 2 2 Y 2k Tn .x/hn D Pn.1=2;1=2/ .x/hn : n n 2k  1 nD1 nD1 kD1 (3.4.53)

Remark 3.4.9. The generating function can also be used to define the corresponding orthogonal polynomials as coefficients of the series expansion. Corollary 3.4.10. For  > 0, x 2 Œ1; 1 and jhj < 1, 1 X nC nD0



hn Cn./ .x/ D

1  h2 : .1 C h2  2hx/C1

(3.4.54)

Proof. Use again (3.4.44) and (3.4.45) of the proof of Theorem 3.4.7 and find that 1 X nC nD0



hn Cn./ .x/ D

1 0 1  h2 h˚ .h/ C ˚.h/ D : (3.4.55)  .1 C h2  2hx/C1 t u

3.4 Ultraspherical Polynomials

89

Theorem 3.4.11. We can derive the following relations for n 2 N0 : .1  x 2 /

 d ./ 1 ./ Cn .x/ D .n C 2  1/.n C 2/Cn1 .x/ dx 2.n C /  ./ n.n C 1/CnC1 .x/ (3.4.56) ./

(3.4.57)

./

(3.4.58)

D nxCn./ .x/ C .n C 2  1/Cn1 .x/ D .n C 2/xCn./ .x/  .n C 1/CnC1 .x/ .C1/

D 2.1  x 2 /Cn1 .x/;

(3.4.59)

d ./ d ./ Cn .x/  C .x/; dx dx n1 d ./ d CnC1 .x/  x Cn./ .x/; .n C 2/Cn./ .x/ D dx dx nCn./ .x/ D x

(3.4.60) (3.4.61)

as well as (for n 2 N)  d  ./ ./ CnC1 .x/  Cn1 .x/ D2.n C /Cn./ .x/ dx   .C1/ D2 Cn.C1/ .x/  Cn2 .x/ :

(3.4.62) (3.4.63)

Proof. Relation (3.4.56) is proved by induction. The case n D 0 is obvious. Taking ./ n to n C 1 we then use the three-term recurrence (3.4.10) for CnC1 and obtain .1  x 2 /

d ./ 2.n C / ./ 2.n C / 2 ./ CnC1 .x/ D Cn .x/  x Cn .x/ (3.4.64) dx nC1 nC1 2.n C / d n C 2  1 d ./ C x.1  x 2 / Cn./ .x/  .1  x 2 / Cn1 .x/: nC1 dx nC1 dx ./

d Now, the induction assumption is applied twice, for .1  x 2 / dx Cn .x/ as well as ./ d for .1  x 2 / dx Cn1 .x/. Next, the three-term recurrence (3.4.10) helps to substitute ./ ./ ./ xCnC1 , xCn1 , and x 2 Cn (here we use it twice). This leads to a formula that only ./ ./ ./ contains the ultraspherical polynomials CnC2 , Cn , and Cn2 . Simplifying their ./ coefficients concludes the induction and yields (3.4.56) (the coefficient of Cn2 turns ./ out to be 0). The three-term recurrence (3.4.10) for .n C 1/CnC1.x/ in (3.4.56) gives us (3.4.58) and the step between (3.4.58) and (3.4.57) is once again only (3.4.56). Equation (3.4.59) is a direct consequence of (3.4.7). To prove (3.4.60), apply the derivative to (3.4.57) and rearrange the terms such that

90

3 Orthogonal Polynomials

nCn./ .x/  x

d ./ d ./ C .x/ C C .x/ D dx n dx n1

.n C 2/

(3.4.65)

d d2 d ./ Cn1 .x/  .n  1/x Cn./ .x/  .1  x 2 / 2 Cn./ .x/: dx dx dx

Using (3.4.7) on each term of the right-hand side as well as (3.4.57), the righthand side becomes 0 as desired. Equation (3.4.61) is shown analogously to (3.4.60), only now (3.4.58) takes the role of (3.4.57) before. Finally, (3.4.62) is obtained by adding (3.4.60), (3.4.61) and (3.4.63) follows directly from (3.4.59) or (3.4.7). u t Remark 3.4.12. For q 2 N, q  3, and t 2 Œ1; 1, 1

q2

Pn .qI t/ D nCq3 Cn

2

n



.n C 1/ .q  2/ q2 2 Cn .t/ D .t/ .n C q  2/

(3.4.66)

denotes the Legendre polynomials of dimension q and degree n (see, e.g., M¨uller 1998 and Sect. 6.4, in particular Remark 6.4.8).

3.5 Application of the Legendre Polynomials in Electrostatics R Let x 2 R3 and let %.x/ be a charge distribution with total charge R3 %.x/ dx. Then, the fundamental equations of electrostatics are the electrostatic pre-Maxwell equations. Let E denote the electric field, i.e., for x 2 R3 , rx  E.x/ D 4%.x/; rx ^ E.x/ D 0:

(3.5.1) (3.5.2)

We introduce the electric potential ˚ and solve the equations by E.x/ D r˚.x/ which gives us the following Poisson equation

˚.x/ D 4%.x/:

(3.5.3)

Example 3.5.1. A point charge q at x0 2 R3 yields %.x/ D qı.x  x0 /

(3.5.4)

with the delta-distribution ı in R3 (equality holds in the weak sense). We obtain

˚.x/ D 0 The solution is given by ˚.x/ D field can be calculated to be

q jxx0 j

for x 2 R3 n fx0 g:

(3.5.5)

for x 2 R3 nfx0 g. The corresponding electric

3.5 Application of the Legendre Polynomials in Electrostatics

E.x/ D q

x  x0 jx  x0 j3

91

for x 2 R3 n fx0 g:

(3.5.6)

Another approach to (3.5.5) introduces polar coordinates .r; '; t/ for x 2 R3 nf0g by p x1 D r 1  t 2 cos ';

p x2 D r 1  t 2 sin ';

x3 D rt;

(3.5.7)

where r D jxj > 0, ' 2 Œ0; 2/, t 2 Œ1; 1 (see also (4.1.2)). This gives us the representation of the Laplace equation in polar coordinates, namely 

@ @r

2

1 @ 1 1 2 @ @ C 2 .1  t 2 / C 2 C r @r r @t @t r 1  t2



@ @'

2 ! ˚.r; '; t/ D 0 (3.5.8)

for x 2 R3 n f0g. By separation of variables we set ˚.r; '; t/ D U.r/V .'/W .t/

(3.5.9)

and insert this in the equation above, i.e., 

@ V .'/W .t/ @r C

2 U.r/ C V .'/W .t/

2 @ U.r/ r @r

1 1 @ @ 1 U.r/V .'/ .1  t 2 / W .t/ C 2 U.r/W .t/ 2 r @t @t r 1  t2

(3.5.10) 

@ @'

2 V .'/ D 0:

Next, we multiply by r 2 .1  t 2 / and divide by U.r/V .'/W .t/: r 2 .1  t 2 /

U 00 .r/ U 0 .r/ C 2r.1  t 2 / U.r/ U.r/

C .1  t 2 /2

(3.5.11)

V 00 .'/ W 0 .t/ W 00 .t/  2t.1  t 2 / C D 0; W .t/ W .t/ V .'/

which can be rewritten as  r 2 .1  t 2 /  .1  t 2 /2

U 00 .r/ U 0 .r/  2r.1  t 2 / U.r/ U.r/

(3.5.12)

V 00 .'/ W 00 .t/ W 0 .t/ C 2t.1  t 2 / D : W .t/ W .t/ V .'/

The left-hand side only depends on r and t, the right-hand side only on '. Thus, the equation can only be fulfilled if both sides are equal to a constant  2 R. Therefore,

92

3 Orthogonal Polynomials

V 00 .'/ D  , V 00 .'/  V .'/ D 0; V .'/

' 2 Œ0; 2/:

(3.5.13)

The non-trivial solutions p of this differential equation p are linear combinations of the functions V1 D exp.i '/ and V2 D exp.i '/. V1 and V2 have to be 2-periodic. This leads to a discretization of , namely  D m2 , m 2 N0 . The linearly independent solutions are given by Vm .'/ D exp.im'/;

' 2 Œ0; 2/; m 2 Z:

(3.5.14)

Now, we consider the left-hand side of (3.5.12) for these values of : 

m2 W 0 .t/ U 0 .r/ U 00 .r/ W 00 .t/  .1  t 2 / C 2t D r2 C 2r : 2 .1  t / W .t/ W .t/ U.r/ U.r/

(3.5.15)

By the same argument as before we find that this equation can only hold if both Q i.e., sides are equal to a constant , 

m2 W 0 .t/ W 00 .t/  .1  t 2 / C 2t D Q 2 .1  t / W .t/ W .t/

(3.5.16)

which is equivalent to   m2 .1  t 2 /W 00 .t/  2tW 0 .t/ C Q C W .t/ D 0; 1  t2

(3.5.17)

for t 2 Œ1; 1 and U 0 .r/ U 00 .r/ Q C 2r , 0 D r 2 U 00 .r/ C 2rU 0 .r/  U.r/ Q D r 2 U.r/ U.r/

(3.5.18)

for r > 0.

Symmetric Problems For the case of a point charge located at x0 we can assume (after a certain rotation of the coordinate system) that x0 is on the "3 -axis, i.e., x0 D Œ0; 0; r0 T , where r0 D jx0 j > 0. In this case it is clear that the solution of (3.5.5) has a rotational symmetry, i.e., ˚ does not dependent on '. We can use ˚.r; '; t/ D U.r/W .t/

(3.5.19)

3.5 Application of the Legendre Polynomials in Electrostatics

93

with r > 0 and t 2 Œ1; 1 (this corresponds to m D 0 before). Therefore, we find the equation for W : Q .t/ D 0; Lt W .t/ C W

t 2 Œ1; 1;

(3.5.20)

where we have introduced the Legendre operator Lt D .1  t 2 /

d2 d  2t : dt 2 dt

(3.5.21)

Equation (3.5.20) is the Legendre differential equation (or the Gegenbauer differential equation (3.4.3) with  D 1=2) which possesses a polynomial solution if and only if Q D n.n C 1/; n 2 N0 : (3.5.22) The solutions are the Legendre polynomials Wn .t/ D Pn .t/, t 2 Œ1; 1, n 2 N0 . Q Now, we consider the second equation for these values of : r 2 U 00 .r/ C 2rU 0 .r/  n.n C 1/U.r/ D 0;

r > 0:

(3.5.23)

This ordinary differential equation possesses two linearly independent solutions given by U1;n .r/ D r n ;

n 2 N0 ;

U2;n .r/ D

1 r nC1

;

n 2 N0 :

(3.5.24)

Together we find the two linearly independent solutions of the Laplace equation in the rotational symmetric case to be ˚1;n .r/ D r n Pn .t/;

˚2;n .r/ D

1 r nC1

Pn .t/;

(3.5.25)

for r > 0, t 2 Œ1; 1, n 2 N0 . Every solution can be expressed as a linear combination of these two, i.e., there exist coefficients An , Bn such that ˚.r; t/ D

 1  X 1 An r n C Bn nC1 Pn .t/: r nD0

(3.5.26)

For the case of a point charge located in x0 D Œ0; 0; r0 T we know a solution, namely ˚.x/ D

q ; jx  x0 j

x 2 R3 n fx0 g:

(3.5.27)

We use this solution to compute the coefficients An and Bn in the expansion (3.5.26). Restricting the solutions to the "3 -axis, the point x possesses the polar coordinates r D jxj, t D 1, ' D 0, i.e.,

94

3 Orthogonal Polynomials

˚.x/ D

q ; jr  r0 j

r > 0; r ¤ r0 :

(3.5.28)

We use the well-known geometric series and obtain

˚.x/ D

8 ˆ q ˆ < rr0 D ˆ q ˆ : r0 r D

q 1 r 1 r0 r q 1 r0 1 r r

q r

D D

0

1

P r0 n

r nD0  n 1 q P r r0 r0 nD0

for r > r0 ; (3.5.29) for r < r0 :

Comparing this to the expansion for t D 1 (remember that Pn .1/ D 1), i.e., ˚.r; 1/ D

 1  X Bn An r n C nC1 ; r nD0

r > 0;

(3.5.30)

results in the following coefficients: An D

8 r0 ; q

if r < r0 ;

r0nC1

( Bn D

qr0n

if r > r0 ;

0

if r < r0 :

(3.5.31)

Thus, we finally obtain

˚.x/ D ˚.r; t/ D

8 1 P ˆ ˆ r0 D jx0 j; for jxj D r < r0 D jx0 j:

0

(3.5.32) From a mathematical point of view there is another interesting aspect. Canceling the charge q leads to (x ¤ x0 ) 81 P ˆ ˆ
r0 D jx0 j; for jxj D r < r0 D jx0 j:

(3.5.33)

0

The value jx  x0 j can be expressed using polar coordinates (see (4.1.2) in Sect. 4.1 for details) by ˇh iT ˇˇ2 p ˇ p jx  x0 j2 D ˇˇ r 1  t 2 cos '; r 1  t 2 sin '; rt  r0 ˇˇ   r2 r0 D r 2 1 C 02  2 t : r r

(3.5.34)

3.6 Hermite Polynomials and Applications

95

Take r > r0 and we obtain, for r > r0 ,

r

1

1

q 1C

r02 r2

D  2 rr0 t

X rn 1 0 D P .t/; nC1 n jx  x0 j r nD0

(3.5.35)

which is equivalent to 1

q 1C

D

r02 r2

 2 rr0 t

1   X r0 n nD0

r

Pn .t/:

(3.5.36)

Pn .t/:

(3.5.37)

Analogously, for r < r0 , q

1 1C

r2 r02

 2 rr0 t

D

1  n X r nD0

r0

Substituting h D r0 =r, respectively h D r=r0 , we get the result of Theorem 3.4.7 for the Legendre polynomials, i.e.,  D 12 . For a more detailed treatment see, e.g., Jackson (1998).

3.6 Hermite Polynomials and Applications The Hermite polynomials Hn are the unique orthogonal polynomials in L2 .R/ with the measure d.x/ D w.x/dx D exp.x 2 /dx. Definition 3.6.1. The Hermite polynomials Hn are defined as follows: (i) H R n is a polynomial of degree n, (ii) R Hn .x/Hm .x/ exp.x 2 / dx D 0 for m ¤ n, p (iii) kHn k2 D 2n nŠ, n 2 N0 . By this definition we can calculate the first few polynomials H0 .x/ D 1, H1 .x/ D 2x, H2 .x/ D 4x 2  2, H3 .x/ D 8x 3  12x. Rodrigues’ representation and the explicit representation are given by 

d Hn .x/ D .1/ exp.x / dx n

bn=2c

Hn .x/ D

X kD0

n

2



exp.x 2 / ;

.1/k nŠ .2x/n2k : kŠ.n  2k/Š

(3.6.1)

(3.6.2)

The following recurrence relation holds with H0  1, H1  0: Hn .x/ D 2xHn1 .x/  2nHn2 .x/

for n 2 N; x 2 R:

(3.6.3)

96

3 Orthogonal Polynomials

The derivative is given by Hn0 .x/ D 2nHn1 .x/ for n 2 N0 , x 2 R, and the following differential equation is solved by the Hermite polynomials: Hn00 .x/  2xHn0 .x/ C 2nHn .x/ D 0;

(3.6.4)

where n 2 N0 , x 2 R. Furthermore, for un .x/ D exp.x 2 =2/Hn .x/ we have u00n .x/ C .2n C 1  x 2 /un .x/ D 0;

n 2 N0 ;

x 2 R:

(3.6.5)

For further details and properties of Hermite polynomials we refer to, e.g., Lebedev (1973), Sneddon (1980), and Szeg¨o (1967).

Application in Quantum Mechanics The starting point of any system is its Hamilton function E D H.pi ; qj ; t/;

(3.6.6)

where qj are (unified) coordinates, pi are the impulse coordinates and t is the time. H is the total energy of the system. The step from classical mechanics to quantum mechanics is performed by substitution rules: E ! i„

@ ; @t

p ! i„r:

(3.6.7)

h Note that „ D 2

1:05457  1034 Js is Plank’s constant. The coordinates qj are substituted by the wave function, where j j2 is the probability density, i.e., for all t,

Z j .x; t/j2 dx D 1:

(3.6.8)

R3

For example, let  be the wave function of a particle in a potential V , the classical Hamilton function is p2 C V .x/: (3.6.9) ED 2m This becomes by our substitutions i„

@ „2  .x; t/ D 

x  .x; t/ C V .x/ .x; t/ @t 2m

or equivalently i„P D H

with H D 

„2

x C V .x/: 2m

(3.6.10)

(3.6.11)

3.6 Hermite Polynomials and Applications

97

If V and, thus, H are time-space separating, we use the approach  .x; t/ D '.x/ exp. iEt „ /;

(3.6.12)

such that for all t; i„

@ iEt iEt  .x; t/ D i„'.x/ iE „ exp. „ / D H'.x/ exp. „ / @t

(3.6.13)

which is equivalent to H'.x/ D E'.x/. This is the time-independent Schr¨odinger equation. H is the time-independent Hamilton operator and E is the energy of the system, which is an unknown. Furthermore, Z Z Z 2 iEt 2 j .x; t/j dx D j'.x/ exp. „ /j dx D j'.x/j2 dx; (3.6.14) 1D R3

R3

R3

i.e., the probability density is time-independent. Therefore, the eigenvalue problem R H' D E' has to be solved with R3 j'.x/j2 dx D 1.

One-Dimensional Oscillation Let m be a mass connected to a wall by a spring with spring constant f . The classical Hamilton function of this one-dimensional system is H.p; x/ D

m! 2 x 2 p2 C 2m 2

(3.6.15)

p with ! D f =m the eigenfrequency of the system. By our substitution rules we obtain the operator „2 d 2 m! 2 x 2 HD : (3.6.16) C 2 2m dx 2 This operator H describes, e.g., the oscillation of a molecule with two atoms and m is the relative mass of the atoms. Thus, the eigenvalue problem for the oscillation is:   „2 d 2 m! 2 x 2 '.x/ D E'.x/;  C 2m dx 2 2

x 2 R:

(3.6.17)

q „ 2E Defining a new coordinate by y D x=b, where b D m! , and setting " D „! this becomes d2 '.by/ C."  y 2 / '.by/ D 0: (3.6.18) „ƒ‚… dy 2 „ƒ‚… u.y/

u.y/

98

3 Orthogonal Polynomials

This equation possesses a solution if and only if " D 2n C 1, where n 2 N0 . The solution is of the form un .y/ D cn exp 

y2

2 Hn .y/

(3.6.19)

for n 2 N0 using the Hermite polynomial Hn of degree n. The constant cn is determined by the condition Z

Z R

j'n .x/j2 dx D 1 ,

R

jun .y/j2 dy D

1 : b

(3.6.20)

Thus, we can summarize 'n .x/ D cn Hn . xb / exp 

x2 2b 2



(3.6.21)

for n 2 N0 and x 2 R. These functions are the eigenfunctions of the onedimensional oscillation corresponding to the eigenvalues En D .n C 12 /„!;

n 2 N0 :

(3.6.22)

This shows that the energy of a quantum-mechanical oscillator can only take discrete values, i.e., the quantization of energy. For further information the reader is referred to, e.g., Landau and Lifshitz (2004).

3.7 Laguerre Polynomials and Applications In this section we consider orthogonal polynomials with respect to the measure d.x/ D w.x/dx D exp.x/x ˛ dx with a parameter ˛ > 1 and the support interval R0 D Œ0; 1/. .˛/

Definition 3.7.1. For ˛ > 1, the generalized Laguerre polynomials Ln , n 2 N0 , are uniquely defined by .˛/

(i) Ln is a polynomial of degree n defined on R0 , R .˛/ .˛/ (ii) R0 Lm .x/Ln .x/ exp.x/x ˛ dx D 0 for n ¤ m,  .˛/ 2

(iii) Ln  D .˛ C 1/ nC˛ D .nC˛C1/ . n

.nC1/

Note that for ˛ D 0 these are the classical Laguerre polynomials. The Laguerre polynomials admit the following Rodrigues’ representation and explicit representation for n 2 N0 and x 2 R0 : L.˛/ n .x/ D exp.x/

x ˛ nŠ



d dx

n



exp.x/x nC˛ ;

(3.7.1)

3.7 Laguerre Polynomials and Applications

L.˛/ n .x/ D

99

n X .n C ˛ C 1/ kD0

.x/k : .k C ˛ C 1/ kŠ.n  k/Š

(3.7.2) .˛/

By these properties we obtain the first three polynomials, namely L0 .x/ D 1,

.˛/ .˛/ L1 .x/ D 1C˛x and L2 .x/ D 12 .1 C ˛/.2 C ˛/  2.2 C ˛/x C x 2 . We find the recursion formula for n 2 N, n  2: .˛/

.˛/

nL.˛/ n .x/ D .2n C ˛  1  x/ Ln1 .x/  .n C ˛  1/ Ln2 .x/;

x 2 R0 (3.7.3)

and the recursion including the derivative for n 2 N; x

d .˛/ .˛/ L .x/ D nL.˛/ n .x/  .n C ˛/Ln1 .x/; dx n

x 2 R0 :

(3.7.4)

The differential equation with ˛ > 1, x 2 R0 , xy 00 C .1 C ˛  x/y 0 C y D 0

(3.7.5)

possesses a polynomial solution if and only if  D n 2 N0 . This solution is given .˛/ by y.x/ D cn Ln .x/, x 2 R0 , with a constant cn 2 R. For further details and properties of Laguerre polynomials we refer to, e.g., Lebedev (1973), Sneddon (1980), and Szeg¨o (1967).

Eigenoscillations of an n-Fold Pendulum Let l be the total length of the n-fold pendulum, each section has length a D l=n. The angles of each section are collected in the vector ' D Œ'1 ; : : : ; 'n T . The dynamic of the pendulum is described by M'R C C' D 0;

(3.7.6)

where we have linearized near a stable equilibrium, i.e., 'j D 0, j D 1; : : : ; n, and M is the mass matrix with entries Mi;j D a2 minfi; j g

(3.7.7)

and C is the matrix of restitutional forces with Ci;j D igaıi;j , g is the gravitational acceleration. Small vibrations are given by ' D 'O sin.!t/. M'R C C' D 0;

i:e:; .C  ! 2 M/'O D 0:

(3.7.8)

100

3 Orthogonal Polynomials

We decompose M by Cholesky decomposition, i.e. M D UT U. This gives us 2

1  6 :: U D a4 : 0

3 1 :: 7 : :5

(3.7.9)

1

Now, we set xO D U', O where xO D ŒxO 0 ; : : : ; xO n1 T . We find that

H) H) H)



0 D C  ! 2 UT U '; O

1 O 0 D CU  ! 2 UT x; T 1 1

0 D .U / CU  ! 2 I x; O   O 0 D ga .UT /1 CU1  ga ! 2 I xO D .A  I/x:

(3.7.10) (3.7.11) (3.7.12) (3.7.13)

This is an eigenvalue problem for the matrix 2

1 61 6 6 60 6 AD6 6 6 6 4

1 0 3 2 0

0

::

3

7 7 7 7 7 7: 0 7 7 7 n C 15

: 2 5 3 :: :: :: :: : : : : :: :: :: : : : 0 0 n C 1 2n  1

(3.7.14)

Writing out the eigenvalue problem explicitly, we obtain 0xO 1 C 1xO 0  1xO 1 D xO 0 ;

(3.7.15)

k xO k1 C .2k C 1/xO k  .k C 1/xO kC1 D xO k ;

(3.7.16)

.n  1/xO n2 C .2n  1/xO n1 D xO n1 ;

(3.7.17)

where k D 1; : : : ; n  2. We have to append xO 1 and xO n such that (3.7.16) holds for k D 0; : : : ; n  1, i.e., xO 1 can be chosen arbitrarily and xO n D 0 to satisfy (3.7.17). We take a look at the recurrence relation for the classical Laguerre polynomials, i.e., the case ˛ D 0:  kLk1 .x/ C .2k C 1/Lk .x/  .k C 1/LkC1 .x/ D xLk .x/:

(3.7.18)

We can identify x with  and Lk .x/ with xO k . Hence, xO k D Lk ./ for k D 0; : : : ; n satisfies (3.7.15)–(3.7.17) for all . Since we have to fulfill xO n D 0, we find Ln ./ D xO n D 0:

(3.7.19)

3.8 Exercises

101

The eigenfrequencies 1 ; : : : ; n of the system AxO D xO

(3.7.20)

are determined by (3.7.19). Therefore, let n;1 ; : : : ; n;n be the zeros of Ln in R0 . Then, the eigenfrequencies of the system are q (3.7.21) !k D ga n;k ; k D 1; : : : ; n: The corresponding eigenmodes to the eigenvalue !k can be calculated from xO j D Lj .n;k /;

j D 0; : : : ; n  1;

(3.7.22)

and 'O D U 1 x. O All interesting properties of the system, such as energy etc., can be expressed using Laguerre polynomials. For further details the reader is referred to Braun (1997).

3.8 Exercises (Gaussian Integration, Legendre Series, Kernel Expansions) Next, we remember the conditions in Gauß quadrature rules which must be satisfied by the weights and knots in order to get an optimal degree of polynomial precision. Restricting ourselves to the weight function w.x/ D 1 and the interval Œ1; 1 we are canonically led to the theory of Legendre polynomials, i.e., Gauß–Legendre integration. Our exercises do not only provide the classical theory of Gaussian integration, but also new types of error estimates involving the Green function with respect to the Legendre operator as an essential tool. Moreover, the Clenshaw algorithm for the staple evaluation of Legendre series and its modified version are discussed with the help of some exercises. Finally, we develop error estimates for kernel expansions that arise in numerical algorithms involving truncations of expansions in terms of ultraspherical polynomials.

Gaussian Integration Revisited Definition 3.8.1. The Green function G.L C I ; / W Œ1; 1 Œ1; 1 ! R with respect to the Legendre operator L C ,  2 R, (see (3.5.21) for a definition of the operator L) is uniquely defined by the following properties: (i) (Boundedness) For arbitrary, but fixed x 2 Œ1; 1, G.L C I x; / is a continuous function on Œ1; 1 satisfying the conditions jG.L C I x; 1/j < 1;

(3.8.1)

jG.L C I x; 1/j < 1:

(3.8.2)

102

3 Orthogonal Polynomials

(ii) (Differential equation) For each fixed x 2 Œ1; 1, G.L C I x; / is twice continuously differentiable in Œ1; 1 n fxg. For  … SpectL , we have .Lt C /G.L C I x; t/ D 0; t 2 Œ1; 1 n fxg: q For n 2 SpectL , we have with PQn D 2nC1 Pn that 2 .Lt C n /G.L C n I x; t/ D PQn .x/PQn .t/;

t 2 Œ1; 1 n fxg;

(3.8.3)

(3.8.4)

where SpectL D fn D n.n C 1/ W n 2 N0 g. (iii) (Characteristic singularity) .1  x 2 /

ˇt DxC0 d ˇ G.L C I x; t/ˇ D 1: t Dx0 dt

(3.8.5)

(iv) (Normalization) For each x 2 Œ1; 1 and n 2 SpectL , Z

1 1

G.L C n I t/PQn .t/ dt D 0:

(3.8.6)

The Green function is uniquely determined by these four conditions. Definition 3.8.1 enables us to develop the following integral formulas: Exercise 3.8.2. Let x be a point in .1; 1/. Suppose that F is of class C.2/ .Œ1; 1/. (a) Use the Green-Lagrange formula to prove for  … SpectL , Z

1

F .x/ D 1

G.L C I x; t/.Lt C /F .t/ dt:

(3.8.7)

Furthermore, for  2 SpectL , i.e.,  D n D n.n C 1/, F .x/ D PQn .x/

Z

1 1

F .t/PQn .t/ dt C

Z

1 1

G.L C I x; t/.Lt C /F .t/ dt: (3.8.8)

(b) Use (a) to show that for  2 R, t 2 Œ1; 1, Z 1 Q .1  ı;n /Pn .t/ D .  n / G.L C I t; u/PQn .u/ du:

(3.8.9)

1

Exercise 3.8.3. (a) Implement the Gaussian integration formula Z

1 1

F .t/ dt Gn .F / D

n X i D1

wi F .xi /;

(3.8.10)

3.8 Exercises

103

with xi , i D 1; : : : ; n, the zeroes of the Legendre polynomial Pn and wi , i D 1; : : : ; n the weights determined by n X

wi PQj .xi / D ı0;j

p 2;

j D 0; : : : ; n  1:

(3.8.11)

i D1

(b) Show that (3.8.10) is exact for all polynomials of degree 2n  1. (c) Give an example of a polynomial of degree 2n such that (3.8.10) is not exact. (d) Test your routine by approximating the following integrals for n D 2; 4; 8: Z

1

Z

1 dx; 1 C x4

1

1

e

x2

Z

1

dx;

1

1

x dx: ex  1

(3.8.12)

Exercise 3.8.4. Calculate the error terms for the Gaussian integration formula to test the exactness of your results in Exercise 3.8.3. Do this by implementing routines for the following two estimates of the error or remainder term: (a) For n  1 and p  1; 22.nC1/ .2p/ .t/j: 2n 2 sup jF t 2Œ1;1 .2n C 1/Š n

Rn .F / 

(3.8.13)

(b) For n  1, p  1 and  ¤ k.k C 1/, k D 2n; 2n C 1; : : : ; q Rn .F /  

.p/ A .n/

s Z

1 1

..Lt C /p F .t//2 dt

p q .p/ 2 A .n/ sup j.Lt C /p F .t/j

(3.8.14)

t 2Œ1;1

with .p/ A .n/

D

n n X X i D1 j D1

2

1 X kD2n

.p/

wi wj

1 X kD2n

1 PQk .xi /PQk .xj / .  k.k C 1//2p

2k C 1 : .  k.k C 1//2p

(3.8.15)

Note that lim A .n/ D 0. n!1 (c) In case of (3.8.13), apply the estimate for different n; p to at least one of the integrals from Exercise 3.8.3. In case (3.8.14), apply the estimate for fixed p D 1 but different n; . For which  does the error reach its minimum?

104

3 Orthogonal Polynomials

Remark 3.8.5. Relation (3.8.13) is a Peano-type error estimate that can be found in every standard textbook on numerical integration (see, e.g., Deuflhard and Hohmann 1991; Freund and Hoppe 2007; H¨ammerlin and Hoffmann 1992). Estimate (3.8.14) is due to Freeden (1980b).

Staple Evaluation of Legendre Series (Involving Derivatives) Exercise 3.8.6. Let fTk gk2N0 satisfy the linear, second order recurrence relation Tk .x/  ak .x/Tk1 .x/  bk .x/Tk2 .x/ D 0;

k D 2; : : : ; N;

(3.8.16)

where T0 .x/ ¤ 0 and T1 .x/ are given and bk .x/ ¤ 0 for all k. Show that for N 2 N0 the sum N X SN .x/ D Ak Tk .x/; (3.8.17) kD0

where fAk gkD0;:::;N are given, can be calculated by the Clenshaw algorithm, i.e., Algorithm 3.8.7 using ak , bk of (3.8.16). Algorithm 3.8.7 (Clenshaw). Set UN C1 D UN C2 D 0. Do for k D N; : : : ; 1 Uk .x/ D akC1 .x/UkC1 .x/ C bkC2 .x/UkC2 .x/ C Ak enddo Return SN .x/ D .A0 C b2 .x/U2 .x//T0 .x/ C U1 .x/T1 .x/. Exercise 3.8.8. Specify Algorithm 3.8.7 for the special case of the Legendre polynomials Pk , k 2 N0 . Consider the following tests: (a) Implement the resulting algorithm and compute the functions Fh;N .x/ D

N X

hk Pk .x/;

x 2 Œ1; 1;

(3.8.18)

kD0

Gh;N .x/ D

N X .2k C 1/hk Pk .x/;

x 2 Œ1; 1;

(3.8.19)

kD0

on an equidistant grid in Œ1; 1 for N D 100; 200; 400; 800 and for the values h D 0:1; 0:5; 0:9; 0:95; 0:99. (b) Calculate the differences ˇ ˇ ˇ ˇ 1 ˇ ˇ; RF .h; N; t/ D ˇFh;N .x/  p (3.8.20) 2 1 C h  2hx ˇ

3.8 Exercises

105

ˇ ˇ ˇ ˇ 1  h2 ˇ: ˇ RG .h; N; t/ D ˇGh;N .x/  2 3=2 .1 C h  2hx/ ˇ .m/

Exercise 3.8.9. Show that for N 2 N0 the sum of derivatives Tk .m/

SN .x/ D

N X

(3.8.21) of order m 2 N;

.m/

Ak Tk .x/;

(3.8.22)

kD0

where fAk gkD0;:::;N are given, can be calculated by the modified Clenshaw algorithm, i.e., Algorithm 3.8.10 using ak , bk of (3.8.16). Algorithm 3.8.10 (Modified Clenshaw). .0/ .0/ Set stage 0, i.e., UN C1 D UN C2 D 0. Do for k D N; : : : ; 0 .0/ .0/ .0/ Uk .x/ D akC1 .x/UkC1 .x/ C bkC2 .x/UkC2 .x/ C Ak enddo Do for l D 1; : : : ; m .l/ .l/ Set stage l, i.e., UN C1 D UN C2 D 0. Do for k D N; : : : ; 0   l P l  .lj / .l/ .j / .lj / .j / akC1 .x/UkC1 .x/ C bkC2 .x/UkC2 .x/ Uk .x/ D j D0 j enddo enddo .m/ .m/ Return SN .x/ D U0 .x/T0 .x/. .m/

Remark 3.8.11. Note that Algorithm 3.8.10 not just calculates SN .x/, but all .l/ SN .x/ for l D 0; : : : ; m. For an implementation the derivatives of ak , bk of (3.8.16) have to be computed in advance up to order m. If ak , bk are polynomials of a low degree, the sum in Algorithm 3.8.10 simplifies drastically. Exercise 3.8.12. Specify Algorithm 3.8.10 for the special case of the Legendre polynomials Pk , k 2 N0 . Exercise 3.8.13. Specify Algorithm 3.8.7 for the second derivatives of the Legendre polynomial Pk00 , k 2 N, by using (3.4.7) for the derivative of the ultraspherical polynomials and their three-term recurrence relation. Compare the results of the summation with the results of Algorithm 3.8.10 for Pk00 , k 2 N, using Exercise 3.8.12. Remark 3.8.14. See Clenshaw (1955), Deuflhard (1976), and Tscherning (1976) for further details and Fengler (2005) for an application of these algorithms in the computation of vectorial/tensorial scaling functions and wavelets for a non-linear Galerkin scheme to solve the Navier–Stokes equation on the sphere.

106

3 Orthogonal Polynomials

Error Estimates of Kernel Expansions Exercise 3.8.15. Compute the first derivative of the hypergeometric function with respect to its last argument, i.e., for z 2 R: ab d F .a; bI cI z/ D F .a C 1; b C 1I c C 1I z/: dz c

(3.8.23)

Exercise 3.8.16. Let s 2 N, p 2 N0 and z 2 R with jzj < 1. Show that   p1  1  X 1 nCs1 n X nCs1 n z z  D s1 s1 .1  z/s nD0 nDp D

zp .1  z/s

(3.8.24)

  pCs1 F .s C 1; pI p C 1I z/; s1 (3.8.25)

where F denotes the hypergeometric function of Definition 3.3.8. Exercise 3.8.17. Let s 2 N, p 2 N0 . Show that F .s C 1; pI p C 1I / D

s1 X kD0

p pCk

  s1 k  k

(3.8.26)

if p > 0. Show also that F .s C 1; 0I 1I / D 1 holds for p D 0. Remark 3.8.18. For more details the standard literature on the hypergeometric function, e.g., Abramowitz and Stegun (1972) and Magnus et al. (1966). A proof and further applications of Exercise 3.8.16 can be found, e.g., in Cherrie et al. (2002). Now, we investigate the error term s EpC1 .; / D 1 

p p sX n s 1 C  2  2  Cn2 . /

(3.8.27)

nD0

for s 2 N, p 2 N0 ,  2 .0; 1/ and 2 Œ1; 1. Our aim is to prove that ˇ ˇ ˇ ˇ ˇ s ˇ ˇ s ˇ .; 1/ˇ : ˇEpC1 .; /ˇ  ˇEpC1

(3.8.28)

Exercise 3.8.19. Use Exercise 3.8.16 to transform the right-hand side of (3.8.28), i.e., show that   ˇ ˇ pCs ˇ ˇ s F .s C 1; p C 1I p C 2I /: ˇEpC1 .; 1/ˇ D  pC1 s1

(3.8.29)

3.8 Exercises

107

This leads us to a new version of (3.8.28), i.e.,   ˇ ˇ ˇ s ˇ pC1 p C s F .s C 1; p C 1I p C 2I /: ˇEpC1 .; /ˇ   s1

(3.8.30)

Before we can go into the details of this estimate and its proof we are concerned with three helpful tools. Exercise 3.8.20. Let s; p 2 N with s  3 and p  1. Prove that the following hypergeometric function at 1 satisfies the estimate F .s C 1; p C 1I p C 2I 1/ 

p

s

2:

(3.8.31)

Exercise 3.8.21. Let s 2 N, p 2 N0 . Show that the sum of Gegenbauer polynomials at 0 satisfies ( p X s 0 ; for p D 4m C 2 or p D 4m C 3; 2 (3.8.32) Cn .0/  . 2s C2m/ ; for p D 4m or p D 4m C 1; . s / .2mC1/ nD0 2

and moreover,

p X nD0

  pCs Cn .0/ < : s1 s 2

(3.8.33)

Exercise 3.8.22. Let s 2 N, p 2 N0 . Show that the sum of Gegenbauer polynomials at 0 satisfies p X

( s 2

Cn .0/ 

nD0

; for p D 4m or p D 4m C 1;

1 1

. 2s C2mC1/ . 2s / .2mC2/

and moreover,

p X nD0

s

; for p D 4m C 2 or p D 4m C 3;

Cn2 .0/ > 1 

  pCs : s1

(3.8.34)

(3.8.35)

Finally, we are prepared to prove the major theorem necessary for error estimate (3.8.30) in three exercises. Exercise 3.8.23. Let s 2 N, p D 0. Let D D .0; 1/ .1; 1/ and the functions h˙ W D ! R be defined as h˙ .; / D pC1

  pCs F .s C 1; p C 1I p C 2I / s1

p p sX n s 2  Cn2 . / ˙ 1  1 C   2 nD0

DsF .s C 1; 1I 2I / ˙ 1 

p s 1 C  2  2 :

(3.8.36)

108

3 Orthogonal Polynomials

Prove that, for all .; / 2 D, it holds that h .; /  0 and hC .; /  0. Exercise 3.8.24. Let s 2 N, s > 1, p 2 N. Let D D .0; 1/ .1; 1/ and the functions h˙ W D ! R be defined as h˙ .; / D pC1

  pCs F .s C 1; p C 1I p C 2I / s1

˙1

p p sX n s 1 C  2  2  Cn2 . /:

(3.8.37)

nD0

Show the following properties of h˙ : (a) h˙ .0; / D 0, (b) h .1; 0/ > 0 and hC .1; 0/ > 0, @ (c) @ h˙ .; / ¤ 0 in D. Hint: Use Exercises 3.8.21 and 3.8.22 for the two inequalities of part (b), Exercise 3.8.17 can be helpful for the first inequality. Calculate the partial derivative @ in part (c) (using Exercise 3.8.15) and assume that @ h˙ .; / D 0 for .; / 2 D. This leads to the equation   pCs F .s C 1; p C 1I p C 2I / (3.8.38) .p C 1/ s1  s1 s C F .s C 2; p C 2I p C 3I / D ˙.1  2 C  2 / 2 1 pC2 ! p p X X s s n 2 2 n1 2 s.  /  Cn . / C .1  2 C  / n Cn . / : p

nD0

nD1

Show now that the bracket on the left-hand side of (3.8.38) containing the hypergeometric functions can be reduced to .1 C /s1 . The bracket on the righthand side of (3.8.38) containing the Gegenbauer polynomials can be transformed s s 2 into .p C s/ pC1 Cp2 . /  .p C 1/ p CpC1 . / using the corresponding recurrence relations of Theorem 3.4.11. Prove the estimate s s     pCs 2 2 .1 C /s2 pCs pCs pC1 Cp . /  CpC1 . / ;  ˙ s s  1 .1  2 C  2 / 2 1 s1 1C (3.8.39)

and find the conclusion that D 1 or  D 0 which gives you the desired @ contradiction to the assumption @ h˙ .; / D 0 in D. Exercise 3.8.25. Let s D 1, p 2 N. Let D D .0; 1/ .1; 1/ and the functions h˙ W D ! R be defined as

3.8 Exercises

109

h˙ .; / D

pC1

  pCs F .s C 1; p C 1I p C 2I / s1

p p sX n s 2 ˙ 1  1 C   2  Cn2 . / nD0

D pC1 ˙ 1 

p X p 1 C  2  2  n Pn . /;

(3.8.40)

nD0 1

where Pn . / D Cn2 . / is the Legendre polynomial of degree n. Show the following properties of h˙ : (a) h˙ .0; / D 0, (b) h .1; 0/ > 0 and hC .1; 0/ > 0, @ (c) @ h˙ .; / ¤ 0 in D. Hint: Note that properties (a) and (b) can be treated as in Exercise 3.8.24. Property (c) also starts out as in Exercise 3.8.24, but instead of inequalities as in (3.8.39) the equation p 1 D ˙ 1  2 C  2 .Pn . /  PnC1 . // (3.8.41) is squared and the resulting quadratic equation in  is shown to possess no realvalued solutions. Thereby, the recurrence relations of Theorem 3.4.11 can be very helpful if they are applied to the case of Legendre polynomials and their derivatives. Exercise 3.8.26. Let s 2 N, p 2 N. Let D D .0; 1/ .1; 1/ and the functions h˙ W D ! R be defined as in Exercise 3.8.24. Use the properties of Exercises 3.8.24 and 3.8.25 to prove that for all .; / 2 D hold h .; /  0 and hC .; /  0. Exercise 3.8.27. Let  2 .0; 1/, p 2 N0 and s 2 N. Use Exercise 3.8.26 to prove that the error term (3.8.27) can be estimated by   ˇ ˇ ˇ ˇ pCs ˇ ˇ s ˇ ˇ s F .s C 1; p C 1I p C 2I /; .; 1/ˇ D  pC1 ˇEpC1 .; /ˇ  ˇEpC1 s1 (3.8.42) where 2 Œ1; 1. Remark 3.8.28. One major application of the results of these exercises lies in error estimates for numerical algorithms involving truncated kernel expansions such that, e.g., the fast multipole method (see also the exercises of Chap. 4 or, e.g., Greengard and Rokhlin 1988) which goes back to Greengard (1988), Greengard and Rokhlin (1987), and Rokhlin (1985). A detailed derivation and error analysis of this kind can be found in Gutting (2008). Note that the case treated in Exercise 3.8.25 goes back to Sauter (1991). For more details the reader is referred to Gutting (2008) and the references therein.

Chapter 4

Scalar Spherical Harmonics in R3

As we have seen in the geomathematical motivation of Chap. 1, spherical functions are an essential tool for all geosciences. In this chapter we develop orthonormal function systems for scalar functions on spheres in the three-dimensional Euclidean space, namely the scalar spherical harmonics, which are then generalized to vectorial functions in Chap. 5 as well as to systems for scalar functions on spheres in the more general, q-dimensional setting in Chap. 6. The main features in each case are the addition theorem, the Funk–Hecke formula, and orthogonal invariance leading to expressions in the terms of Legendre polynomials. Scalar spherical harmonics have many fields of applications such as, e.g., geodesy and geophysics (see Sects. 1.1 or 1.2 for two examples), quantum mechanics (see, e.g, Landau and Lifshitz 2004), or numerical algorithms such as the fast multipole method of Greengard and Rokhlin (1988) (see Beatson and Greengard (1997) and the references therein for an overview or Gutting (2008) for an application in gravitational field modeling). All in all, they are essential for any analysis of spherical functions. The scalar spherical harmonics also provide the foundation for vector spherical harmonics (see Chap. 5). There is an extensive literature on spherical harmonics starting from Gauß (1801) and we only give a few recommendations: for a more mathematical point of view we refer to, e.g., Freeden et al. (1998), Freeden and Schreiner (2009), Hobson (1955), Hochstadt (1971), Lebedev (1973), Lense (1954), M¨uller (1966, 1998) and for physical aspects, in particular in quantum mechanics and in angular momentum theory, see Edmonds (1964), Jones (1985), Landau and Lifshitz (2004), Var˘salovi˘c et al. (1988) and Zare (1988). The layout of this chapter is as follows: In Sect. 4.1 we provide some basic notation for the sphere in R3 , in particular, we introduce zonal functions on the sphere. Orthogonal invariance is discussed for spaces of spherical functions in Sect. 4.2. We also lay the groundwork on orthogonal invariance for spherical vector fields which are elaborated in Chap. 5. As we will see in Sect. 4.3, we are naturally led to spherical harmonics by investigating polynomial basis systems on the sphere. We show their essential properties such as the addition theorem, orthogonality, and their role as eigenfunctions of the Beltrami operator. For their completeness in the W. Freeden and M. Gutting, Special Functions of Mathematical (Geo-)Physics, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-0348-0563-6 4, © Springer Basel 2013

113

4 Scalar Spherical Harmonics in R3

114

space of square-integrable functions on the sphere we propose and analyze two summability methods in Sect. 4.4, before the irreducibility of the space of spherical harmonics of a fixed degree is shown in Sect. 4.5. This yields the Funk–Hecke Formula. In Sect. 4.6 Green’s function on the sphere with respect to the Beltrami operator is introduced and corresponding integral theorems are established. This is another result that is required for vector spherical harmonics in Chap. 5. Moreover, we briefly present the quantum-mechanical modeling of the hydrogen atom in Sect. 4.7 as a well-known example of use of spherical harmonics in physics (see also Landau and Lifshitz 2004). Finally, the reader is encouraged to solve the exercises in Sect. 4.8 which are concerned with some important topics in approximation and numerical analysis on the sphere such as the low discrepancy method, locally supported spherical wavelets, function systems for the ball, and the fast multipole method.

4.1 Basic Notation We start by introducing some basic spherical notation. The unit sphere in R3 is defined by S2 D f 2 R3 W jj D 1g: (4.1.1) We use Greek letters for elements of S2 . The polar coordinate representation of a point x 2 R3 is given by 2 p 3 r p1  t 2 cos.'/ x.r; '; t/ D 4 r 1  t 2 sin.'/ 5 ; rt

(4.1.2)

where r D jxj 2 R0 is the distance to the origin, ' 2 Œ0; 2/ the longitude, t D cos.#/ 2 Œ1; 1 the polar distance and # 2 Œ0;  the latitude. The canonical basis in R3 is denoted by "1 ; "2 ; "3 . Another orthonormal basis is given by the moving frame consisting of the following three vectors (depending on the spherical coordinates ' and t): 2p 2 3 3  sin.'/ 1  t 2 cos.'/ p "r .'; t/ D 4 1  t 2 sin.'/ 5 ; "' .'; t/ D 4 cos.'/ 5 ; 0 t 2 3 t cos.'/ 5 "t .'; t/ D 4 t p sin.'/ : 1  t2

(4.1.3)

"' and "t are tangential vectors. Note that "r ^ "' D "t , where ^ denotes the vector product (see Fig. 4.1 for an illustration). The gradient r in R3 can be

4.1 Basic Notation

115

Fig. 4.1 The perpendicular vectors "r , "' , "t on the unit sphere S2

decomposed into a radial and an angular part, namely r D "r

1 @ C r; @r r

(4.1.4)

where the tangential operator r  is called the surface gradient and can be written in polar coordinates of (4.1.2) as follows: p 1 @ @ r  D "' p C "t 1  t 2 : 2 @' @t 1t

(4.1.5)

Another essential tangential operator is the surface curl gradient L which is defined by L F ./ D  ^ r F ./ for F 2 C.1/ .S2 /,  2 S2 . In the polar coordinates of (4.1.2) we have: p @ 1 @ : L D "' 1  t 2 C "t p 2 @t @' 1t

(4.1.6)

L F is a tangential vector field perpendicular to r  F , i.e., for  2 S2 , r F ./  L F ./ D 0:

(4.1.7)

We canonically define the surface divergence r  f ./ D

3 X

r Fi ./  "i

(4.1.8)

L Fi ./  "i ;

(4.1.9)

i D1

and the surface curl L  f ./ D

3 X i D1

where f D ŒF1 ; F2 ; F3 T 2 c.1/ .S2 /. Note that we use lower-case letters for vector fields and capital letters for scalar fields. The same convention applies to

4 Scalar Spherical Harmonics in R3

116

the corresponding function spaces such as c.1/ .S2 / for the space of continuously differentiable vector fields and C.1/ .S2 / for the space of continuously differentiable scalar fields on the sphere. The surface curl represents a scalar-valued function on the unit sphere: L  f ./ D r  .f ./ ^ /: (4.1.10) Finally, the Laplace operator  can be decomposed into D

1 2 @ @2 C 2  ; C @r 2 r @r r

(4.1.11)

where  denotes the Beltrami operator (sometimes also called Laplace–Beltrami operator) and 1 @2 @2 1 @ 2 @ .1  t C D / : 1  t 2 @' 2 @t @t 1  t 2 @' 2

 D Lt C

(4.1.12)

Note that Lt denotes the Legendre operator of (3.5.21), which is that part of the Beltrami operator depending on the polar distance t. Moreover, for F 2 C.2/ .S2 /, it holds that r  r F ./ D  F ./

and L  L F ./ D  F ./:

(4.1.13)

Definition 4.1.1 (Regular Region on the Sphere). A bounded region G  S2 is called regular, if its boundary @G is an orientable piecewise smooth Lipschitzian manifold of dimension 1. An example is a spherical cap of radius r around  2 S2 , i.e., C.; r/ D f 2 S2 W 1  r      1g. The following spherical versions of the theorems of Gauß and Stokes are wellknown (see, e.g., Freeden and Schreiner 2009 and the references therein). Theorem 4.1.2 (Surface Theorems of Gauß and Stokes). Suppose that   S2 is a regular region with continuously differentiable boundary curve @ . Let f be a .1/ tangential vector field of class ctan . /, i.e., f ./   D 0 for all  2  . Then, Z 

Z



r

Z  f ./ dS./ D

L  f ./ dS./ D

Z

  f ./ d ./;

(4.1.14)

  f ./ d ./;

(4.1.15)

@

@

where d is the arc element.

 denotes the positively oriented unit tangential vector of the boundary curve @ at  2 @ . The unit normal vector  points into the exterior of  and is perpendicular to  and , i.e.,  is perpendicular to @ at the point  2 @ , but tangential to S2 (see Fig. 4.2 for an illustration).

4.1 Basic Notation

117

Fig. 4.2 A region  on the sphere S2 and the vectors  ,  at a point  2 @

It is important to point out the assumption of the vector field f being tangential in Theorem 4.1.2. This causes an additional term in Green’s formulas involving r  , but it does not affect those for L , which is due to the fact that r    D 2, but L   D 0 for  2 S2 . The normal derivative is given by @F @

./ D   r F ./ D   L F ./

(4.1.16)

for F 2 C.1/ . /. Lemma 4.1.3. Let   S2 be a regular region with continuously differentiable boundary @ . Suppose that F , G are of class C.1/ . /. Then, Z 

G./r F ./ dS./ C

F ./r G./ dS./

G./L F ./ Z D

(4.1.17)

Z

 .F ./G.// d ./ C 2 @





Z D

Z

Z

.F ./G.// d ./; @

Z dS./ C 

F ./L G./ dS./

(4.1.18)

 .F ./G.// d ./: @

Theorem 4.1.4. Let G 2 C.2/ . /,   S2 be a regular region with a continuously differentiable boundary @ and a unit outward normal vector field . Then, we have (i) Green’s first surface identity for F 2 C.1/ . /, i.e., Z  Z      r G./  r F ./ dS./ C F ./ G./ dS./ 



Z D @

@ F ./ G./ d ./; @

(4.1.19)

4 Scalar Spherical Harmonics in R3

118

(ii) Green’s second surface identity for F 2 C.2/ . /, i.e., Z 

F ./ G./  G./ F ./ dS./ Z D

F ./ @

@ @ G./  G./ F ./ d ./: @ @

(4.1.20)

Proof. This is an immediate consequence of the surface Gauß theorem (Theorem 4.1.2). For details see, e.g., Freeden and Schreiner (2009) and the references therein. t u The aforementioned statements (Lemma 4.1.3 and Theorem 4.1.4) hold as well for the entire sphere S2 instead of a subregion  , thereby observing that the occurring boundary integrals vanish. For functions F 2 C.1/ .S2 / and tangential vector fields .1/ f 2 ctan .S2 /, this implies the following identities: Z S2

f ./ 

r F ./

f ./ 

L F ./

Z

S2

Z S2

Z dS./ D  Z

S2

dS./ D 

r  f ./ dS./ D

S2

Z S2

F ./r  f ./ dS./;

(4.1.21)

F ./L  f ./ dS./;

(4.1.22)

L  f ./ dS./ D 0:

(4.1.23)

Definition 4.1.5. Let  be a point on S2 . A function of the form G W S2 ! R, where  7! G ./ D G.  / with G W Œ1; 1 ! R, is called (-)zonal function on S2 . The application of our spherical differential operators to zonal functions yields, for F 2 C.1/ .Œ1; 1/, that r F .  / D F 0 .  /.  .  //;

(4.1.24)

L F .  / D F 0 .  /. ^ /;

(4.1.25)

while, for F 2 C.2/ .Œ1; 1/,  F .  / D 2.  /F 0 .  / C .1  .  /2 /F 00 .  /:

(4.1.26)

Theorem 4.1.6. Let G 2 L2 .Œ1; 1/. Then, for all  2 S2 , Z

Z S2

G.  / dS./ D 2

1 1

G.t/ dt:

(4.1.27)

4.2 Orthogonal Invariance

119

Proof. First, consider the case  D "3 . Note that p p  D Π1  t 2 cos.'/; 1  t 2 sin.'/; tT

(4.1.28)

with 1  t  1, 0  ' < 2. Therefore, "3   D t and Z

Z S2

G."3  / dS./ D

1 1

Z

2 0

ˇ ˇ ˇ @ @ ˇ ^ ˇˇ d' dt: G.t/ ˇˇ @' @t

(4.1.29)

We compute 2p 3 1  t 2 . sin.'// p @ D4 1  t 2 cos.'/ 5 ; @' 0 and find that

@ @ @'  @t 2 2

2 @ D @t

pt 2 6 p1t t 4 1t 2

cos.'/

3

7 sin.'/ 5 ;

(4.1.30)

1

ˇ ˇ p ˇ ˇ ˇ @ ˇ ˇ @ ˇ D 0, ˇ @' ˇ D 1  t 2 and ˇ @t ˇ D

p1 . 1t 2

Together with the rule

jx ^ yj D jxj jyj  .x  y/ for x; y 2 R this yields that 2

2

3

ˇ ˇ ˇ @ @ ˇ ˇ ˇ ˇ @' ^ @t ˇ D 1:

(4.1.31)

Thus, Z

Z S2

G"3 ./ dS./ D

1

1

Z

2

Z G.t/ d' dt D 2

0

1 1

G.t/ dt:

(4.1.32)

Now, let t 2 SO.3/ D fs 2 R33 W det s D 1g be a rotation with t D "3 , i.e.,  D tT "3 . Then, Z

Z S2

G.  / dS./ D Z

Z S2

D S2

G..tT "3 /  / dS./ D

S2

G."3  .t// dS./

G."3  ˛/ det tT dS.˛/ D 2

where  D tT ˛.

Z

1 1

G.t/ dt;

(4.1.33) t u

4.2 Orthogonal Invariance Systems of equations which maintain their form when the coordinate axes are subjected to an arbitrary rotation are said to be rotationally, or orthogonally, invariant. The orthogonal invariance is, of course, closely related to the group

4 Scalar Spherical Harmonics in R3

120

O.3/ of all orthogonal transformations, i.e., the group of all t 2 R33 such that ttT D tT t D i, i D .ıi;j /i;j D1;2;3 denotes the unit matrix of R33 . The set of all rotations, i.e., SO.3/ D ft 2 O.3/ W det t D 1g is a subgroup called the special orthogonal group. We briefly recapitulate some properties of these groups (see, e.g., M¨uller (1998) and Vilenkin (1968) and many others): 1. Let ;  be members of S2 . Then, there exists an orthogonal transformation t 2 O.3/ with  D t and an orthogonal transformation s 2 SO.3/ with  D s. 2. For every t 2 O.3/, t  t D   ; ;  2 S2 : (4.2.1) 3. Suppose that  2 S2 . The set O .3/ D ft 2 O.3/ W t D g is a subgroup of O.3/. Analogously, the set SO .3/ D ft 2 SO.3/ W t D g is a subgroup of SO.3/. 4. For every t 2 O.3/, we have det t D ˙1. If det t D 1, t is called a rotation, while for det t D 1, t is called a reflection . 5. Let t; t0 2 O.3/ with det t D 1, det t0 D 1. Then, t ^ t D t. ^ /; t0  ^ t0  D t0 . ^ /;

;  2 S2 ; ;  2 S2 :

(4.2.2) (4.2.3)

The following definitions will prove useful for our later considerations. Definition 4.2.1. Let F 2 L2 .S2 /, f 2 l2 .S2 /, and t 2 O.3/. For scalar and vector fields the operator Rt is defined by Rt W L2 .S2 / ! L2 .S2 /; Rt W l2 .S2 / ! l2 .S2 /;

Rt F ./ D F .t/;

(4.2.4)

Rt f ./ D tT f .t/;

(4.2.5)

respectively. Rt F and Rt f are called the t-transformed fields. Remark 4.2.2. Note that l2 .S2 / denotes the Hilbert space of square-integrable vector fields on the unit sphere S2 with the canonical scalar product. The Banach space of continuous vector fields is given by c.S2 / with the canonical norm. For further details on the notation for spherical vector fields, see Sect. 5.1. For examples illustrating how the operators Rt act on functions and vector fields, see Figs. 4.3 and 4.4, respectively. Definition 4.2.3. Let X be a subspace of L2 .S2 / or l2 .S2 /. X is called invariant with respect to orthogonal transformations or, equivalently, orthogonally invariant if, for all F 2 X and for all orthogonal transformations t 2 O.3/, the function Rt F is of class X . An orthogonally invariant space X is called reducible if there exists a proper subspace X 0  X which itself is invariant with respect to orthogonal transformations.

4.2 Orthogonal Invariance

121

Fig. 4.3 The operator Rt acting on a function

T

Fig. 4.4 The definition of the operator Rt for vector fields (note that it is necessary not only to substitute  by t, but also to transform the directions of the vectors)

Note that the expressions invariant with respect to rotations and invariant with respect to reflections are understood in analogy to the aforementioned definition. A linear, orthogonally invariant space which is not reducible is called irreducible. It should be noted that each orthogonally invariant space of dimension 1 is irreducible. Lemma 4.2.4. Let .X; h; i/ be an orthogonally invariant Hilbert subspace of the space L2 .S2 /. Let X1 be an orthogonally invariant subspace of X . Then, the orthogonal complement X1? of X1 is orthogonally invariant as well. Proof. For all F 2 X1 , F ? 2 X1? , and for all orthogonal transformations t 2 O.3/, we have Z hF; Rt F ? i D F ./F ? .t/ dS./ (4.2.6) Z

S2

D S2

F .tT /F ? ./ dS.tT / Z

D j det tT j S2

Z D S2

F .tT /F ? ./ dS./

RtT F ./F ? ./ dS./ D 0;

4 Scalar Spherical Harmonics in R3

122

since RtT F 2 X1 . This implies that Rt F ? 2 X1? and, therefore, X1? is invariant with respect to orthogonal transformations. t u Analogous results can be formulated for Hilbert spaces of square-integrable vector and tensor fields (cf. Freeden and Schreiner 2009). Lemma 4.2.4 shows that each orthogonally invariant Hilbert space can be completely decomposed into invariant parts. In view of the last result, we are particularly interested in irreducible spaces, i.e., spaces that provide us with elements that are invariant with respect to certain orthogonal transformations. The following results (see, e.g., Gervens 1989) help us to analyze the structure of such rotationally invariant functions. Lemma 4.2.5. Let F be a function of class C.S2 / with Rt F ./ D F ./ for all transformations t 2 SO.3/ and all  2 S2 . Then, F D F ."3 / D C D const:

(4.2.7)

Proof. For all  2 S2 , there exists a rotation t 2 SO.3/ with t D "3 . Consequently, for every  2 S2 , we have F ./ D Rt F ./ D F .t/ D F ."3 / D C D const. t u Theorem 4.2.6. Let  2 S2 be fixed. Furthermore, suppose F is of class C.S2 / with Rt F ./ D F ./ for all t 2 SO .3/ and for all  2 S2 . Then, F can be represented in the form F ./ D ˚.  /;  2 S2 ; (4.2.8) ˚ being a function ˚ W Œ1; 1 ! R. Proof. Without loss of generality, let  D "3 (if this were not true, we could use the function G./ D Rt0 F ./, where t0 2 O.3/ with t0 "3 D ). With  D t "3 C p 1  t 2 0 we have, by assumption, that F .t "3 C

p p 1  t 2 0 / D F .t "3 C 1  t 2 00 /;

(4.2.9)

for all points 0 ; 00 of the unit circle. Hence, F depends only on t D   "3 and is, therefore, a function of t alone, as desired. u t Lemma 4.2.7. Let  2 S2 be fixed. Let F 2 C.S2 / with Rt F ./ D .det t/F ./ for all t 2 O .3/ and all  2 S2 . Then, F D 0. Proof. Suppose that  is an element of S2 . There exists a reflection t 2 O .3/ with t D , but then—by assumption—we have F ./ D Rt F ./ D F ./, hence, F ./ D 0. t u Note that in Lemma 4.2.5 and Theorem 4.2.6, the rotations can as well be replaced by reflections, i.e., in the scalar case, we need not distinguish between rotations and reflections. In the vectorial case, however, this is not true anymore. In what follows, f is supposed to be a spherical vector field, i.e., f W S2 ! R3 . Let  be an element of S2 . In every point  ¤ ˙, we are able to introduce the

4.2 Orthogonal Invariance

123

so-called moving frame at the point , namely "1 D ;

(4.2.10)

1 .  .  //; "2 D p 1  .  /2

(4.2.11)

1 "3 D p  ^ ; 1  .  /2

(4.2.12)

such that there exist functions F1 ; F2 ; F3 W S2 ! R with f D F1 "1 C F2 "2 C F3 "3 :

(4.2.13)

For further investigations, the following lemma is helpful. Lemma 4.2.8. Let  2 S2 be fixed. Let the moving frame "i , i D 1; 2; 3, be defined as in (4.2.10)–(4.2.12). Then, for all t 2 O.3/, Rt "i D "i ;

i D 1; 2;

Rt "3 D .det t/"3 :

(4.2.14)

Proof. For t 2 O .3/, Rt "1 D tT "1t D tT t  D  D "1 :

(4.2.15)

For the tangential fields, we show the case i D 2. We have 1 tT .  .t  /t/ Rt "2 D tT "2t D p 1  .t  /2

(4.2.16)

1 D p .tT   .  tT /tT t/ 1  .  tT /2 1 D p .  .  // D "2 : 1  .  /2 The case i D 3 follows similarly.

t u

We now extend our results for rotationally invariant functions to the vector case. Lemma 4.2.9. Let f 2 c.S2 / with Rt f ./ D f ./ (or equivalently, f .t/ D tf ./) for all t 2 SO.3/ and  2 S2 . Then, there exists a constant C 2 R such that (4.2.17) f ./ D C ;  2 S2 :

4 Scalar Spherical Harmonics in R3

124

Proof. Consider the orthogonal matrix 2

1 0 t D 4 0 1 0 0

3 0 05 : 1

(4.2.18)

Then, t"3 D "3 and, by assumption, f ."3 / D tf ."3 /. Hence, in connection with (4.2.18), we have f ."3 / D C "3 , C 2 R. For  2 S2 , there exists a rotation t0 with t0 "3 D . Consequently, we have f ./ D f .t0 "3 / D t0 f ."3 / D C t0 "3 D C . t u Theorem 4.2.10. Let  2 S2 . Let f 2 c.S2 /, i.e., f is a continuous vector field on the unit sphere S2 , with Rt f ./ D f ./ for all t 2 SO .3/. Then, for  ¤ ˙, f has the representation: f ./ D ˚1 .  /"1 C ˚2 .  /"2 C ˚3 .  /"3 ;

(4.2.19)

where ˚i , i D 1; 2; 3, are functions ˚i W Œ1; 1 ! R. Proof. From Lemma 4.2.8, it follows that the functions Fi in (4.2.13) fulfill Rt Fi ./ D Fi ./;

 2 S2 ;

(4.2.20)

provided that t D . Therefore, via Theorem 4.2.6, we know that, for the functions Fi , we have Fi ./ D ˚i .  /. t u Corollary 4.2.11. Suppose that  2 S2 . Let f be of class c.S2 / with Rt f ./ D f ./ for all t 2 O .3/. Then, for  ¤ ˙, f has the representation, f ./ D ˚1 .  /"1 C ˚2 .  /"2 ;

(4.2.21)

˚i , i D 1; 2, being functions ˚i W Œ1; 1 ! R. Proof. Starting from Theorem 4.2.10, we now have to consider reflections, as well. By our assumption and Lemma 4.2.8, we get ˚3 .  / D ˚3 .  / and, therefore, ˚3 .  / D 0. t u Corollary 4.2.12. Suppose that  2 S2 . Let f be of class c.S2 / with Rt f ./ D .det t/ f ./ for all t 2 O .3/. Then, for  ¤ ˙, the field f can be represented as follows: f ./ D ˚3 .  /"3 (4.2.22) with ˚3 being a function ˚3 W Œ1; 1 ! R. Proof. Using the same reasoning as in the proof of Corollary 4.2.11, but now considering the change in sign under reflections, we end up with ˚1 .  / D ˚1 .  /; ˚2 .  / D ˚2 .  /; hence, ˚1 .  / D ˚2 .  / D 0.

(4.2.23) t u

4.3 Homogeneous Polynomials on the Unit Sphere in R3

125

4.3 Homogeneous Polynomials on the Unit Sphere in R3 In this section we consider spherical polynomials by first analyzing homogeneous polynomials on R3 , then requiring them to be harmonic and, finally, restricting these homogeneous harmonic polynomials to the unit sphere. This defines spherical harmonics and shows us how they inherit many of their properties. Such an approach to spherical harmonics is motivated by some hints in Stein and Weiss (1971). In its present form it can be found in Freeden (1979). Remark 4.3.1. Note that we present the theory of spherical harmonics in this chapter in a real framework. It can easily be carried over to the complex-valued case which is chosen in Chap. 6 for the generalized q-dimensional setting. Definition 4.3.2. A polynomial P on Rn , n 2 N, is called homogeneous of degree m 2 N if P .x/ D m P .x/ for all  2 R and all x 2 Rn . The space of all homogeneous polynomials of degree m on Rn is denoted by Homm .Rn / D

8 < :

P W P .x/ D

X Œ˛Dm

9 =

C˛ x ˛ ; x 2 Rn ; ˛ 2 Nn0 : ;

(4.3.1)

Pn Note that for the multi-index it holds that Œ˛ D i D1 ˛i (see the last part of Sect. 6.1 for a general overview on multi-indices). The set of restrictions of homogeneous polynomials to a set D  Rn is defined by Homm .D/ D fP jD W P 2 Homm .Rn /g : (4.3.2) Example 4.3.3. P .x/ D x12 C x22 C x32 2 Hom2 .R3 /, therefore, P jS2 2 Hom2 .S2 / although P jS2  1, i.e., deg.P jS2 / D 0. The index m of Homm .S2 / refers to the degree of the original polynomial on R3 . Theorem 4.3.4. The set of functions fx 7! x ˛ gŒ˛Dn is a basis of Homn .R3 / and M.n/ D dim.Homn .R3 // D .nC1/.nC2/ . 2 Proof. Homogeneous polynomials of degree n in R3 have the form X

P .x/ D

Œ˛Dn

C˛ x ˛ D

X ˛1 C˛2 C˛3 Dn

C˛1 ;˛2 ;˛3 x1˛1 x2˛2 x3˛3 ;

(4.3.3)

where ˛1 ; ˛2 ; ˛3 2 N0 . Let P .x/ D 0. Then, n X ˛1 D0

0 @

n˛ X1 ˛2 D0

1 C˛1 ;˛2 ;n˛1 ˛2 x2˛2 x3n˛1 ˛2 A x1˛1 D 0:

(4.3.4)

For x2 ; x3 2 R fixed, we have a polynomial in R with respect to x1 which is zero for all x1 2 R. Therefore,

4 Scalar Spherical Harmonics in R3

126 n˛ X1 ˛2 D0

C˛1 ;˛2 ;n˛1 ˛2 x2˛2 x3n˛1 ˛2 D 0:

(4.3.5)

Now, keep only x3 fixed and we obtain a polynomial in x2 which is zero for all x2 2 R. Thus, C˛1 ;˛2 ;n˛1 ˛2 x3n˛1 ˛2 D 0

)

C˛1 ;˛2 ;n˛1 ˛2 D 0

(4.3.6)

for all ˛ 2 N30 with Œ˛ D n. Therefore, the set fx 7! x ˛ gŒ˛Dn is a basis of Homn .R3 /. The dimension is equal to the number of monomials x ˛ with Œ˛ D n, i.e., #fx 7! ˛ x gŒ˛Dn . We have n C 1 choices for ˛1 2 f0; : : : ; ng and n C 1  ˛1 choices for ˛2 2 f0; : : : ; n  ˛1 g. In the end, there remains one choice for ˛3 D n  ˛1  ˛2 . This gives us M.n/ D dim.Homn .R3 // D

n X

.n C 1  ˛1 / D

˛1 D0

.n C 1/.n C 2/ : 2

(4.3.7) t u

Thus, we have found the desired basis properties in Homn .R3 /.

Remark 4.3.5. In (4.3.7), we have introduced the notation M.n/ D M.3I n/ D dim.Homn .R3 // in accordance with the more general q-dimensional setting in Chap. 6 which uses M.qI n/ D dim.Homn .Rq //. Formally, we can insert the gradient operator in R3 into a homogeneous polynomial Hn 2 Homn .R3 /, i.e., Hn .r/ D

X

C˛

Œ˛Dn

If Un 2 Homn .R3 / with Un .x/ D Hn .r/Un .x/ D

X X

P

C˛ CQ ˇ

Œ˛Dn ŒˇDn

ŒˇDn



@Œ˛ @x1˛1 @x2˛2 @x3˛3

:

(4.3.8)

CQ ˇ x ˇ , it holds that

@ @x1

˛1

 ˇ x1 1

@ @x2

˛2

 ˇ x2 2

@ @x3

X

C˛ CQ ˛

Œ˛Dn

ˇ

x3 3 : (4.3.9)

Now, all summands are zero except the ones with ˛ D ˇ such that Hn .r/Un .x/ D

˛3

 3  Y X @ ˛i ˛i xi D C˛ CQ ˛ ˛1 Š ˛2 Š ˛3 Š 2 R: „ ƒ‚ … @x i i D1 Œ˛Dn

˛Š

(4.3.10)

Lemma 4.3.6. The mapping h;  iHomn .R3 / W Homn .R3 /  Homn .R3 / ! R

(4.3.11)

4.3 Homogeneous Polynomials on the Unit Sphere in R3

127

given by hHn ; Un iHomn .R3 / D Hn .rx /Un .x/

(4.3.12)

3

defines an inner product on Homn .R /. The set of monomials 

 1 ˛ ˇˇ 3 x 7! p x ˇ˛ 2 N0 ; Œ˛ D n (4.3.13) ˛Š  is an orthonormal system in the Hilbert space Homn .R3 /; h;  iHomn .R3 / . Proof. If Hn ; Un 2 Homn .R3 /, then hHn ; Hn iHomn .R3 / D Hn .rx /Hn .x/ D

X

C˛2 ˛Š  0

(4.3.14)

Œ˛Dn

and the scalar product of a homogeneous P polynomial Hn of degree n with itself is hHn ; Hn iHomn .R3 / D 0 if and only if Œ˛Dn C˛2 ˛Š D 0. However, this is equivalent to C˛2 D 0 for all multi-indices ˛ 2 N30 with Œ˛ D n, i.e., with Hn  0. For the symmetry we find that X

hHn ; Un iHomn .R3 / D

C˛ CQ ˛ ˛Š D hUn; Hn iHomn .R3 / :

(4.3.15)

Œ˛Dn

Let also Vn 2 Homn .R3 / and ; 2 R. Then hHn ; Un C Vn iHomn .R3 / D

X

C˛ .CQ ˛ C D˛ /˛Š

Œ˛Dn

D hHn ; Un iHomn .R3 / C hHn ; Vn iHomn .R3 / :

(4.3.16)

For the monomials X 1 1 0 M˛.n/.x/ D p x ˛ D p ı˛;˛0 x ˛ ; ˛Š ˛Š Œ˛ 0 Dn

(4.3.17)

we obtain hM˛.n/; M˛.n/iHomn .R3 / D

2 X  1 p ı˛;˛0 ˛ 0 Š D 1; ˛Š Œ˛ 0 Dn

(4.3.18)

and for ˛ ¤ ˛ 0 ; .n/

hM˛.n/; M˛Q iHomn .R3 / D

X

1 1 p ı˛;˛0 p ı˛0 ;˛Q ˛ 0 Š D 0: ˛Š ˛Š Q Œ˛ 0 Dn

(4.3.19)

4 Scalar Spherical Harmonics in R3

128

Since the monomials form a basis of the finite dimensional space Homn .R3 / by Theorem 4.3.4 we also have completeness. t u By the fundamental theorem of Fourier analysis an orthonormal expansion of the homogeneous polynomial Hn 2 Homn .R3 / is given by X

Hn .x/ D

hHn ; M˛.n/ iHomn .R3 / M˛.n/.x/ D

Œ˛Dn

D Hn .ry /

X

1 1 Hn .ry / p y ˛ p x ˛ ˛Š ˛Š Œ˛Dn

1 X nŠ ˛ ˛ y x : nŠ ˛Š

(4.3.20)

Œ˛Dn

Thus, we have Hn .x/ D Hn .ry /

 D E 1 .x  y/n D Hn ; .x /n : Homn .R3 / nŠ nŠ

1

(4.3.21)

Definition 4.3.7. Let V ¤ ; and X be a Hilbert space on the set V . A kernel K.; / W V  V ! R

(4.3.22)

is called reproducing kernel for X if (i) K.x; / 2 X for all x 2 V , (ii) hF; K.x; /iX D F .x/ for all x 2 V and all F 2 X . If such a kernel exists in X , then X is called a reproducing kernel Hilbert space. It should be noted that a reproducing kernel is always unique (see, e.g., Aronszajn 1950; Davis 1963). Theorem 4.3.8. The mapping KHomn .R3 / W R3  R3 ! R; 1 .x  y/n nŠ  is the reproducing kernel in Homn .R3 /; h;  iHomn .R3 / . .x; y/ 7! KHomn .R3 / .x; y/ D

(4.3.23)

Proof. The second property of Definition 4.3.7 is shown in (4.3.21). The first property holds since .x /n is a homogeneous polynomial of degree n for any fixed x 2 R3 . Thus, KHomn .R3 / .x; / 2 Homn .R3 / for any fixed x 2 R3 . t u Theorem 4.3.9 (Addition Theorem for Homn .R3 /). Let fHn;j gj D1;:::;M.n/ , where M.n/ D 12 .nC1/.nC2/, be an orthonormal system in Homn .R3 /. For all x; y 2 R3 , it holds that X

M.n/ j D1

Hn;j .x/Hn;j .y/ D

1 .x  y/n D KHomn .R3 / .x; y/: nŠ

(4.3.24)

4.3 Homogeneous Polynomials on the Unit Sphere in R3

129

Proof. Since the reproducing kernel in a Hilbert space is unique, we have to show PM.n/ that j D1 Hn;j .x/Hn;j .y/ is a reproducing kernel in Homn .R3 /. Clearly, X

M.n/

X

M.n/

Hn;j .x/Hn;j 2 Homn .R3 / and

j D1

Hn;j Hn;j .y/ 2 Homn .R3 /

(4.3.25)

j D1

for any fixed x; y 2 R3 . Let Hn 2 Homn .R3 /, then X

M.n/

Hn .x/ D

hHn ; Hn;j iHomn .R3 / Hn;j .x/

(4.3.26)

j D1

and we obtain M.n/ D E X Hn ; Hn;j .x/Hn;j j D1

X

M.n/

Homn .R3 /

D

Hn;j .x/hHn ; Hn;j iHomn .R3 / D Hn .x/:

j D1

(4.3.27) PM.n/ Therefore, the kernel defined by j D1 Hn;j .x/Hn;j .y/ is the reproducing kernel in the Hilbert space Homn .R3 / with the inner product h;  iHomn .R3 / . t u

Interpolation Problem Given measurements .xj ; bj / 2 R3  R, j D 1; : : : ; M.n/, determine F 2 Homn .R3 / with the property F .xj / D bj ;

j D 1; : : : ; M.n/ D 12 .n C 1/.n C 2/:

(4.3.28)

When does a solution exist? If a solution exists, how does it look like? Definition 4.3.10. A system of M.n/ points fx1 ; : : : ; xM.n/ g  R3 is called a fundamental system relative to Homn .R3 / if the matrix A with the entries Aj;k D Hn;j .xk /, j; k D 1; : : : ; M.n/, is regular, where fHn;j g is an orthonormal system in Homn .R3 /. Lemma 4.3.11. For each n 2 N0 , there exists a fundamental system relative to the space Homn .R3 /. Proof. Let n 2 N0 be fixed. We construct the points fx1 ; : : : ; xM.n/ g inductively. Clearly there exists a point x1 2 R3 with Hn;1 .x1 / ¤ 0:

(4.3.29)

4 Scalar Spherical Harmonics in R3

130

Now, assume there exist fx1 ; : : : ; xm g, m < M.n/, such that

 Hn;j .xk / j;kD1;:::;m

(4.3.30)

is regular. Furthermore, assume that for every y 2 R3 n fx1 ; : : : ; xm g the matrix 2

3 Hn;1 .x1 / : : : : : : Hn;1 .xm / Hn;1 .y/ 6 7 :: :: :: 6 7 : : : 6 7 6 7 :: :: :: 4 5 : : : Hn;mC1 .x1 / : : : : : : Hn;mC1 .xm / Hn;mC1 .y/

(4.3.31)

is singular. This means that there exists a row i 2 f1; : : : ; m C 1g such that this row can be represented as a linear combination of the other rows

 Hn;j .x1 /; : : : ; Hn;j .xm /; Hn;j .y/ ; j 2 f1; : : : ; m C 1g n fi g: (4.3.32) In particular, Hn;i .y/ can be represented by a linear combination of the functions fHn;j .y/gj 2f1;:::;mC1gnfi g for all y 2 R3 . But this is a contradiction to the linear independence of fHn;1 ; : : : ; Hn;mC1 g. This inductive construction can be continued until we reach m D M.n/ D 12 .n C 1/.n C 2/. t u If the points of the interpolation problem are a fundamental system relative to Homn .R3 /, the problem can be solved uniquely. Theorem 4.3.12. Let fHn;j gj D1;:::;M.n/ be an orthonormal system in Homn .R3 / and fxk gkD1;:::;M.n/ be a fundamental system relative to Homn .R3 /. Then, each homogeneous polynomial F 2 Homn .R3 / is uniquely representable in the form (x 2 R3 ) X

M.n/

F .x/ D

X

M.n/

aj KHomn .R3 / .xj ; x/ D

j D1

j D1

aj

.xj  x/n : nŠ

(4.3.33)

Proof. It is clear that F can be written as X

M.n/

F D

bk Hn;k ;

(4.3.34)

kD1

where bk D hF; Hn;k iHomn .R3 / , k D 1; : : : ; M.n/. Let faj gj D1;:::;M.n/ be the solution of the system of linear equations X

M.n/ j D1

aj Hn;k .xj / D bk ;

k D 1; : : : ; M.n/:

(4.3.35)

4.3 Homogeneous Polynomials on the Unit Sphere in R3

131

The system is uniquely solvable since the points xk form a fundamental system (thus, the matrix is regular). Then, XX

M.n/ M.n/

F D

X

M.n/

aj Hn;k .xj /Hn;k D

kD1 j D1

aj KHomn .R3 / .xj ; /:

(4.3.36)

j D1

t u Note that the system can be very ill-conditioned.

Harmonic Polynomials Definition 4.3.13. Let D  R3 be open and connected. F 2 C.2/ .D/ is called harmonic if x F .x/ D 0 for all x 2 D. The set of all harmonic functions in C.2/ .D/ is denoted by Harm.D/. Definition 4.3.14. The set of all homogeneous harmonic polynomials on R3 with degree m 2 N0 is denoted by ˚ Harmm .R3 / D P 2 Homm .R3 / W P D 0 :

(4.3.37)

Moreover, we define for m 2 N0 ; Harm0;:::;m .R3 / D

m M

Harmi .R3 /;

(4.3.38)

Harm0;:::;i .R3 /:

(4.3.39)

i D0

Harm0;:::;1 .R3 / D

1 [ i D0

Let H 2 Homn .R3 /. There is a representation H.x/ D

n X

j

Anj .x1 ; x2 /x3 ;

(4.3.40)

j D0

where Anj 2 Homnj .R2 /. Now, let H 2 Harmn .R3 /: 0 D x H.x/ D

 n  2 n X X @ @2 j @2 j A C .x ; x /x C x Anj .x1 ; x2 /: nj 1 2 3 2 2 @x1 @x2 @x32 3 j D0 j D0 (4.3.41)

4 Scalar Spherical Harmonics in R3

132

Note that  2  @ @2 C 2 Ar .x1 ; x2 / D x;2 Ar .x1 ; x2 / D 0 @x12 @x2

for r 2 f0; 1g:

(4.3.42)

Therefore,  n2  2 n X X @ @2 j j 2 0D C 2 Anj .x1 ; x2 /x3 C j.j  1/x3 Anj .x1 ; x2 / 2 @x @x 1 2 j D0 j D2 D

n2 X j D0

j x;2 Anj .x1 ; x2 /x3

n2 X j C .j C 2/.j C 1/x3 Anj 2 .x1 ; x2 /: (4.3.43) j D0

Theorem 4.3.15. Let n 2 N0 . Let An and An1 be homogeneous polynomials of degree n and n  1 in R2 . For j D 0; : : : ; n  2, we define recursively Anj 2 .x1 ; x2 / D 

1 x;2 Anj .x1 ; x2 /: .j C 1/.j C 2/

(4.3.44)

Then, H W R3 ! R given by H.x1 ; x2 ; x3 / D

n X j D0

j

Anj .x1 ; x2 /x3

(4.3.45)

is in Harmn .R3 /. Moreover, dim.Harmn .R3 // D 2n C 1:

(4.3.46)

Proof. We already know that H 2 Harmn .R3 / from our considerations before. The degrees of freedom for H occur in the choices of An and An1 . Since Aj is of class Homj .R2 /, we know it can be represented by Aj .x/ D

X

C˛ x ˛ ;

(4.3.47)

Œ˛Dj

where ˛ 2 N20 . Thus, dim.Homj .R2 // D j C 1. This yields dim.Harmn .R3 // D dim.Homn .R2 // C dim.Homn1 .R2 //

(4.3.48)

D .n C 1/ C ..n  1/ C 1// D 2n C 1; which shows the remaining part of Theorem 4.3.15.

t u

Theorem 4.3.16. For n  2, the space Homn .R3 / can be decomposed into the orthogonal and direct sum

4.3 Homogeneous Polynomials on the Unit Sphere in R3 3 Homn .R3 / D Harmn .R3 / ˚ Harm? n .R /

133

(4.3.49)

with 3 2 3 Harm? (4.3.50) n .R / D j  j Homn2 .R / ˚ D Ln .x/ D jxj2 Hn2 .x/ W Hn2 2 Homn2 .R3 / :

Proof. (4.3.49) is just the standard decomposition of functional analysis. For the proof of (4.3.50), let n  2, Hn2 2 Homn2 .R3 / and Kn 2 Harmn .R3 / be arbitrary. Then, hj  j2 Hn2 ; Kn iHomn .R3 / D hHn2 j  j2 ; Kn iHomn .R3 / D Hn2 .rx /x Kn .x/ D 0: (4.3.51) 3 Therefore, j  j2 Hn2 2 Harm? n .R /. Because of Theorem 4.3.4 and (4.3.46) in Theorem 4.3.15 we easily calculate that 3 3 3 dim.Harm? n .R // D dim.Homn .R //  dim.Harmn .R //

D dim.Homn2 .R3 //; which gives us the desired equality of the spaces.

(4.3.52) t u

Corollary 4.3.17. Applying Theorem 4.3.16 iteratively, we obtain the decomposition bn=2c M j  j2i Harmn2i .R3 /; (4.3.53) Homn .R3 / D i D0

where

b n2 c

is the largest integer that is less than or equal to n2 .

Remark 4.3.18. Let fHn;j gj Dn;:::;n be an orthonormal system in Harmn .R3 /. This system can be completed to an orthonormal system in Homn .R3 /, i.e., fHn;j gj Dn;:::;n [ fUn;j gj D1;:::;dn2 ;

(4.3.54)

3 where fUn;j gj D1;:::;dn2 , dn2 D 12 n.n1/, is an orthonormal system in Harm? n .R /. Note that in view of the typical indexing of spherical harmonics we also use the indices n; : : : ; n for the 2n C 1 orthonormal functions of Harmn .R3 /.

Lemma 4.3.19. For Hn 2 Homn .R3 /, we have ProjHarmn .R3 / Hn .x/ D

 bn=2c X sD0

.1/s

 nŠ.2n  2s/Š jxj2s s Hn .x/: .2n/Š.n  s/ŠsŠ

(4.3.55)

4 Scalar Spherical Harmonics in R3

134

Proof. First, we prove Hobson’s formula by induction (cf. Hobson 1955) 

 bn=2c  X nŠ.2n  2s/Š jxj2s s xin .1/s .2n/Š.n  s/ŠsŠ sD0 (4.3.56) for x 2 R3 n f0g, i 2 f1; 2; 3g, n 2 N0 . To prove this, we need to remember that @ @xi

n

1 .2n/Š 1 D .1/n n jxj 2 nŠ jxj2nC1

( s xin

D

nŠ x n2s .n2s/Š i

;

n  2s;

0

;

else;

(4.3.57)

and @  2s2n1 n2s jxj D.1/.2n  2s C 1/jxj2s2.nC1/1 xinC12s xi @xi

(4.3.58)

C .n  2s/jxj2s2n1 xin2s1 : Everything else consists only of lengthy, but uneventful transformations to get (4.3.56) in the end. Because of the invariance of the operator  with respect to orthogonal transformations we find from (4.3.56) that 1 .2n/Š 1 .y  rx / D .1/n n jxj 2 nŠ jxj2nC1 n

 bn=2c X

 nŠ.2n  2s/Š 2s s jxj  .y  x/n .1/ .2n/Š.n  s/ŠsŠ sD0 (4.3.59) s

for all y 2 R3 , x 2 R3 n f0g, n 2 N0 . In more detail, we choose t 2 O.3/ such that t"i D y and set z D tx. This yields @ D "i  rx D "i  tT rz D y  rz ; @xi

(4.3.60)

z D x ; p jzj D x T tT tx D jxj; "i  x D ."i /T tT tx D y  z: Together this gives us (4.3.59). We know from Theorem 4.3.12 that each Hn 2 Homn .R3 / may be represented by M.n/ X Hn .x/ D aj .xj  x/n ; x 2 R3 ; (4.3.61) j D1

where the aj 2 R, j D 1; : : : ; M.n/, are suitable coefficients and the points xj 2 R3 , j D 1; : : : ; M.n/, form a fundamental system relative to Homn .R3 /. Therefore, we use (4.3.59) with xj as y:

4.3 Homogeneous Polynomials on the Unit Sphere in R3

Hn .rx /

135

M.n/ X 1 1 D aj .xj  rx /n jxj jxj j D1

X

M.n/

D

aj .1/n

j D1

D .1/n

(4.3.62)

.2n/Š 1 2n nŠ jxj2nC1

.2n/Š 1 2n nŠ jxj2nC1

 bn=2c X

 bn=2c X

.1/s

sD0

.1/s

sD0

 nŠ.2n  2s/Š jxj2s s .xj  x/n .2n/Š.n  s/ŠsŠ

 nŠ.2n  2s/Š jxj2s s Hn .x/: .2n/Š.n  s/ŠsŠ

For x ¤ 0, we have by Theorem 4.3.16 that Hn D Kn C j  j2 Hn2 , where the first function Kn 2 Harmn .R3 / and the second function Hn2 2 Homn2 .R3 /. This provides us with 1 .2n/Š 1 1 1 D Kn .rx / C Hn2 .rx /x D .1/n Kn .x/; jxj jxj jxj nŠ2n jxj2nC1 (4.3.63) 1 since x jxj D 0, x Kn .x/ D 0 and only the summand for s D 0 remains in (4.3.62). Finally, we obtain Hn .rx /

nŠ2n 1 jxj2nC1 Hn .rx / (4.3.64) .2n/Š jxj 0 1 bn=2c X nŠ.2n  2s/Š D@ jxj2s s A Hn .x/: .1/s .2n/Š.n  s/ŠsŠ sD0

ProjHarmn .R3 / Hn .x/ D Kn .x/ D .1/n

t u

This concludes the proof of Lemma 4.3.19. Remark 4.3.20. Additionally, we find in the course of the previous proof that

ProjHarm? (4.3.65) 3 Hn .x/ D Hn .x/  Kn .x/ n .R / 1 0 bn=2c X nŠ.2n  2s/Š jxj2s s A Hn .x/: .1/s1 D@ .2n/Š.n  s/ŠsŠ sD1 Note that the sum here starts at s D 1. Theorem 4.3.21 (Addition Theorem in Harmn .R3 /). Let fHn;j gj Dn;:::;n be an orthonormal system in Harmn .R3 / with respect to h ;  iHomn .R3 / . For every x; y 2 R3 with x D jxj, y D jyj (;  2 S2 ), we obtain n X j Dn

Hn;j .x/Hn;j .y/ D

2n nŠ n n jxj jyj Pn .  /; .2n/Š

where Pn denotes the Legendre polynomial of degree n.

(4.3.66)

4 Scalar Spherical Harmonics in R3

136

Proof. Let fKn;j gj D1;:::;M.n/ be the orthonormal system in Homn .R3 / which completes the basis from fHn;j g. Due to the uniqueness of the reproducing kernel in Homn .R3 / we know that X

M.n/

Kn;j .x/Kn;j .y/ D

j D1

.x  y/n nŠ

(4.3.67)

for x; y 2 R3 , where M.n/ D 12 .nC1/.nC2/. We project both sides to Harmn .R3 /: ProjHarmn .R3 /

 M.n/ X

 Kn;j .x/Kn;j .y/ D

j D1

n X

Hn;j .x/Hn;j .y/

(4.3.68)

j Dn

and using Lemma 4.3.19 together with x .x  y/n D n.n  1/jyj2 .x  y/n2 :  ProjHarmn .R3 /

.x  y/n nŠ

D

 D

bn=2c 1 X .1/s .2n  2s/Š.nŠ/2 jxj2s jyj2s .x  y/n2s nŠ sD0 .n  2s/Š.n  s/ŠsŠ.2n/Š

bn=2c 2n nŠ n n X .1/s .2n  2s/Š jxj jyj .  /n2s n .n  2s/Š.n  s/ŠsŠ .2n/Š 2 sD0 „ ƒ‚ …

(4.3.69)

DPn ./

due to the explicit representation of the Legendre polynomials. For details see Corollary 3.4.4 with  D 12 and (by use of Lemma 2.3.3) 2N m

 .N  m C 12 /  . 12 /

D

 .2N  2m/ 2N 2m .2N  2m/Š : D N 2N 2m1  .N  m/ 2 2 .N  m/Š

(4.3.70)

This yields the addition theorem for homogeneous harmonic polynomials in R3 . u t Theorem 4.3.22. For Hm 2 Harmm .R3 / and Kn 2 Harmn .R3 /, we have hHm jS2 ; Kn jS2 iL2 .S2 / D where n D

ın;m hHm ; Kn iHomn .R3 / ; n

(4.3.71)

.2nC1/Š 42n nŠ .

Proof. We only present the ideas of the proof. By the fundamental theorem of potential theory (Theorem 6.2.11, see also, e.g., Freeden 1979 and the references therein) it holds, for all x 2 R3 with jxj < 1, that 1 Kn .x/ D 4

Z  S2

@ 1 1 @ Kn .y/  Kn .y/ jx  yj @ @y jx  yj

 dS.y/;

(4.3.72)

4.3 Homogeneous Polynomials on the Unit Sphere in R3

137

where  is the outward pointing normal to the surface S2 . Therefore, 1 Hm .rx /Kn .x/ D 4

Z 

@ 1 Kn .y/ jx  yj @  @ 1 dS.y/: (4.3.73) Kn .y/ Hm .rx / @y jx  yj S2

Hm .rx /

Now, we have Hm .rx /

.2m/Š Hm .x  y/ 1 .2m/Š Hm .y  x/ D .1/m m D m : 2mC1 jx  yj 2 mŠ jx  yj 2 mŠ jx  yj2mC1

(4.3.74)

At x D 0 we have ( Hm .rx /Kn .x/jxD0 D

0

for m ¤ n;

hHm ; Kn iHomn .R3 /

for n D m:

(4.3.75)

Therefore, 1 4

Z  S2

D

(

Hm .y/ @ @ Hm .y/ Kn .y/  Kn .y/ 2mC1 jyj @ @ jyj2mC1

0

for m ¤ n;

2m mŠ hHm ; Kn iHomn .R3 / .2m/Š

for n D m:

 dS.y/

(4.3.76)

Next, we see that on the sphere S2 with y D r, jj D 1: @ @ n @ Kn .y/ D   ry Kn .y/ D Kn .r/ D r Kn ./ D nr n1 Kn ./ (4.3.77) @ @r @r which gives us ˇ ˇ @ Kn .y/ˇˇ D nKn ./ @ y2S2

ˇ ˇ @ Hm .y/ˇˇ D mHm ./: @ y2S2

(4.3.78)

1 1 @ Hm .y/ D mHm .y/ 2mC1 C Hm .y/.2m  1/ 2mC2 ; 2mC1 @ jyj jyj jyj

(4.3.79)

and

Moreover,

and, therefore, ˇ @ Hm .y/ ˇˇ D mHm ./  .2m C 1/Hm ./ D .m C 1/Hm ./: @ jyj2mC1 ˇy2S2

(4.3.80)

4 Scalar Spherical Harmonics in R3

138

If we put these results together, this yields 1 4

Z 

 Hm .y/ @ @ Hm .y/ K dS.y/ .y/  K .y/ n n jyj2mC1 @ @ jyj2mC1 S2 Z 1 D .nHm ./Kn ./ C .m C 1/Hm ./Kn .// dS./ 4 S2 Z nCmC1 D Hm ./Kn ./ dS./: (4.3.81) 4 S2

Altogether, we find the desired result.

t u

We reformulate the addition theorem (Theorem 4.3.21). Corollary 4.3.23 (Addition Theorem in Harmn .R3 /). Let fHn;j gj Dn;:::;n be an orthonormal system in Harmn .R3 / with respect to h ;  iHomn .R3 / , i.e., the system p f n Hn;j gj Dn;:::;n is orthonormal in Harmn .R3 / with respect to h ;  iL2 .S2 / . For every x; y 2 R3 with x D jxj, y D jyj (;  2 S2 ), we find that n X 2n C 1 n n p p jxj jyj Pn .  /; n Hn;j .x/ n Hn;j .y/ D 4 j Dn

where n D of degree n.

.2nC1/Š 42n nŠ

(4.3.82)

as in Theorem 4.3.22 and Pn denotes the Legendre polynomial

Harmonic Polynomials on the Sphere In analogy to (4.3.37)–(4.3.39) we introduce harmonic polynomials on subsets of R3 , in particular the sphere S2 . Definition 4.3.24. For m 2 N0 and a set D  R3 , ˚ Harmm .D/ D P jD W P 2 Harmm .R3 / ; ˚ Harm0;:::;m .D/ D P jD W P 2 Harm0;:::;m .R3 / ; ˚ Harm0;:::;1 .D/ D P jD W P 2 Harm0;:::;1 .R3 / :

(4.3.83) (4.3.84) (4.3.85)

The elements of the space Harmn .S2 /, n 2 N0 , are called (scalar) spherical harmonics of degree n. From the comparison of h ;  iL2 .S2 / and h ;  iHomn .R3 / (see Theorem 4.3.22) it immediately follows that, for Yn 2 Harmn .S2 / and Ym 2 Harmm .S2 /,

4.3 Homogeneous Polynomials on the Unit Sphere in R3

139

Z hYn ; Ym iL2 .S2 / D

S2

Yn ./Ym ./ dS./ D 0

(4.3.86)

if n ¤ m. Theorem 4.3.25. If n 2 N0 , then dim.Harmn .R3 // D dim.Harmn .S2 // D 2n C 1:

(4.3.87)

Proof. It is clear that dim.Harmn .R3 //  dim.Harmn .S2 //:

(4.3.88)

Assume that m D dim.Harmn .S2 // < dim.Harmn .R3 // D 2n C 1. Let fHn;j g  Harmn .R3 / be a linearly independent system in Harmn .R3 / and let Yj D Hn;j jS2 , i.e., Hn;j .x/ D r n Yj ./. Suppose that F 2 C.S2 / with n X

F ./ D

aj Yj ./ D 0:

(4.3.89)

j Dn

Consider the following boundary value problem: find U 2 C.2/ .B31 / \ C.B31 / such that U D 0 in B31 ; U jS2 D F: (4.3.90) Note that B31 D fx 2 R3 W jxj < 1g denotes the open unit ball in R3 around the origin and its boundary @B31 is the unit sphere S2 . This boundary value problem is uniquely solvable by n X U.x/ D aj r n Yj ./; (4.3.91) j Dn

where x D r and we use the same coefficients aj as in (4.3.89), i.e., U.x/ D 0 for x 2 S2 . Thus, n X aj r n Yj ./ D 0 (4.3.92) j Dn

for all r < 1, or

n X

aj Hn;j .x/ D 0

(4.3.93)

j Dn

for all x 2 B31 . The left-hand side is a polynomial, i.e., n X

aj Hn;j .x/ D 0

for all x 2 R3 :

(4.3.94)

j Dn

Since the polynomials Hn;j are linearly independent, aj D 0 for all j .

t u

4 Scalar Spherical Harmonics in R3

140

Lemma 4.3.26. Any spherical harmonic Yn 2 Harmn .S2 /, n 2 N0 , is an infinitely often differentiable eigenfunction of the Beltrami operator  corresponding to the eigenvalue n.n C 1/, i.e.,  Yn ./ C n.n C 1/Yn./ D 0 for all  2 S2 :

(4.3.95)

The sequence fn.n C 1/gn2N0 is called the eigenspectrum of the Beltrami operator. Proof. Yn 2 Harmn .S2 / is the restriction of a polynomial Hn 2 Harmn .R3 / to the x sphere, i.e., x Hn .x/ D 0 for all x D r 2 R3 (r D jxj,  D jxj 2 S2 ) and  2  @ 1  2 @ C Hn .r/ D 0: C  (4.3.96) @r 2 r @r r2  Due to the homogeneity of Hn we know that Hn .x/ D r n Yn ./ with r > 0. Thus, 2 1 0 D x Hn .x/ D n.n  1/r n2 Yn ./ C nr n1 Yn ./ C 2 r n  Yn ./ r r   (4.3.97) D r n2 n2  n C 2n C  Yn ./: Since r > 0, it follows that  Yn ./ C n.n C 1/Yn ./ D 0.

t u

Definition 4.3.27. We denote a complete orthonormal system in the Hilbert space .Harmn .S2 /; h; iL2.S2 / / by fYn;j gj Dn;:::;n , i.e., (i) hYn;j ; Yn;k iL2 .S2 / D ıj;k for all j; k 2 fn; : : : ; ng. (ii) If hF; Yn;j iL2 .S2 / D 0 for all j 2 fn; : : : ; ng with F 2 Harmn .S2 /, then F D 0. We call Yn;j a spherical harmonic of degree n and order j . Remark 4.3.28. The family fYn;j gn2N0 ;j Dn;:::;n forms an L2 .S2 /-orthonormal system in Harm0;:::;1 .S2 /. Theorem 4.3.29 (Addition Theorem of Spherical Harmonics). If fYn;j gj Dn;:::;n is an L2 .S2 /-orthonormal system in Harmn .S2 /, n 2 N0 , then n X

Yn;j ./Yn;j ./ D

j Dn

2n C 1 Pn .  / 4

(4.3.98)

for all .; / 2 S2  S2 , where Pn is the Legendre polynomial of degree n. Moreover, n X  2 2n C 1 Yn;j ./ D 4 j Dn



Yn;j 2 D sup jYn;j ./j  C.S / 2S2

r

2n C 1 4

for all  2 S2 ;

for all j D n; : : : ; n:

(4.3.99)

(4.3.100)

4.3 Homogeneous Polynomials on the Unit Sphere in R3

141

Proof. This result follows immediately from the addition theorem in Harmn .R3 /, see Theorem 4.3.21, respectively Corollary 4.3.23 in combination with Theorem 4.3.22. t u Remark 4.3.30. Using the addition theorem (Theorem 4.3.29), it is easy to show that the mapping .; / 7! KHarmn .S2 / .; / D

2n C 1 Pn .  / 4

(4.3.101)

is the reproducing kernel in Harmn .S2 /, i.e., 2n C 1 4

Z S2

Yn ./Pn .  / dS./ D Yn ./:

(4.3.102)

Lemma 4.3.31. For every Yn 2 Harmn .S2 /, we have r sup jYn ./j 

2S2

2n C 1 kYn kL2 .S2 / : 4

(4.3.103)

Proof. Due to Fourier theory it is clear that by Parseval’s identity kYn k2L2 .S2 /

n X

D

jhYn ; Yn;j iL2 .S2 / j2

(4.3.104)

j Dn

and we have the representation n X

Yn D

hYn; Yn;j iL2 .S2 / Yn;j

(4.3.105)

j Dn

on the sphere S2 . Therefore, by the Cauchy–Schwarz inequality we obtain jYn ./j2 

n X

hYn ; Yn;j i2L2 .S2 /

j Dn

n X  j Dn

Yn;j ./

2

D

2n C 1 kYn k2L2 .S2 / : (4.3.106) 4

Taking the square root provides us with the desired result.

t u

While we have presented the theory of spherical harmonics in a real framework, we now introduce the complex-valued spherical harmonics which are well-known and widely used in physics. In Example 4.3.33 we offer a system of real-valued spherical harmonics of equal importance (in particular in geosciences). Example 4.3.32 (Complex-Valued Spherical Harmonics). Let n 2 N0 , j 2 Z with n  j  n. The function

4 Scalar Spherical Harmonics in R3

142 C Yn;j W S2 ! C

(4.3.107) s

C  7! Yn;j ./ D .1/j

2n C 1 .n  j /Š Pn;j .cos.#//eij' 4 .n C j /Š

is called the (complex) spherical harmonic of degree n and order j , where #; ' are the spherical coordinates of  given by (4.1.2). Pn;j W Œ1; 1 ! R;

j

t 7! Pn;j .t/ D .1  t 2 / 2

dj Pn .t/ dt j

(4.3.108)

is the associated Legendre function of degree n and order j , 0  j  n. For negative orders, we have the symmetry relation Pn;j .t/ D .1/j

.n  j /Š Pn;j .t/: .n C j /Š

(4.3.109)

These spherical harmonics are orthonormal with respect to the canonical scalar product of the space L2C .S2 / of complex-valued square-integrable functions on the unit sphere S2 . Their addition theorem reads as follows: n X j Dn

C C Yn;j ./Yn;j ./ D

2n C 1 Pn .  /: 4

(4.3.110)

For further details on this representation of spherical harmonics the reader is referred to, e.g., Edmonds (1964), Jones (1985), Var˘salovi˘c et al. (1988) and Zare (1988). Example 4.3.33 (Real Fully Normalized Spherical Harmonics). Let n 2 N0 , j 2 Z with n  j  n. The function R W S2 ! R; Yn;j

s  7!

R Yn;j ./

D

8p ˆ ˆ < 2 cos.j'/

2n C 1 .n  jj j/Š Pn;jj j .cos.#// 1 ˆ 4 .n C jj j/Š ˆ :p2 sin.j'/

(4.3.111) ; j 0

is called the (real) fully normalized spherical harmonic of degree n and order j , where #; ' are the spherical coordinates of  given by (4.1.2) and Pn;j , n 2 N0 , 0  j  n, are defined by (4.3.108). R The Yn;j are orthonormal with respect to the scalar product of L2R .S2 / (the space of real-valued, square-integrable functions on the unit sphere S2 ) and the addition R theorem holds as written in Theorem 4.3.29. The real spherical harmonics Yn;j C (cf. Fig. 4.5–4.7) and the complex spherical harmonics Yn;j (of Example 4.3.32) are related via

4.3 Homogeneous Polynomials on the Unit Sphere in R3

143

Fig. 4.5 All spherical harmonics Y3;j of degree 3 (upper row: Y3;3 to Y3;0 , lower row: Y3;1 to Y3;3 )

Fig. 4.6 Spherical harmonics Y7;0 (left), Y7;5 (middle), and Y7;7 (right). Note that if the order is j D ˙n, Yn;˙n are called sectorial spherical harmonics. The spherical harmonics Yn;0 of order 0 are called zonal spherical harmonics. All other basis functions Yn;j , i.e., j ¤ 0 and j ¤ ˙n, are called tesseral spherical harmonics

Fig. 4.7 Spherical harmonics Y10;2 (left), Y10;6 (middle), and Y10;9 (right)

8p   < 2ı0;j Y C ./ C Y C ./ ; n;j n;j 2 R  p  Yn;j ./ D j : .1/ 2 Y C ./  Y C ./ ; n;j n;j 2i for all  2 S2 , n 2 N0 , and j 2 Z with n  j  n.

j  0; j > 0;

(4.3.112)

4 Scalar Spherical Harmonics in R3

144

Note that in the case of real-valued spherical harmonics, indexing with negative integers is just one way to distinguish the two types with sine and cosine. In the literature, in particular in geodetic literature, often two different symbols are used to achieve this (see, e.g., Heiskanen and Moritz 1967; Hofmann-Wellenhof and Moritz 2005; Lense 1954; Morse and Feshbach 1953).

4.4 Closure and Completeness of Spherical Harmonics We consider Fourier expansions in terms of a set of L2 .S2 /-orthonormal spherical harmonics and use two summability methods: Abel-Poisson summability and Bernstein summability. The latter turns out to be also very helpful for vectorial and tensorial spherical harmonics (see Chap. 5 for the vectorial case and Freeden and Schreiner 2009 for the tensorial case). Moreover, it can easily be generalized to q-dimensional spherical harmonics as performed in Chap. 6. Lemma 4.4.1. For all t 2 Œ1; 1 and all h 2 .1; 1/, the following power series converge uniformly: 1 X

hn Pn .t/ D p

nD0 1 X

.2n C 1/hn Pn .t/ D

nD0

1 1 C h2  2ht

;

1  h2 3

.1 C h2  2ht/ 2

(4.4.1)

:

(4.4.2)

Proof. The right-hand side of (4.4.1) is the generating function which we have in the more general setting of Gegenbauer polynomials in Theorem 3.4.7. Relation (4.4.2) directly results from Corollary 3.4.10. t u Remark 4.4.2. Using (4.4.1) of Lemma 4.4.1, we obtain with x D jxj, y D jyj, ;  2 S2 :  1   12  1 jxj 1 X jxj n jxj2 D jyj1 1 C     2 D Pn .  /; (4.4.3) jx  yj jyj2 jyj jyj nD0 jyj provided that jxj < jyj. Moreover, we know that 1

X .1/n 1 1 D jxjn .  ry /n : jx  yj nŠ jyj nD0

(4.4.4)

By comparison we obtain the so-called Maxwell representation .  ry /n

1 Pn .  / D .1/n nŠ : jyj jyjnC1

(4.4.5)

4.4 Closure and Completeness of Spherical Harmonics

145

As y 7! jyj1 ; jyj ¤ 0; is (apart from a multiplicative constant) the fundamental solution of the Laplace equation in three dimensions, Maxwell’s representation tells us that the Legendre polynomials may be obtained by repeated differentiation of the fundamental solution in the direction of . The potential on the right-hand side may be regarded as the potential of a pole of order n with the axis  at the origin. Theorem 4.4.3 (Poisson Integral Formula). If F is continuous on S2 , then for h < 1: ˇ ˇ Z ˇ 1 ˇ 1  h2 ˇ (4.4.6) F ./ dS./  F ./ˇˇ D 0: lim sup 3 h!1 2S2 ˇ 4 S2 .1 C h2  2h.  // 2 Proof. Since jhj < 1 and kPn kC.Œ1;1/ D 1, we use Lemma 4.4.1 and find that Z

1 1

1  h2 .1 C h2  2ht/

3 2

dt D

Z 1 X .2n C 1/hn nD0

1 1

Pn .t/  1 dt D 2:

(4.4.7)

Therefore, we use Theorem 4.1.6: Z

1 1 2 2

1  h2 3

.1 C h2  2h.  // 2

S2

dS./ D 1;

 2 S2 :

(4.4.8)

Choose  2 S2 arbitrary, but fixed. Then, 1 4

Z

.1  h2 /.F ./  F .//

dS./ 3 .1 C h2  2h.  // 2 Z 1 .1  h2 /F ./ D dS./  F ./: 4 S2 .1 C h2  2h.  // 32

(4.4.9)

S2

For h 2 Π12 ; 1/, we split the sphere into two parts, i.e., the spherical cap of radius p 3 1  h around  2 S2 ; C.;

p p 3 3 1  h/ D f 2 S2 W 1  1  h      1g;

(4.4.10)

and its complementary set S2 n C.;

p p 3 3 1  h/ D f 2 S2 W 1      1  1  hg:

(4.4.11)

This allows us to split the integral (4.4.9) in the same way, i.e., Z

Z S2

:::

D S2 nC.;

Z ::: p 3 1h/

C C.;

p 3 1h/

::::

(4.4.12)

4 Scalar Spherical Harmonics in R3

146

For t 2 Œ1; 1 

p 3 1  h, we can estimate that

p 3 1 C h2  2ht D .1  h/2 C 2h.1  t/  2h.1  t/  2h 1  h:

(4.4.13)

This leads to 1  h2



3

.1 C h2  2ht/ 2

p 1  h2 1Ch 1h D p  2 1  h; p 3 3 3 1h .2h 1  h/ 2 .2h/ 2

(4.4.14)

which gives us for the first integral of the splitting (4.4.12) the following estimate: ˇZ ˇ ˇ ˇ

S2 nC.;

.1  h2 /.F ./  F .//

ˇ ˇ dS./ ˇˇ

(4.4.15) 3 .1 C h2  2h.  // 2 Z .1  h2 /.kF kC.S2 / C kF kC.S2 / / dS./  p 3 3 S2 nC.; 1h/ .1 C h2  2h.  // 2 3 Z 1 p 1h 1  h2 D 2 kF kC.S2 / 2 dt 3 1 .1 C h2  2ht/ 2 3 Z 1 p 1h p 2 1  h dt  4 kF kC.S2 /

p 3 1h/

1

p p 3 D 4 kF kC.S2 / 2 1  h .1  1  h C 1/ ƒ‚ … „ 2

p h!1  16 kF kC.S2 / 1  h ! 0; where the convergence is uniform with respect to  2 S2 . Since F 2 C.S2 / and S2 is compact, F is uniformly continuous, i.e., there exists a function W h ! .h/ with lim .h/ D 0;

h!1

(4.4.16)

such that jF ./  F ./j  .h/ (4.4.17) p 3 for all  2 S2 that satisfy 1  1  h      1. The upper bound .h/ is independent of . Now, we consider the second part of the integral in (4.4.12): ˇZ ˇ ˇ ˇ

C.;

p 3 1h/

.1  h2 /.F ./  F .// 3

ˇ ˇ dS./ ˇˇ

.1 C h2  2h.  // 2 Z .1  h2 /jF ./  F ./j dS./  p 3 3 C.; 1h/ .1 C h2  2h.  // 2

(4.4.18)

4.4 Closure and Completeness of Spherical Harmonics

Z  .h/

S2

147

1  h2 .1 C

h2

h!1

 2h.  //

3 2

dS./ D .h/4 ! 0:

Altogether, we find the desired convergence result.

t u

Theorem 4.4.4. Let F 2 C.S /. The series 2

1 X

n X

hn

nD0

F ^ .n; j /Yn;j ./

(4.4.19)

j Dn

with the Fourier coefficients F ^ .n; j / D hF; Yn;j iL2 .S2 / converges uniformly with respect to all  2 S2 for h 2 .0; h0 / with h0 < 1 and lim

h!1

1 X nD0

hn

n X

F ^ .n; j /Yn;j ./ D F ./;

(4.4.20)

j Dn

where this convergence is uniform. Proof. We first show uniform convergence of the series using the definition of the Fourier coefficients and the addition theorem (Theorem 4.3.29). We estimate the function values with the C.S2 /-norm and apply jPn .t/j  1 for t 2 Œ1; 1. ˇX ˇ n Z ˇ 1 n X ˇ ˇ h F ./Yn;j ./ dS./Yn;j ./ˇˇ ˇ 2 nD0

(4.4.21)

j Dn S

 kF kC.S2 /

1 X

Z n

h

nD0

 kF kC.S2 /

1 X

S2

2n C 1 jPn .  /j dS./ 4

hn .2n C 1/ < 1:

nD0

For the limit process, (4.4.2) of Lemma 4.4.1 and the addition theorem (Theorem 4.3.29) are used to obtain a representation as in Theorem 4.4.3. The dominated convergence theorem allows us to interchange the infinite series and the integral. Finally, the actual convergence is established by the Poisson integral formula (Theorem 4.4.3). t u Corollary 4.4.5. The system fYn;j gn2N0 ;nj n is closed in .C.S2 /; kkC.S2 / /, i.e., for all F 2 C.S2 / and for all " > 0, there exists a finite linear combination n N X X nD0 j Dn

dn;j Yn;j ;

(4.4.22)

4 Scalar Spherical Harmonics in R3

148

such that

n N X

X

dn;j Yn;j

F 

C.S2 /

nD0 j Dn

 ":

(4.4.23)

D C.S2 /:

(4.4.24)

In other words: spanfYn;j jn 2 N0 ; n; : : : ; ng

kkC.S2 /

Proof. Let F 2 C.S2 / and " > 0 be arbitrary. According to Theorem 4.4.4 there exists h D h."/ < 1 fixed, such that 1 n

X X

hn F ^ .n; j /Yn;j  F

nD0

C.S2 /

j Dn



" : 2

(4.4.25)

Moreover, the theorem tells us that there exists N D N."/, such that 1 N n n

X

X X X

hn F ^ .n; j /Yn;j  hn F ^ .n; j /Yn;j

nD0

j Dn

nD0

C.S2 /

j Dn



" ; 2

(4.4.26)

since the series converges uniformly for fixed h < 1 (here h D h."/). Together this gives us n N X

X " "

hn F ^ .n; j / Yn;j 2  C D "

F  „ ƒ‚ … C.S / 2 2 nD0 j Dn

(4.4.27)

Ddn;j

and, therefore, the closure of the spherical harmonics in .C.S2 /; kkC.S2 / /.

t u

Now, we consider Bernstein summability to obtain this result. The point of departure for Bernstein summability is the so-called Bernstein kernel of degree n. Definition 4.4.6. The Bernstein kernel of degree n is given by KB;n W Œ1; 1 ! R t 7! KB;n .t/ D

nC1 4



1Ct 2

n :

(4.4.28)

See Fig. 4.8 for an illustration of it. Remark 4.4.7. The name Bernstein is motivated by the fact that the kernel is proportional tothe Bernstein polynomial of degree n with the parameter  D n, i.e., B;n .t/ D n t  .1  t/n scaled to the interval Œ1; 1 (t is the polar distance

4.4 Closure and Completeness of Spherical Harmonics

149

3.5 n=10 n=20 n=40

3

2.5

2

1.5

1

0.5

0

−3

−2

−1

0

1

2

3

Fig. 4.8 Illustration of the kernel KB;n .cos #/; # 2 Œ; , for the degrees n D 10, n D 20, n D 40

between  and , i.e., t D   ). For further details we refer to Fengler et al. (2006) and Freeden and Gutting (2008). First, we mention some important properties of the Bernstein kernel that can easily be verified by the reader. Lemma 4.4.8. The Bernstein kernel (4.4.28) possesses the following properties: (i) For all t 2 Œ1; 1 and n D 0; 1; : : :, we have ^ .0/ D 2 KB;n

(ii) For all t 2 Œ1; 1, (iii) For all t 2 Œ1; 1/,

Z

1 1

KB;n .t/ dt D 1:

(4.4.29)

KB;n .t/  0:

(4.4.30)

lim KB;n .t/ D 0:

(4.4.31)

n!1

(iv) For k D 0; : : : ; n, Z 2

1 1

KB;n .t/Pk .t/ dt D

^ KB;n .k/

n .n C 1/Š nŠ k D nCkC1 D ; .n  k/Š .n C k C 1/Š k (4.4.32)

where Pk is the Legendre polynomial of degree k.

4 Scalar Spherical Harmonics in R3

150

(v) For k 2 N fixed,

^ ^ KB;n .k/ < KB;nC1 .k/:

(vi) For k 2 N0 fixed,

^ KB;n .k/

(4.4.33)

! 1 as n ! 1, i.e., ^ lim KB;n .k/ D 1:

(4.4.34)

n!1

Now, suppose that F is continuous on S2 . We use the property .i / to guarantee that Z

Z S2

KB;n .  /F ./ dS./ D F ./ C

S2

KB;n .  / .F ./  F .// dS./;

(4.4.35) where  2 S2 . We split S2 into two parts depending on a parameter 2 .0; 1/, i.e., into the spherical cap of radius around  2 S2 : C.; / D f 2 S2 W 1       1g

(4.4.36)

and the remaining sphere S2 n C.; /. The corresponding split of the integral (4.4.35) yields Z

Z S2

::: D

Z S2 nC.; /

::: C

::::

(4.4.37)

C.; /

On the one hand, we find with (ii) ˇZ ˇ ˇ ˇ

S2 nC.; /

ˇ Z ˇ KB;n .  /F ./ dS./ˇˇ  4 kF kC.S2 /

1 1

KB;n .t/ dt



nC1  2 kF kC.S2 / 1  : 2

(4.4.38)

On the other hand, F is uniformly continuous on S2 . Thus, there exists a positive function W 7! . / with lim . / D 0;

(4.4.39)

!0C

such that jF ./  F ./j  . /

(4.4.40)

for all  2 S2 with 1       1. Thus, it follows that ˇZ ˇ ˇ ˇ

C.; /

ˇ Z ˇ KB;n .  /.F ./  F .// dS./ˇˇ  . / KB;n .  / dS./ 2 S

D . /:

(4.4.41)

4.4 Closure and Completeness of Spherical Harmonics

151

Summarizing our results, we obtain the estimate ˇZ ˇ ˇ ˇ sup ˇˇ KB;n .  /F ./ dS./  F ./ˇˇ

2S2

S2

ˇZ ˇ ˇ ˇ ˇ D sup ˇ KB;n .  / .F ./  F .// dS./ˇˇ 2 2 2S

S

2S

S2 nC.; /

ˇZ ˇ  sup ˇˇ 2

ˇZ ˇ C sup ˇˇ 2 2S

ˇZ ˇ  sup ˇˇ 2 2S

Z

C.; /

S2 nC.; /

KB;n .  /F ./ dS./  F ./

S2 nC.; /

ˇ ˇ KB;n .  / dS./ˇˇ

ˇ ˇ KB;n .  / .F ./  F .// dS./ˇˇ ˇ ˇ KB;n .  /F ./ dS./ˇˇ

Z

CkF kC.S2 /

S2 nC.; /

KB;n .  / dS./ C . /

 nC1 C . /:  2kF kC.S2 / 1  2

(4.4.42)

Now, we choose D n1=2 , such that ! 0 for n ! 1, i.e.,   1

1 p 2 n

p1 n



nC1

!0

for n ! 1;

(4.4.43)

!0

for n ! 1:

(4.4.44)

This shows us that ˇZ ˇ ˇ ˇ lim sup ˇˇ KB;n .  /F ./ dS./  F ./ˇˇ D 0 n!1 2 2 2S

(4.4.45)

S

and we obtain the following result. Theorem 4.4.9. For F 2 C.S2 /, ˇZ ˇ ˇ ˇ ˇ lim sup ˇ KB;n .  /F ./ dS./  F ./ˇˇ D 0: n!1 2 2 2S

(4.4.46)

S

Now, it can be readily seen that the Legendre series of the Bernstein kernel is given by n X 2k C 1 ^ KB;n .t/ D Pk .t/ KB;n .k/ (4.4.47) 4 kD0

4 Scalar Spherical Harmonics in R3

152

for all t 2 Œ1; 1. This shows us that Z S2

KB;n .  /F ./ dS./ D

n X

C1 4

2k ^ KB;n .k/

kD0

D

n X

Z S2

Pk .  /F ./ dS./

^ KB;n .k/ ProjHarmk .F /./:

(4.4.48)

kD0

Thus, we finally have the ‘Bernstein summability’ of a Fourier series expansion in terms of spherical harmonics. Theorem 4.4.10. For F 2 C.S2 /, ˇ n ˇ k X ˇX ^ ˇ lim sup ˇˇ KB;n .k/ F ^ .k; j /Yk;j ./  F ./ˇˇ D 0: n!1 2 2S

(4.4.49)

j Dk

kD0

Theorem 4.4.10 also enables us to prove the closure of the system of spherical harmonics in the space C.S2 /. Corollary 4.4.11. The system fYk;j gk2N0 ; j Dk;:::;k is closed in C.S2 /, i.e., for any given " > 0 and each F 2 C.S2 /, there exists a linear combination N X k X

dk;j Yk;j

(4.4.50)

kD0 j Dk

such that

N X k

X

dk;j Yk;j

F 

C.S2 /

kD0 j Dk

 ":

(4.4.51)

Proof. Given F 2 C.S2 /. Then, for any given " > 0, there exists an integer N D N."/ such that ˇX ˇ k ˇ N X ˇ ^ ^ ˇ sup ˇ KB;N .k/F .k; j / Yk;j ./  F ./ˇˇ  ": ƒ‚ … 2S2 kD0 j Dk „

(4.4.52)

Ddk;j

t u

This proves Corollary 4.4.11. Since for all F 2 C.S / we have the inequality 2

kF kL2 .S2 / 

p 4 kF kC.S2 / ;

(4.4.53)

we immediately find that fYn;j gn2N0 ;nj n is also closed in .C.S2 /; kkL2 .S2 / /.

4.4 Closure and Completeness of Spherical Harmonics

153

Corollary 4.4.12. The system fYn;j gn2N0 ;nj n is closed in .L2 .S2 /; kkL2 .S2 / /. Proof. We know that C.S2 /

kkL2 .S2 /

D L2 .S2 /:

(4.4.54)

Let F 2 L .S / and " > 0. There exists G 2 C.S / such that kF  GkL2 .S2 /  2" . Due to Corollary 4.4.5 and estimate (4.4.53) there exists a finite linear combination 2

2

2

n N X X

an;j Yn;j

(4.4.55)

nD0 j Dn

corresponding to G such that n N X

X

an;j Yn;j

G  nD0 j Dn

" : 2

(4.4.56)

" " C D ": 2 2

(4.4.57)

L2 .S2 /



Combining these two results yields: n N X

X

an;j Yn;j

F  nD0 j Dn

L2 .S2 /



t u

This completes the proof of Corollary 4.4.12.

Now, we can apply Theorem 3.0.7 to the system of spherical harmonics. This gives us L2 -Fourier approximation on the of the system  sphere, i.e., completeness fYn;j gn2N0 ;nj n in the Hilbert space L2 .S2 /; k  kL2 .S2 / :

n N X X

lim F  hF; Yn;j iL2 .S2 / Yn;j

N !1 nD0 j Dn

D0

for all F 2 L2 .S2 /:

L2 .S2 /

(4.4.58) Remark 4.4.13. (i) Let X 2 fC.S2 /; Lp .S2 /gp2Œ1;1/ . If F; G 2 X with

n N X X

G  hF; Y i Y lim 2 2 n;j L .S / n;j D 0;

N !1 nD0 j Dn

(4.4.59)

X

then F D G almost everywhere on S2 . (ii) For X 2 fC.S2 /; Lp .S2 /gp2Œ1; 4 [Œ4;1/ it is not known whether the truncated 3 Fourier series converges for all F 2 X , i.e., a truncated Fourier series with L2 -Fourier coefficients might not yield an approximation in the X -topology.

4 Scalar Spherical Harmonics in R3

154

(iii) If F W S2 ! R is Lipschitz continuous, then

n N X X

hF; Yn;j iL2 .S2 / Yn;j lim F 

N !1

D 0;

(4.4.60)

C.S2 /

nD0 j Dn

i.e., the Fourier series is uniformly convergent in this case. Remark 4.4.14. Using the completeness property of the spherical harmonics system, we can prove that the spherical harmonics Yn 2 Harmn .S2 / are the only C.2/ .S2 /-eigenfunctions of the Beltrami operator.  only has the eigenvalues n.n C 1/ with n 2 N0 .

4.5 The Funk–Hecke Formula and the Irreducibility of Scalar Harmonics In this section we present a central result in the theory of spherical harmonics, i.e., the Funk–Hecke formula (see, e.g., Funk 1916; Hecke 1918; M¨uller 1998 and the literature therein). It is closely related to the irreducibility of the space of spherical harmonics which is also shown here. Lemma 4.5.1. Let Hn W R3 ! R be a homogeneous harmonic polynomial of degree n with the following properties: (i) Hn .tx/ D Hn .x/ for all orthogonal transformations t 2 SO"3 .3/, i.e., all rotations that leave "3 invariant, (ii) Hn ."3 / D 1. Then, Hn is uniquely determined and coincides with the so-called Legendre harmonic Ln of degree n, i.e., Hn .x/ D Ln .x/ D r n Pn .t/;

x D r.t "3 C

p 1  t 2 .cos '"1 C sin '"2 //; (4.5.1)

where Pn is the Legendre polynomial of degree n. Proof. We already know from Theorem 4.3.15 that Hn as a homogeneous harmonic polynomial of degree n can be written in the form Hn .x/ D

n X

Ank .x1 ; x2 /x3k ;

(4.5.2)

kD0

where the coefficients Ank .x1 ; x2 / fulfill (4.3.44) for k D 0; : : : ; n  2. Therefore, Hn is uniquely determined by the homogeneous polynomials An W .x1 ; x2 / 7! An .x1 ; x2 / and An1 W .x1 ; x2 / 7! An1 .x1 ; x2 /. Condition .i / implies that

4.5 The Funk–Hecke Formula

155

these polynomials depend only on x12 C x22 . Thus, we find with a constant C nk , 2 such that ( 0 ; n  k odd; Ank .x1 ; x2 / D (4.5.3) nk ; n  k even: C nk .x12 C x22 / 2 2

For x D "3 in (4.5.3), we get x12 C x22 D 0 and x3 D 1 such that C0 D 1. In order to determine C nk for even integers n  k, we see that 2



@2 @2 C @x12 @x22

 .x12

C

nk x22 / 2



nk D4 2

2

.x12 C x22 /

nk2 2

:

(4.5.4)

In connection with (4.3.44) we find the recursion relation  4C nk 2

nk 2

2

.x12 Cx22 /

nk2 2

D .kC2/.kC1/C nk2 .x12 Cx22 /

nk2 2

2

(4.5.5)

such that .n  k/2 C nk C .k C 2/.k C 1/C nk2 D 0; 2

2

(4.5.6)

k D 0; 2; : : : ; n  2. In other words, since C0 D 1, C nk D 1 for k D n;

(4.5.7)

2

C nk D .1/ 2

nk 2

nŠ 2kn  2 . nk /Š kŠ 2

for n  k even;

for k D 0; : : : ; n1. This shows us that Hn is uniquely determined by the conditions .i / and (ii). x It remains to prove that Ln .x/ D r n Pn .t/ (with r D jxj, t D "3  , and  D jxj ) is a homogeneous harmonic polynomial that fulfills .i / and (ii). Obviously, Ln 2 Homn .R3 / and by the addition theorem (Theorem 4.3.21) we find that Ln .x/ D jxjn Pn ."3  / D

n .2n/Š X Hn;j ."3 /Hn;j .x/ 2 Harmn .R3 /; 2n nŠ j Dn

(4.5.8)

where Hn;j , j D n; : : : ; n, is an orthonormal basis of Harmn .R3 /. Condition .i i / is also obvious, since Pn .1/ D 1. For condition .i /, consider any t 2 SO"3 .3/; Ln .tx/ D jxjn Pn ."3  t/ D jxjn Pn .tT "3  / D jxjn Pn ."3  / D Ln .x/; (4.5.9) since SO"3 .3/ is a group and, therefore, tT also leaves "3 invariant.

t u

4 Scalar Spherical Harmonics in R3

156

Corollary 4.5.2. Let Hn 2 Harmn .R3 / with the following properties: (i) Hn .tx/ D Hn .x/ for all orthogonal transformations t 2 SO .3/, i.e., all rotations that leave  2 S2 invariant, (ii) Hn ./ D 1. Then, Hn is uniquely determined by Hn .x/ D jxjn Pn .  /;

x D jxj;  2 S2 ;

(4.5.10)

where Pn is the Legendre polynomial of degree n. t u

Proof. This follows from Lemma 4.5.1 by simply rotating  to "3 .

For any function F 2 L .S /, the transformation  7! t;  2 S , produces a change in the functional values of F . Let F; G 2 L2 .S2 /, t 2 O.3/. Observing the change of coordinates  D t, we find that 2

2

2

Z

Z S2

F .t/G./ dS./ D j det tj

S2

F ./G.tT / dS./

(4.5.11)

(observe that dS.t/ D j det tj dS./). Thus, it follows that hRt F; GiL2 .S2 / D hF; RtT GiL2 .S2 / :

(4.5.12)

Theorem 4.5.3. The space Harmn .S2 / of spherical harmonics of degree n is irreducible. Proof. Assume that there exists an invariant subspace Y  Harmn .S2 / of dimension dim.Y / < dim.Harmn .S2 // D 2n C 1. Then, we would be able to show that the orthogonal complement Y ? of Y in Harmn .S2 / (with respect to h; iL2.S2 / ) is an invariant subspace (see Lemma 4.2.4). Now, because of the invariance of Y and Y ? , G and G ? being represented in terms of the L2 .S2 /-orthonormal system fYn;j g, respectively, X

dim.Y /

GD

Yn;j ."3 /Yn;j 2 Y;

(4.5.13)

j D1

G

?

dim.Harmn .S2 //

D

X

Yn;j ."3 /Yn;j 2 Y ?

(4.5.14)

j Ddim.Y /C1

satisfy G.t/ D G, G ? .t/ D G ? for all t 2 SO"3 .3/. Moreover, G; G ? do not vanish identically (note that there exist elements in Y , Y ? different from zero at "3 ). Thus, not all values of Yn;j ."3 /, j D 1; : : : ; dim.Y / or j D dim.Y / C 1; : : : ; dim.Harmn .S2 // are zero. By Lemma 4.5.1 there exists a constant a 2 Rnf0g such that G D aG ? , in contradiction to our assumption. t u

4.5 The Funk–Hecke Formula

157

The irreducibility of Harmn .S2 / leads us to simple representations of spherical harmonics (see, e.g., Freeden et al. 1998). Lemma 4.5.4. Let Zn be a member of Harmn .S2 /. There exist 2n C 1 orthogonal transformations tn ; : : : ; tn such that any Yn 2 Harmn .S2 / can be represented with real numbers an ; : : : ; an in the form n X

Yn D

aj Zn .tj /:

(4.5.15)

j Dn

Proof. Consider the set X D fZn .t/ W t 2 O.3/g:

(4.5.16)

Clearly, there exist tn ; : : : ; tn 2 O.3/ such that X D spanfZn .tn /; : : : ; Zn .tn /g:

(4.5.17)

Therefore, Zn 2 X implies Zn .t/ 2 X . From the irreducibility of Harmn .S2 / (see Theorem 4.5.3), it follows that X is an invariant non-void space of dimension dim.X / D dim.Harmn .S2 //. Moreover, the 2n C 1 linearly independent spherical harmonics Zn .tn /; : : : ; Zn .tn / form a basis. t u Lemma 4.5.5. There exist 2n C 1 points n ; : : : ; n of the unit sphere S2 such that any Yn 2 Harmn .S2 / can be represented with real numbers an ; : : : ; an in the form Yn ./ D

n X

aj Pn .j  /;

 2 S2 :

(4.5.18)

j Dn

Proof. From Lemma 4.5.4 we know that there exist 2n C 1 orthogonal transformations tn ; : : : ; tn such that any Yn 2 Harmn .S2 / admits the form Yn ./ D

n X

aj Pn ."3  tj /;

 2 S2 :

(4.5.19)

j Dn

But this means that Lemma 4.5.5 follows with j D tTj "3 ; j D n; : : : ; n.

t u

Theorem 4.5.6 (Funk–Hecke Formula). Let G 2 L1 .Œ1; 1/, n 2 N0 . Then, Z G.  /Pn .  / dS./ D 2 S2 „

Z

1 1

G.t/Pn .t/ dt Pn .  / ƒ‚ …

DG ^ .n/

for all ;  2 S2 . G ^ .n/ is called the Legendre coefficient of G.

(4.5.20)

4 Scalar Spherical Harmonics in R3

158

R Proof. Let Vn .; / D S2 G.  ˛/Pn .  ˛/ dS.˛/ for ;  2 S2 and t 2 O.3/ an orthogonal transformation. Then, Z Vn .t; t/ D Z

G.t  ˛/Pn .t  ˛/ dS.˛/

S2

D

(4.5.21)

G.t  tˇ/Pn .t  tˇ/ dS.tˇ/

S2

Z

D j det tj „ƒ‚ … D1

S2

G.  ˇ/Pn .  ˇ/ dS.ˇ/ D Vn .; /:

Furthermore, Vn .; / with fixed  2 S2 is a spherical harmonic by the addition theorem, i.e., for any  2 S2 , Z Vn .; / D

S2

G.  ˛/Pn .  ˛/ dS.˛/

Z

n X

D S2

D

G.  ˛/

j Dn

(4.5.22)

4 Yn;j .˛/Yn;j ./ dS.˛/ 2n C 1

n X

4 Yn;j ./G^ .n; j /: 2n C 1 j Dn

This means that Vn .; / is a harmonic polynomial that is invariant under orthogonal transformations that leave  invariant. Hence, by Lemma 4.5.1 or, more precisely, by Corollary 4.5.2, Vn .; / D ˛n Pn .  /: (4.5.23) Setting  D  we obtain (using Theorem 4.1.6) Z ˛n D ˛n Pn .  / D Z

1

S2

G.  ˛/Pn .  ˛/ dS.˛/

G.t/Pn .t/ dt D G ^ .n/

(4.5.24)

which corresponds to the desired Legendre coefficient.

t u

D 2

1

Corollary 4.5.7. Let G 2 L1 .Œ1; 1/. For every Yn 2 Harmn .S2 /, we have Z S2

G.  /Yn ./ dS./ D G ^ .n/Yn ./;

 2 S2 ;

where G ^ .n/ is the Legendre coefficient as defined in Theorem 4.5.6.

(4.5.25)

4.6 Green’s Function with Respect to the Beltrami Operator

159

Proof. This is a direct consequence of the Funk–Hecke formula (Theorem 4.5.6) and the fact that the Legendre polynomial Pn is the reproducing kernel in Harmn .S2 / (see Remark 4.3.30). t u

4.6 Green’s Function with Respect to the Beltrami Operator One of the most essential ingredients for our approach to vector spherical harmonics is Green’s function on the sphere S2 with respect to the Beltrami operator which we define here (cf. Freeden 1979, 1980a). Several integral theorems involving the Green function are included. It should be noted that the Green function of this section also plays an important role in the theory of tensor spherical harmonics for which we refer to Freeden and Gutting (2008) and Freeden and Schreiner (2009) and the many references therein. Definition 4.6.1. The function G. I ; / W .; / 7! G. I ; /, 1     < 1, is called Green’s function on S2 , or sphere function, with respect to the operator  , if it satisfies the following properties: (i) For every fixed  2 S2 , the function  7! G. I ; / is twice continuously differentiable on the set S2 n fg such that  G. I ; / D 

1 ; 1     < 1; 4

(4.6.1)

1 ln.1    / 4

(4.6.2)

(ii) For every  2 S2 , the function  7! G. I ; / 

is continuously differentiable on S2 . (iii) For all orthogonal transformations t 2 O.3/, G. I t; t/ D G. I ; /; (iv) For every  2 S2 , 1 4

Z S2

G. I ; / dS./ D 0:

(4.6.3)

(4.6.4)

Because of Condition (iii) we find that G. I ; /P is a zonal function and we usually write G. I   / instead of G. I ; /. Note that, for 1     < 1, ln j  j D

1 1 1 ln.2  2  / D ln.1    / C ln.2/: 2 2 2

(4.6.5)

4 Scalar Spherical Harmonics in R3

160

Lemma 4.6.2. G. I ; / is uniquely determined by its defining properties (i)–(iv). The explicit representation of the Green function, for 1     < 1, is G. I ; / D

1 1 1 ln.1    / C  ln.2/: 4 4 4

(4.6.6)

Proof. We first show uniqueness. Let D.; / be the difference between two Green functions. Then, .i / D.; / is twice continuously differentiable for all  2 S2 with 1     < 1 and  D.; / D 0, (ii) D.; / is a continuously differentiable function for every  2 S2 , (iii) For R all orthogonal transformations t 2 O.3/, D.t; t/ D D.; /, (iv) S2 D.; / dS./ D 0 for every  2 S2 . Without loss of generality, set  D "3 . Now, we know that D."3 ; / D D.t/, where @ t is one of the spherical coordinates of  D .t; '/. Thus, @' D D 0 and by .i /  and (4.1.12) (coordinate representation of  ) we find that @ @ .1  t 2 / D.t/ D 0 @t @t

(4.6.7)

for all 1  t < 1. Hence, .1  t 2 / @t@ D.t/ D C0 2 R for all 1  t < 1. D is continuously differentiable by (ii), i.e., @t@ D.t/ tends to a constant C1 2 R for t ! 1. Therefore, @ lim .1  t 2 / D.t/ D 0 D C0 (4.6.8) t !1 @t and .1  t 2 / @t@ D.t/ D 0 for all t 2 Œ1; 1. This gives us @t@ D.t/ D 0 on .1; 1/, i.e., D.t/ D C2 on .1; 1/ and by continuity on Œ1; 1. Together with .i v/, this yields Z Z 1 0D D."3 ; / dS./ D 2 D.t/ dt D 4 C2 : (4.6.9) S2

1

Thus, C2 D 0, i.e., D  0, which shows the uniqueness. Now, all we have to prove is that the explicit representation (4.6.6) fulfills the properties .i /–.i v/. Obviously, we have (iii) and, for (ii), we see that G. I ; / 

1 1 1 ln.1    / D  ln.2/ 4 4 4

(4.6.10)

is arbitrarily often differentiable. Consider now the integral Z S2

G. I ; / dS./ D 2

Z

1

1

1 .ln.1  t/ C 1  ln.2// dt D 1; 4

(4.6.11)

4.6 Green’s Function with Respect to the Beltrami Operator

161

which gives us .i v/. For .i /, we calculate by setting t D    and using (4.1.26) 4 G. I ; / D 4 G. I t/ D 2tG 0 . I t/ C .1  t 2 /G 00 . I t/ D 2t

1 1 C .1  t 2 / D 1: 1t .1  t/2

This concludes the proof for the representation (4.6.6).

(4.6.12) t u

By applying Green’s second surface theorem, i.e., (4.1.20) of Theorem 4.1.4, we see that Z  n.n C 1/ G. I ; /Yn ./ dS./ D .1  ı0;n /Yn ./: (4.6.13) S2

Therefore, we can determine the spectral representation of the Green function as the following bilinear expansion, for 1     < 1, G. I ; / D

n 1 X X nD1 kDn

D

1 Yn;k ./Yn;k ./ n.n C 1/

1 X 2n C 1 nD1

4

(4.6.14)

1 Pn .  /: n.n C 1/

Theorem 4.6.3 (Third Green’s Surface Theorem for r  ). Let  be a fixed point of S2 . Suppose that F is a continuously differentiable function on S2 . Then, F ./ D

1 4

Z S2

F ./ dS./ 

Z     r G. I   /  r F ./ dS./: (4.6.15) S2

Proof. Let  2 S2 be fixed. For each sufficiently small % > 0, Green’s first surface identity, i.e., (4.1.19) of Theorem 4.1.4, gives us with  .; %/ D S2 n C.; %/, where C.; %/ D f 2 S2 W 1  %      1g: Z

  .;%/

 F ./ G. I   / C r F ./  r G. I   / dS./ Z

D

F ./ @ .;%/

@ G. I   / d ./; @

(4.6.16)

where  is the unit normal to the circle @ .; %/ that consists of all points  2 S2 with    D 1  %.  points outward  .; %/ D S2 n C.; %/, i.e., into C.; %/. Thus,   D p ^ . ^ /: 1  .  /2

(4.6.17)

4 Scalar Spherical Harmonics in R3

162

We use the differential equation for Green’s function and obtain Z  .;%/

F ./ G. I   / dS./ D 

1 4

Z F ./ dS./:

(4.6.18)

 .;%/

Moreover, by noting that @ .; %/ D f 2 S2 W    D 1  %g, we find that Z F ./ @ .;%/

@ G. I   / d ./ @

(4.6.19)

    .1  %/ 1 .  .1  %// d ./ D F ./ p   4% 1  .1  %/2 @ .;%/ p Z 1 1  .1  %/2 D d ./ F ./ 4 @ .;%/ % p 1 2% 1  .1  %/2 p D 2 1  .1  %/2 F .% / D  F .% /; 4 % 2 Z

where the mean value theorem with % 2 f 2 S2 W 1     D %g has been used. The continuity of F yields that F .% / ! F ./ as % !  for % ! 0 such that Z lim

%!0 @ .;%/

F ./

@ G. I   / d ./ D F ./: @

(4.6.20) t u

This gives us the desired result.

Corollary 4.6.4 (Third Green’s Surface Theorem for L ). Under the assumptions of Theorem 4.6.3, F ./ D

1 4

Z S2

F ./ dS./ 

Z     L G. I   /  L F ./ dS./: (4.6.21) S2

Proof. We use Green’s surface identity for L which is analogous to (4.1.19) of Theorem 4.1.4. Note that instead of the normal vector the tangential vector  ^

 D p 1  .  /2

(4.6.22)

is required. The same reasoning as in Theorem 4.6.3 is used to prove this corollary. t u Theorem 4.6.5 (Third Green’s Surface Theorem for  ). Let  be a fixed point of S2 . Suppose that F is a twice continuously differentiable function on S2 . Then,

4.6 Green’s Function with Respect to the Beltrami Operator

F ./ D

1 4

Z

Z S2

F ./ dS./ C

S2

G. I   / F ./ dS./:

163

(4.6.23)

Proof. Theorem 4.6.5 can be seen as a consequence of Theorem 4.6.3 and relation (4.1.13). t u Remark 4.6.6. The Green function   G . /2 I ;  W .; / 7! G . /2 I    ;

(4.6.24)

where 1      1, with respect to the iterated Beltrami operator . /2 is given by the spherical convolution which is defined by Z .K F /./ D

S2

K.  /F ./ dS./

(4.6.25)

for K 2 L1 .Œ1; 1/ and F 2 L1 .S2 /. For the Green function this yields   G . /2 I    D G. I  / G. I  / ./ Z G. I   /G. I   / dS./: D

(4.6.26)

S2

The bilinear expansion of G . /2 I    can be easily calculated by 

 G . /2 I    D

(4.6.27)

Z X 1 X 1 1 1 2k C 1 2l C 1 Pk .  /Pl .  / dS./ 4 4 .k.k C 1// .l.l C 1// S2 kD1 lD1

D

Z 1 X 1 X 1 1 2k C 1 2l C 1 Pk .  /Pl .  / dS./ 4 4 .k.k C 1// .l.l C 1// S2 kD1 lD1

D

1 X 1 X 2k C 1

4

kD1 lD1

1 1 ık;l Pk .  /; .k.k C 1// .l.l C 1//

such that 1 X 1 2k C 1 G . / I    D Pk .  /: 4 .k.k C 1//2



 2

(4.6.28)

kD1

For .; / 2 S2  S2 with 1     < 1 we obtain   G . /2 I    D G. I   /:

(4.6.29)

4 Scalar Spherical Harmonics in R3

164

This differential equation  enables us to give an explicit representation of the Green function G . /2 I    , 1      1,  G . /2 I    D D

Z S2

G. I   /G. I   / dS./

1 X 2n C 1 nD1

D

4

1 Pn .  / .n.n C 1//2

8 1 ˆ ˆ ˆ 4 ˆ < 1 .1  .ln.2//2 /  4

(4.6.30)

ˆ C ln.2/ ˆ 4 ln.1 ˆ ˆ :1  4  24

;    D 1;

ln.1    / ln.1 C   /  1 ;    ¤ ˙1;  .  / /  4 L2 1 2 1 4 2

;    D 1;

where L2 denotes the dilogarithm, i.e., L2 .t/ D

1 X 1 k t ; k2

t 2 R:

(4.6.31)

kD1

For the proof the reader is referred to Freeden and Schreiner (2009). In addition, the third Green surface theorem for  (Theorem 4.6.5) admits the following reformulation. Corollary 4.6.7 (Third Green’s Surface Theorem for  ). Let  be a point of S2 . Assume that F 2 C.2/ .S2 /. Then, 1 F ./ D 4

Z

Z S2

F ./ dS./ C

S2

  G . /2 I     F ./ dS./:

(4.6.32)

The third Green surface theorems are the points of departure for a palette of methods and procedures in spherical theory. Applications in constructive approximation are equidistribution of point sets (see Cui and Freeden 1997; Freeden and Hesse 2002; Hesse et al. 2010; Hlawka 1982), (best-)approximate integration (cf. Freeden 1980a; Freeden and Reuter 1982; Freeden et al. 1998) and spline theory (cf. Freeden 1981; Freeden and Hermann 1986; Freeden et al. 1997, 1998). The essential ideas of these approaches in constructive approximation are highlighted by exercises in our book (see Sects. 4.8, 6.7 and 10.9). Furthermore, wavelet solvers for the surface gradient equation as well as the surface curl gradient equation are derivable from the third Green surface theorems (for more details the reader should consult Freeden and Gerhards 2012). Remark 4.6.8. Theorem 4.6.5 and (4.6.30) can be used to find the following integral estimate with the help of the Cauchy–Schwarz inequality for F 2 C.2/ .S2 /:

4.7 The Hydrogen Atom

165

ˇ ˇ ˇZ ˇ Z ˇ ˇ ˇ ˇ   ˇ ˇF ./ 1 ˇ F ./ dS./ˇ D ˇ G. I   / F ./ dS./ˇˇ ˇ 2 4 2 S

Z 

(4.6.33)

S

1  Z  1 2 2 2  G. I   /G. I   / dS./  F ./ dS./ : 2 S2 „S ƒ‚ … 



1 D 4

For more details we refer to Freeden and Schreiner (2009). Theorem 4.6.9 (Differential Equation for  on S2 ). Let H 2 C.S2 / with 1 4

Z S2

H./ dS./ D 0:

(4.6.34)

Let F 2 C.2/ .S2 / satisfy the Beltrami differential equation  F D H: Z

Then, F ./ D

S2

(4.6.35)

G. I   /H./ dS./;

 2 S2 ;

(4.6.36)

and F is uniquely determined with 1 4

Z S2

F ./ dS./ D 0:

(4.6.37)

Other solutions take the form FQ D F C C with a constant C 2 R. Proof. Theorem 4.6.9 is a direct consequence of Theorem 4.6.5.

t u

4.7 The Hydrogen Atom The hydrogen atom is the simplest atom we can think of. It consists of an electron with the charge e and mass m in the Coulomb potential of a nucleus with charge e and mass M where M m. The starting point of the quantum-mechanical discussion of the hydrogen atom is the Hamilton function of an electron with charge e in the Coulomb potential of a nucleus with charge e, i.e., H.p; x/ D

p2 e2  ; 2 4ε0 jxj

x  0;

(4.7.1)

4 Scalar Spherical Harmonics in R3

166

where x is the relative distance of the electron and the nucleus and is the reduced mass given by mM D  m; (4.7.2) mCM since M m. Applying the substitution rules of (3.6.7) (in this case we only need pj ! i„ @x@j , j D 1; 2; 3), we obtain the Hamilton operator H D

e2 „2 x  : 2m 4ε0 jxj

(4.7.3)

Thus, the stationary Hamilton equation is (x 2 R3 ) H .x/ D E .x/

,



„2 e2 x .x/  .x/ D E .x/; 2m 4ε0 jxj

(4.7.4)

where 2 L2 .R3 / with k kL2 .R3 / D 1 and E < 0 such that we obtain bounded states. By separation of variables .x/ D U.r/F ./, x D r, r D jxj > 0 and  2 S2 . We use the representation of the Laplace operator in polar coordinates (4.1.11) and get: 

 1  e2 2 @ @2 C U.r/F ./  E U.r/F ./ D 0: C   U.r/F ./  2 2 @r r @r r 4ε0 r (4.7.5) The radial derivatives only apply to U and the Beltrami operator applies to F such that  2  „2 @ „2 U.r/  2 @  U.r/F ./  C  F ./ 2 2m @r r @r 2m r 2 e2 U.r/F ./  E U.r/F ./ D 0: (4.7.6)  4ε0 r 

„2 2m

„ Dividing by U.r/, F ./, and  2m as well as multiplying by r 2 , this is equivalent to 2

r2

U 0 .r/ 2m U 00 .r/ C 2r C 2 U.r/ U.r/ „



  F ./ e2r C Er 2 D  4ε0 F ./

(4.7.7)

for all r > 0,  2 S2 . This can only be true if both sides are equal to a constant  2 R. Thus, we obtain for the right-hand side ( 2 S2 ): 

 F ./ F ./

D

,

 F ./ D F ./:

(4.7.8)

4.7 The Hydrogen Atom

167

This is the Beltrami differential equation which has a polynomial solution if and only if  D l.l C 1/, l 2 N0 . In this case the solution is a spherical harmonic of degree l: l X F ./ D Yl ./ D am Yl;m ./; (4.7.9) mDl

i.e., the basis system of the orthonormal spherical harmonics Yl;m , m D l; : : : ; l, describes all solutions. For the left-hand side we get with  D l.l C 1/ and after dividing by r 2 and multiplying with U.r/:   2mE l.l C 1/ 2 0 2m e 2 00 C U.r/ D 0; r > 0: (4.7.10)  U .r/C U .r/C r 4ε0 „2 r „2 r2 To solve this equation, we use P .r/ D rU.r/;

(4.7.11)

0

0

P .r/ D U.r/ C rU .r/; 00

0

(4.7.12)

0

00

0

00

P .r/ D U .r/ C U .r/ C rU .r/ D 2U .r/ C rU .r/;

(4.7.13)

such that we have (r > 0) 2 P 00 .r/ : U 00 .r/ C U 0 .r/ D r r Using the abbreviation aB D P 00 .r/ C

4ε0 „2 m e2

(4.7.14)

(Bohr’s atomic radius), we find that

 2 2E 4ε0 l.l C 1/  P .r/ D 0: C  aB r e 2 aB r2

(4.7.15)

Next, we introduce the Rydberg energy ER D

„2 m e4 D 2 2 2„ .4ε0 /2 2maB

(4.7.16)

and multiply (4.7.15) by aB2 such that aB2 P 00 .r/ C

 2a

B

r

C

E a2 l.l C 1/  P .r/ D 0:  B 2 ER r

(4.7.17)

Q We p introduce dimensionless coordinates by % D r=aB , P .%/ D P .r/ and  D E=ER (observe that E < 0). This yields   2 l.l C 1/ d2 Q 2 Q P .%/ D 0;  P .%/ C   d%2 % %2

% > 0:

(4.7.18)

4 Scalar Spherical Harmonics in R3

168

To solve this equation, we set (% > 0): PQ .%/ D V .%/%lC1 exp.%/:

(4.7.19)

We obtain an equation for V given by   l C1 2 d2 d V .%/   C V .%/ .1  .l C 1// D 0: V .%/ C 2 d%2 d% % %

(4.7.20)

This equation is multiplied by % and we set y D 2%, VQ .y/ D V .%/ such that d2 d y 2 VQ .y/ C .2l C 2  y/ VQ .y/ C dy dy



 1  l  1 VQ .y/ D 0: 

(4.7.21)

Comparing this equation to the differential equation of the Laguerre polynomials .˛/ Ln (see Sect. 3.7) given by y

d d2 .˛/ Ln .y/ C .˛ C 1  y/ L.˛/ .y/ C nL.˛/ n .y/ D 0; 2 dy dy n

y > 0;

(4.7.22)

we see that ˛ D 2l C 1 and that we have a polynomial solution if and only if 1  l  1 D nr ; 

nr 2 N0 :

(4.7.23)

The solution is then given by .2lC1/ VQ .y/ D C2 L.2lC1/ .y/ D C2 Lnl1 .y/ nr

(4.7.24)

with n D nr C l C 1 D 1=, n 2 N. This means that for 1= only values in N are allowed which gives the discretization of energy by 1 En 1 e2 m e4 Dn , D 2 , En D  D  ;  ER aB 4ε0 2n2 2„2 .4ε0 /2 n2 Resubstituting V into P , then P into U , and finally U into

n 2 N: (4.7.25)

, we obtain:

U.r/ D 1r P .r/ D 1r PQ .%/ D 1r V .%/%lC1 exp.%/ 1 .2lC1/ 2r /. arB /lC1 exp. nar B / D C2 Lnl1 . na B r .2lC1/

2r 2r l D C2 a1B . n2 /l Lnl1 . na /. na / exp. nar B /: B B „ ƒ‚ … CQ2

(4.7.26)

4.8 Exercises

169

n=1, l=0 n=2, l=0 n=2, l=1 n=3, l=0 n=3, l=1 n=3, l=2

0.5

0.4

0.3

0.2

0.1

0

0

5

10

15

20

25

30

2 Fig. 4.9 Radial probability densities r 2 Un;l .r/ for n D 1; 2; 3 and corresponding l

We get after normalizing k kL2 .R3 / D 1: n;l;m .x/

D C Un;l .r/Fl;m ./ (4.7.27) s       2r l .2lC1/ 2r r .n  l  1/Š 2 1 exp  Yl;m ./: D 3=2 Lnl1 .n C l/Š n2 naB naB naB a B

n 2 N is called the main quantum number, l D 0; : : : ; n  1 is called the angular momentum quantum number and m D l; : : : ; l is called the magnetic quantum number. For n D 1, we get for the energy E1 the well-known ionization energy 1 of hydrogen, i.e., 4ε E1 D 13:6 eV (see Fig. 4.9 for an illustration of the radial 0 parts).

4.8 Exercises (Low Discrepancy Method, Locally Supported Wavelets, Up Function, Anharmonic Functions for the Ball, Fast Multipole Method, Wigner Matrices, Quaternionic Generation of Spherical Harmonics) The following exercises discuss some important features in constructive approximation and numerical analysis. The essential tool is the theory of spherical harmonics of this chapter.

4 Scalar Spherical Harmonics in R3

170

Numerical Integration by Partitioning the Sphere Partitioning the sphere provides a numerical integration rule, which is useful for application whenever the high complexity of a (geo-)physical process demands simply structured integration techniques. Exercise 4.8.1. Let the sphere S2 be partitioned into a set T of N open domains Tj  S2 , j D 1; : : : ; N , such that Tj \ Tk D ;; N [

k 6D j;

Tj D S2 ;

(4.8.1) (4.8.2)

j D1

where Tj is the closure of Tj . If we choose a point j 2 Tj for j D 1; : : : ; N , then we are led to the approximate integration rule QN .F / D

N X

kTj kF .j /;

(4.8.3)

j D1

where kTj k denotes the area of Tj . Show that ˇZ ˇ N X ˇ ˇ ˇ kTj kF .j /ˇˇ  4 max sup jF ./  F ./j: ˇ 2 F ./ dS./  j D1;:::;N S

(4.8.4)

;2Tj

j D1

Exercise 4.8.2. Let F be Lipschitz-continuous with Lipschitz constant CF . Prove that ˇ ˇZ N X ˇ ˇ ˇ ˇ F ./ dS./  kT kF . / diam .Tj /; (4.8.5) j j ˇ  4 CF max ˇ S2

j D1;:::;N

j D1

where diam.Tj / is the diameter of Tj , i.e., diam.Tj / D sup j  j:

(4.8.6)

;2Tj

Spherical Low Discrepancy Method The spherical low discrepancy method provides a selection principle for the quality of nodal sets in numerical integration with respect to their equidistribution over the sphere. Definition 4.8.3. Assume that F is of class C.2/ .S2 /. Let XN D f1N ; : : : ; NN g be a system of points on S2 . Let

4.8 Exercises

171

"XN .F / D .F /  XN .F /;

(4.8.7)

where .F / is the integral mean of F on S2 , i.e., .F / D

1 4

Z S2

F ./ dS./;

(4.8.8)

and XN .F / is the arithmetic sum with respect to XN ; XN .F / D

N 1 X F .jN /: N j D1

(4.8.9)

Exercise 4.8.4. Verify that for F 2 C.2/ .S2 /; j"XN .F /j  D.XN / V .F /;

(4.8.10)

where the L2 -discrepancy D.XN / of the system of points XN is given by 11=2 N N X   X 1 D.XN / D @ 2 G . /2 I iN  jN A N i D1 j D1 0

(4.8.11)

using the Green function with respect to . /2 of Remark 4.6.6. The L2 -variance of F reads 1=2 Z  2 j F ./j dS./ : (4.8.12) V .F / D S2

Definition 4.8.5. A sequence fXN g, XN D f1N ; : : : ; NN g  S2 , is called equidistributed or an equidistribution, if .F / D lim XN .F / N !1

(4.8.13)

holds for all F 2 C.2/ .S2 /. Exercise 4.8.6. Prove that fXN g is equidistributed if and only if lim D.xN / D 0:

N !1

(4.8.14)

Note that G.. /2 /I ; ) is explicitly available in accordance with Remark 4.6.6. Exercise 4.8.7. Consider the following point sets XN  S2 . (a) Depending on a parameter 2 N the spherical coordinates of the Reuter grid (cf. Reuter 1982) are given by

4 Scalar Spherical Harmonics in R3

172 3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

1

2

3

4

5

0

6

0

1

1

0.5

0.5

0

0

−0.5

−0.5

−1 1

−1 1 1

0.5 0.5

0

0

−0.5 −1

−1

1

2

3

4

5

6

1

0.5 0.5

0

−0.5

0

−0.5 −1

−1

−0.5

Fig. 4.10 Coordinates and points of the Reuter grid of Exercise 4.8.7(a) for D 25 (left column) and the Hammersley sequence of Exercise 4.8.7(b) (right column). The number of points is N D 786 in both cases

.#0 ; '0;1 / D .0; 0/ (North Pole); (4.8.15)   i D 1; : : : ;  1; j D 1; : : : ; i ; (4.8.16) .#i ; 'i;j / D i  ; .j  12 / 2

i .# ; ' ;1 / D .; 0/ where

(South Pole);

(4.8.17)

$

1 %    cos. /cos2 .#i /

:

i D 2 arccos sin2 .#/

(4.8.18)

This leads to a system of points XN. / with N. /  2 C 4 2 . See Fig. 4.10 (left column) for an illustration. (b) For a given N 2 N define the system of points by their spherical coordinates  .#n ; 'n / D arccos.2˚2 .n/  1/; 2 n1 ; N

n D 1; : : : ; N;

(4.8.19)

4.8 Exercises

173

where we use the mapping ˚p .n/ D

s X

aj p j 1

(4.8.20)

j D0

with the coefficients aj 2 f0; : : : ; p  1g resulting from the representation n1 D

s X

aj p j

with p 2 f2; 3; : : :g:

(4.8.21)

j D0

The sequence of these point systems is called a Hammersley sequence. See Fig. 4.10 (right column) for an illustration. (c) For a given number of points N 2 N the following system of points is generated by their spherical coordinates .#n ; 'n / D .arccos.2˚3 .n/  1/; 2˚2 .n// ;

n D 1; : : : ; N;

(4.8.22)

where we use the mappings ˚p .n/ of part (b) with p D 2 and p D 3. The resulting sequence of point systems is called a Corput–Halton sequence. See Fig. 4.11 (left column) for an illustration. (d) Given N 2 N, N > 1, we define the spiral grid XN using the spherical coordinates .#1 ; '1 / D .; 0/ (South Pole); (4.8.23)     .#j ; 'j / D arccos.hj /; 'j 1 C p3:6 .1  h2j /1=2 .mod .2// ; (4.8.24) N

.#N ; 'N / D .0; 0/

(North Pole);

(4.8.25)

2 where hj D 1 C 2j and j D 2; : : : ; N  1. See Fig. 4.11 (right column) for N 1 an illustration. (e) Given N 2 N, N > 3, we introduce the improved spiral grid XN using the spherical coordinates

(South Pole); .#1 ; '1 / D .; 0/   .#j ; 'j / D arccos.hj /; 'j 1 C p3:6

2 N rj Crj 1

.#N ; 'N / D .0; 0/



 .mod .2// ;

(North Pole);

(4.8.26) (4.8.27) (4.8.28)

where /2 hj D  1 C 2k.j with k.j / D .1  N 1 q rj D 1  h2j ; r1 D 0;

1 /.j N 3

C 1/ C

1 N C1 ; 2 N 3

and j D 2; : : : ; N  1. See Fig. 4.12 (left column) for an illustration.

(4.8.29) (4.8.30)

4 Scalar Spherical Harmonics in R3

174 3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

1

2

3

4

5

0

6

1

1

0.5

0.5

0

0

−0.5

−0.5

-1 1

−1 1 0.5

1

0

1

−1 −1

4

5

6

1 0.5

0

0

−0.5

3

0.5

0.5

0

2

0

−0.5

−0.5

−0.5 −1

−1

Fig. 4.11 Coordinates and points of the Corput–Halton sequence of Exercise 4.8.7(c) (left column) and the spiral grid of Exercise 4.8.7(d) (right column). The number of points is N D 786 in both cases

(f) Depending on the parameter Ns 2 N we construct the so-called HEALPix grid consisting of N D 12Ns2 points on the sphere in three steps. At first the northern polar cap is considered:   .#j ; 'j / D arccos 1 

k2 3Ns2



   ; 2k j  2k.k  1/  12 ;

(4.8.31)

where j D 1; : : : ; Ncap D 2Ns .Ns  1/ and $r kD

j 2



q

% b j2 c

C 1:

(4.8.32)

For the second step the remaining northern hemisphere and the equator are considered:

4.8 Exercises

175

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

1

2

3

4

5

0

6

1

1

0.5

0.5

0

0

−0.5

−0.5

−1 1

−1 1

0

1

0.5

−1 −1

3

4

5

6

1 0.5

0

0

−0.5

2

0.5

0.5

0

1

0

−0.5

−0.5

−1 −1

−0.5

Fig. 4.12 Coordinates and points of the improved spiral grid of Exercise 4.8.7(e) (left column) and the HEALPix grid of Exercise 4.8.7(f) for Ns D 8 (right column). The number of points is N D 786 for the improved spiral grid and N D 768 in the HEALPix case

where j D 1; : : : ; Nbelt



   m ; 2N .j  1/.mod .4N ; // C 1  s 2 s (4.8.33) D 4Ns .Ns C 1/, m D .k  Ns C 1/.mod 2/, and

  .#Ncap Cj ; 'Ncap Cj / D arccos 43 

kD

2k 3Ns

j

j 1 4Ns

k

C Ns :

(4.8.34)

Finally, the points of the southern hemisphere are constructed by .#N j C1 ; 'N j C1 / D .  #j ; 'j /;

(4.8.35)

where j D 1; : : : ; Nsouth D Ncap C Nbelt  4Ns . See Fig. 4.12 (right column) for an illustration. Calculate and compare the discrepancy D.XN / for each of the point systems. Generate uniformly distributed random points for an additional comparison.

4 Scalar Spherical Harmonics in R3

176

Remark 4.8.8. For more details on the point systems in (a)–(c) see Freeden et al. (1998). The system of part (a) goes back to Reuter (1982). The idea of the construction using the mappings ˚p goes back to van der Corput (1935a,b). For the theory of uniform distributions in Euclidean spaces we refer to Kuipers and Niederreiter (1974) and Niederreiter (1992). For the spiral grids we refer to Rakhmanov et al. (1994) and Saff and Kuijlaars (1997), for the improvement of part (e) see Thomson (2007). The HEALPix (Hierarchical Equal Area isoLatitude Pixelization) point system is related to cosmic microwave background experiments and the reader is referred to G´orski et al. (2005) for a detailed introduction. Note that the number of points in parts (a) and (f) cannot take all values of N.

Locally Supported Wavelets on the Sphere In constructive approximation locally supported functions are nothing new, having been discussed already by Haar (1910). The primary importance of spherically locally supported Haar kernels is the generation of an entire “basis family” by means of two operations, namely dilation and rotation. In other words, an entire set of approximations is available from one locally supported “Haar mother kernel”. Exercise 4.8.9. Let fLh gh2.1;1/  L2 .Œ1; 1/ be a non-negative, Œh; 1-locally supported family. Suppose that fLh F gh2.1;1/ given by the spherical convolution Z .Lh F /./ D Lh .  /F ./ dS./ (4.8.36) S2

is an approximate identity generated by fLh gh2.1;1/ , i.e., lim kF  Lh F kC.0/ .S2 / D 0

h!1

(4.8.37)

for all F 2 C.0/ .S2 /. Show that Z

!

hQ

jF ./  Lh F ./j  2

Lh .t/ dt h

 Z C 2



1 hQ

sup jF ./  F ./j 2S2 hhQ

Lh .t/ dt

sup jF ./  F ./j(4.8.38) 2S2 Q h1

holds for every hQ 2 Œh; 1, for every F 2 C.0/ .S2 /, and for all  2 S2 . If F is additionally Lipschitz-continuous with Lipschitz-constant CF , i.e., jF ./  F ./j  CF j  j D CF

p 1    ;

(4.8.39)

4.8 Exercises

177

then kF  Lh F kC.0/ .S2 / Z

p  2 2 CF

hQ

! 1

Z

Lh .t/ dt .1  h/ 2 C

h

1 hQ



(4.8.40) !

Q 12 Lh .t/ dt .1  h/

for every hQ 2 Œh; 1. .k/

Exercise 4.8.10. The (smoothed) Haar scaling function fLh gh2.1;1/ , k 2 N0 , is .k/ .k/ defined by Lh W Œ1; 1 ! R, t 7! Lh .t/; .k/

Lh .t/ D

( D

1

.k/

.k/ 1 Bh .t/

2

where .k/ Bh .t/

R1

dt

Bh .t/;

0

if t 2 Œ1; h/;

.t h/k .1h/k

if t 2 Œh; 1:

(4.8.41)

(4.8.42)

(a) Show that the Legendre coefficients .Bh /^ .n/ D 2 .k/

Z

1 1

.k/

Bh .t/Pn .t/ dt;

n 2 N0 ;

(4.8.43)

satisfy the recursion relation for n 2 N; 2n C 1 k C 1  n .k/ ^ .k/ h.Bh /^ .n/ C .B / .n  1/; nCkC2 nCk2 h (4.8.44) where the initial values are .Bh /^ .n C 1/ D .k/

1h 6 0; D kC1   1h 1h .k/ ^ 1 : .Bh / .1/ D 2 kC1 kC2 .Bh /^ .0/ D 2 .k/

(b) Verify that

ˇ ˇ   3 ˇ .k/ ^ ˇ ˇ.Lh / .n/ˇ D O .n.1  h// 2 k

(4.8.45) (4.8.46)

(4.8.47)

for n ! 1. (c) Show that

.k/

Lh

C.0/ .Œ1;1/

.k/

D Lh .1/ D

1 kC1 ; 2 1  h

k 2 N0 :

(4.8.48)

4 Scalar Spherical Harmonics in R3

178 .k/

(d) Verify that for k  2 the kernels Lh Lipschitz-constant takes the value CL.k/ h



.k/ 0

D Lh

C.0/ .Œ1;1/

o n .k/ (e) Verify that Lh

h2.1;1/

are Lipschitz-continuous and the

0  1 k.k C 1/ .k/ D Lh .1/ D : 2 .1  h/2

(4.8.49)

is an approximate identity as introduced in (4.8.37).

Exercise 4.8.11. Suppose fhj gj 2N0  .1; 1 is a strict monotonically increasing .k/ sequence with lim hj D 1. Let fj gj 2N0 be the spherical difference wavelet j !1

given by .k/

j

.k/

.k/

D Lhj C1  Lhj ;

k 2 N0 :

(4.8.50)

Set Z   .k/ .k/ .k/ Lhj .  /F ./ dS./; Tj .F I / D Lhj F ./ D

(4.8.51)

Z   .k/ .k/ .k/ Rj .F I / D j F ./ D j .  /F ./ dS./:

(4.8.52)

S2

S2

Prove that

.k/

lim kF  TJ .F I  /kC.0/ .S2 / D 0

(4.8.53)

J !1

holds for all F 2 C.0/ .S2 /, where .k/

.k/

.k/

.k/

TJ .F I  / D TJ 1 .F I  / C RJ 1 .F I  / D TJ0 .F I  / C

J 1 X

.k/

Rj .F I  / (4.8.54)

j DJ0

for all J; J0 2 N0 and 0  J0 < J  1. Exercise 4.8.12. Show that, for hj D 1  2j , ( .k/ Lhj .t/

D

if t 2 Œ1; 1  2j /;

0 1 .k 2

C 1/2j.kC1/.t  1 C 2j /k

if Œ1  2j ; 1;

(4.8.55)

and

.k/

j .t/ D

8 ˆ 0 ˆ ˆ ˆ ˆ 1 if not mentioned otherwise (see Fig. 4.13 for an illustration varying the parameters h and ). Exercise 4.8.15. Show that for n ! 1; L^ h; .n/ Hint: Use

R1 x

Z D 2

1

1

1=2

Pn1 .t/ dt D Cn

3

Lh; .t/Pn .t/ dt D O.n 2  /: 1=2

.x/, where Cn

(4.8.63)

is the Gegenbauer polynomial.

Exercise 4.8.16. Verify that for all  >  12 ; 1 X 2 2n C 1  ^ Lh; .n/ < 1: 4 nD0

(4.8.64)

Exercise 4.8.17. Show that .2/ Lh; .

Z  / D .Lh; Lh; / .  / D

S2

Lh; .  /Lh; .  / dS./

(4.8.65)

.2/

possesses the support supp.Lh; / D f 2 S2 W 2h  1      1g. Exercise 4.8.18. Prove that for n ! 1; Z 1 ^  .2/ .2/ Lh; .t/Pn .t/ dt D O.n32 /: Lh; .n/ D 2 1

(4.8.66)

4.8 Exercises

181 1.4

0.4

−2/3 −1/3 0 1 2

0.35 0.3

−0.99 0 1

1.2 1

0.25

0.8

0.2

0.6

0.15 0.4 0.1 0.2

0.05 0

0

−0.05

−0.2 −3

−2

−1

0

1

2

3

−3

−2

−1

0

1

2

3

.2/

Fig. 4.14 Iterated smoothed Haar functions Lh; .cos.#// for h D 0:2 and  D 2=3; 1=3; 0; 1; 2 (left) and the up function Uph; .cos.#// for h D 0:5 and  D 0:99; 0; 1 (right)

Exercise 4.8.19. Suppose that h 2 .1; 1/. Prove that the following statements hold true: .2/

(a) If  > 1, then  7! Lh; .  /,  2 S2 , is of class L2 .S2 / for each  2 S2 . .2/

(b) If  >  12 , then  7! Lh; .  /,  2 S2 , is of class C.0/ .S2 / for each  2 S2 . .2/

(c) If  > k1 , k 2 N0 , then  7! Lh; .  /,  2 S2 , is of class C.k/ .S2 / for each 2  2 S2 (Fig. 4.14).

Exercise 4.8.20. Suppose that h 2 .1; 1/ and  > 1. We set '0 D arccos.h/ and 'k D 2k '0 , hk D cos. '2k /, k 2 N. Then, the up function Uph; is defined by .2/ .2/ the infinite spherical convolution of Lh1 ; ; Lh1 ; ; : : :, i.e., .2/

1

*

.2/

Uph; D Lh1 ; Lh2 ; : : : D

k D1

.2/

Lhk ;

(4.8.67)

Verify the following properties: (a) The Legendre symbols of the up function fulfill for every n 2 N0 ; 

Uph;

^

.n/ D

1  Y

.2/

^

Lhk ;

.n/

(4.8.68)

kD1

and lim

hD!1

 ^ Uph; .n/ D 1:

Even more, the speed of convergence fulfills for n ! 1;  ^ Uph; .n/ D O.nk / for every k 2 N.

(4.8.69)

(4.8.70)

4 Scalar Spherical Harmonics in R3

182

(b) Uph; W Œ1; 1 ! R is locally supported with supp Uph; D Œh; 1. (c) The function  7! Uph; .  /,  2 S2 , is of class C.1/ .S2 / for all  2 S2 . (d) Uph; W Œ1; 1 ! R admits the uniformly convergent orthogonal expansion in terms of Legendre polynomials Uph; D

1 X 2n C 1 

4

nD0

Uph;

^

.n/Pn ;

(4.8.71)

 ^ where Uph; .0/ D 1 and, for all n 2 N0 , 1  ^ Y  ^ .2/ 0  Uph; .n/ D Lhk ; .n/  1:

(4.8.72)

kD1

(e) For all t 2 Œ1; 1, we have the estimate 0  Uph; .t/  Uph; .1/ D

1 X 2n C 1  nD0

4

Uph;

^

.n/:

Exercise 4.8.21. Show that for all  > 1 and F 2 L2 .S2 /;

lim Up F  F 2 2 D 0: h!1

h;

L .S /

(4.8.73)

(4.8.74)

j

Exercise 4.8.22. The scaling function ˚h; W Œ1; 1 ! R is defined by j

˚h; D

1

*

kDj

.2/

Lhk ; ;

j 2 N:

(4.8.75)

j

(a) Show that supp ˚h; D Œhj 1 ; 1. (b) Verify the refinement equation: j C1

.2/

j

˚h; Lhj ; D ˚h; ;

j 2 N:

(4.8.76)

Exercise 4.8.23. The scale spaces Vj , j 2 N, are given by n o j Vj D ˚h; F W F 2 L2 .S2 / :

(4.8.77)

Verify that the sequence of scale spaces fVj gj 2N constitutes a multi-resolution analysis of L2 .S2 / : (a) Every Vj , j 2 N, is a linear subspace with Vj  C.1/ .S2 /  L2 .S2 /. (b) The spaces Vj are nested, i.e., Vj  Vj C1 , j 2 N.

4.8 Exercises

183

14

j=1 j=2 j=3

12

10

j=1 j=2

8

10 6

8

4

6 4

2

2 0

0 −2

−3

−2

−1

0

1

2

3

−2

−3

−2

−1

0

1

2

3

j

Fig. 4.15 The scaling functions ˚h; .cos.#// for j D 1; 2; 3,  D 0:9, and h D 1 (left) and the corresponding wavelets

j h; .cos.#//

for j D 1; 2,  D 0:9, and h D 1 (right)

(c) For the intersection and the union of all spaces it holds that 1 \

Vj D V1

and

j D1

1 [

kk2L .S2 /

D L2 .S2 /:

Vj

(4.8.78)

j D1

(d) There exists a family fKj gj 2N of L2 .S2 /-kernels such that (Fig. 4.15) Vj D fKj F W F 2 Vj C1 g:

(4.8.79)

Exercise 4.8.24. We now set the spherical difference wavelets j

and

j C1

j

h; D ˚h;  ˚h;

(4.8.80)

n o j Wj D h; F W F 2 L2 .S2 / :

(4.8.81)

Prove that there exist zonal functions Zj for all j 2 N that satisfy the scale relation j C1

j

h; Zj D h; :

(4.8.82)

Show that these functions Zj are given by the Legendre coefficients ^  .2/ ^ 1  Lhj ; .n/  .2/ Lhj C1 ; .n/; Zj^ .n/ D ^  .2/ 1  Lhj C1 ; .n/ where Zj^ .0/ is arbitrary.

n 2 N;

(4.8.83)

4 Scalar Spherical Harmonics in R3

184

Remark 4.8.25. Note that for numerical calculations Zj .0/ should be chosen in such a way that the kernel Zj is strongly (space) localized. This can be achieved by setting Zj .1/ D 0 such that the Legendre coefficient of order 0 should be chosen in accordance with Zj^ .0/ D 4

1 X 2n C 1 nD1

4

.1/n Zj^ .n/:

(4.8.84)

Remark 4.8.26. The up function enables a multi-resolution analysis on S2 by use of locally supported zonal kernels. The infinite smoothness of the scaling functions allows us to apply up functions as trial functions in a large class of differential equations (see Rvachev 1990) and integral equations of satellite technology (see Freeden 1999). More details about the multi-resolution analysis by spherical up functions can be found in Freeden and Schreiner (2006).

Function Systems for the Ball B3R Exercise 4.8.27. Prove that the space L2 .B3R / can be orthogonally decomposed as L2 .B3R / D Harm.B3R / ˚ Anharm.B3R /; ?

(4.8.85)

3

where Anharm.B3R / D Harm.B3R / L .BR / is the space of anharmonic functions, i.e., functions that are not harmonic in the ball of radius R in three dimensions B3R . 2

I gn;m2N0 Ij Dn;:::;n forms a Exercise 4.8.28. Show that the function system fGm;n;j 2 3 complete orthonormal system in L .BR /, where

r I Gm;n;j

D

    n 4m C 2n C 3 .0;nC1=2/  jxj2 jxj x 2 P  1 Y n;j jxj ; m R R2 R3

x 2 B3R ;

(4.8.86) .0;nC1=2/ of degree m 2 N0 (see Sect. 3.3) and the with the Jacobi polynomial Pm orthonormal spherical harmonics Yn;j , n 2 N0 , j D n; : : : ; n. II Exercise 4.8.29. Show that the function system fGm;n;j gn;m2N0 Ij Dn;:::;n also forms 2 3 a complete orthonormal system in L .BR /, where

II Gm;n;j

8q     .0;2/ jxj x < 2mC3 2 P  1 Y m n;j 3 R jxj D q R : 3 R3

.0;2/

;

x 2 B3R n f0g;

;

x D 0;

(4.8.87)

with the Jacobi polynomial Pm of degree m 2 N0 (see Sect. 3.3) and the orthonormal spherical harmonics Yn;j , n 2 N0 , j D n; : : : ; n.

4.8 Exercises

185

I Exercise 4.8.30. Verify that the system fGm;n;j gn;m2N0 Ij Dn;:::;n can be split into harmonic and anharmonic functions. II Exercise 4.8.31. Show that the functions of fGm;n;j gn;m2N0 Ij Dn;:::;n are not continuous in x D 0 for n > 0. I the radial and angular parts of Remark 4.8.32. Note that in contrast to Gm;n;j II Gm;n;j are decoupled which can be exploited numerically (cf. Michel 2010 and the references therein). II Exercise 4.8.33. Show that for the product series of the functions Gm;n;j of (4.8.86)

K.x; y/ D

1 X n 1 X X

II II K ^ .m; n/Gm;n;j .x/Gm;n;j .y/

(4.8.88)

mD0 nD0 j Dn

with x; y 2 B3R n f0g it holds that ! 1    X am .2m C 3/ .0;2/  jxj jyj 2 R  1 Pm.0;2/ 2 R  1 Pm K.x; y/ D 3 R mD0 2n C 1  x Pn jxj  bn  4 nD0 1 X

y jyj

!  ;

(4.8.89)

where Pn denotes the Legendre polynomial of degree n 2 N0 and K ^ .m; n/ D am bn with fam gm2N0 , fbn gn2N0 real sequences. I Exercise 4.8.34. Show that the product series of the functions Gm;n;j of (4.8.86)

K.x; y/ D

1 X n 1 X X

I I K ^ .m; n/Gm;n;j .x/Gm;n;j .y/;

x; y 2 B3R ;

(4.8.90)

mD0 nD0 j Dn

(a) Converges in L2 .B3R  B3R / if and only if 1 1 X X  2 n K ^ .n; m/ < 1;

(4.8.91)

mD0 nD0

(b) Converges uniformly on B3R  B3R if 1 1 X X ˇ ^ ˇ .n C m C 12 /2m ˇK .n; m/ˇ n.2m C n/ < 1: .mŠ/2 mD0 nD0

(4.8.92)

4 Scalar Spherical Harmonics in R3

186

II Exercise 4.8.35. Show that the product series of the functions Gm;n;j of (4.8.87)

K.x; y/ D

1 X n 1 X X

II II K ^ .m; n/Gm;n;j .x/Gm;n;j .y/;

x; y 2 B3R n f0g;

mD0 nD0 j Dn

(4.8.93) (a) Converges in L2 .B3R  B3R / if and only if the sequence fK ^ .n; m/gn;m2N0 fulfills 1 1 X X  2 n K ^ .n; m/ < 1;

(4.8.94)

mD0 nD0

(b) Converges uniformly on B3R  B3R if the sequence fK ^.n; m/gn;m2N0 fulfills 1 1 X X ˇ ˇ ^ ˇK .n; m/ˇ nm5 < 1:

(4.8.95)

mD0 nD0

Remark 4.8.36. The system of (4.8.86) goes back to Ballani et al. (1993) and Dufour (1977) (see also Freeden and Michel 2004; Michel 1999) and the second system of (4.8.87) is due to Tscherning (1996). Both systems can be used to construct splines and wavelets for B3R in the same way as known from the sphere S2 . These functions find many applications in modeling the Earth’s interior, e.g., in gravimetry, normal mode variations or seismic body wave tomography. For further details we refer to Michel (2010, 2012) and the references therein.

The Fast Multipole Method The fast multipole method for the Laplace equation in three dimensions (see, e.g., Greengard and Rokhlin 1988) allows the computation of the interaction of many particles which is described by the fundamental solution of the Laplace equation as a sum of the form N X j D1

aj

1 ; jxj  yj

(4.8.96)

which is evaluated in O.N / points y. The direct approach takes a numerical effort of O.N 2 / operations which is reduced by the method to O.N /. The main idea of the method consists for one part of the subdivision of the computational domain into hierarchical sets of nested cubes which are arranged in an octal tree data structure. The other part is the possibility to consider the interaction of cubes instead of each single point by summarizing all points of

4.8 Exercises

187

one cube and representing their information by a multipole expansion. Such an expansion results from the decomposition of the single pole in terms of inner and outer harmonics. For the interaction of the different cubes two different paths are taken: the direct way, i.e., all points in one cube interact with all points in another cube, and the approximate way, i.e., by translation of the expansion coefficients. The key is to apply the approximation as often as possible and on the coarsest possible level of the tree. The direct evaluation is only to be used to the closest boxes, where the approximation is not applicable. At first, we define outer and inner harmonics based on the orthonormal system of spherical harmonics defined in Example 4.3.32 (cf. Epton and Dembart 1995 or Gutting 2008). Definition 4.8.37 (Complex Outer Harmonics). Let n 2 N0 , m 2 Z, n  m  n. Then, the complex-valued function r On;m .x/ D

4 p 1 .n C m/Š.n  m/Š nC1 Yn;m 2n C 1 jxj

D .1/m



x jxj



.n  m/Š Pn;m .cos.#//eim' jxjnC1

(4.8.97) (4.8.98)

denotes the outer harmonic of degree n and order m, where .#; '/ are the spherical x and x ¤ 0. Yn;m is the complex-valued spherical harmonic coordinates of jxj of degree n and order m of Example 4.3.32 and Pn;m is the associated Legendre function of (4.3.108). Definition 4.8.38 (Complex Inner Harmonics). Let n 2 N0 , m 2 Z, n  m  n. Then, the complex-valued function r In;m .x/ D

4 1 jxjn Yn;m p 2n C 1 .n C m/Š.n  m/Š

D .1/m



x jxj



jxjn Pn;m .cos.#//eim' .n C m/Š

(4.8.99) (4.8.100)

denotes the inner harmonic of degree n and order m, where x 2 R3 . .#; '/, Yn;m , Pn;m are the same as in Definition 4.8.37. Exercise 4.8.39. Show the symmetries On;m .x/ D .1/n On;m .x/; On;m .x/ D .1/m On;m .x/;

In;m .x/ D .1/n In;m .x/; In;m .x/ D .1/m In;m .x/;

(4.8.101) (4.8.102)

where n 2 N0 , m 2 Z with n  m  n, x 2 R3 n f0g for the outer harmonics and x 2 R3 in case of the inner harmonics.

4 Scalar Spherical Harmonics in R3

188

Exercise 4.8.40. Prove that for jxj > jyj the following expansion holds: n 1 X X 1 In;m .y/On;m .x/; D jx  yj nD0 mDn

(4.8.103)

and for jx  x0 j > jy  x0 j; n 1 X X 1 D In;m .y  x0 /On;m .x  x0 /; jx  yj nD0 mDn

(4.8.104)

where the expansion center is x0 2 R3 . Exercise 4.8.41. Consider the following differential operators which are closely related to the so-called ladder operators in quantum mechanics (see, e.g., Biedenharn and Louck 1981; Edmonds 1964; Var˘salovi˘c et al. 1988): @C D

@ @ Ci ; @x1 @x2

(4.8.105)

@ D

@ @ i ; @x1 @x2

(4.8.106)

@z D

@ : @x3

(4.8.107)

Show that these operators possess the following representation in terms of spherical coordinates:   cos.#/ @ i @ @ @C D ei' sin.#/ C ; (4.8.108) C @r r @# r sin.#/ @'   cos.#/ @ i @ @ i'  ; (4.8.109) @ D e sin.#/ C @r r @# r sin.#/ @' @z D cos.#/

sin.#/ @ @  : @r r @#

(4.8.110)

Exercise 4.8.42. Prove that the complex outer harmonic of degree n 2 N0 and order m D n; : : : ; n can be written as 8 jyj. Then, the outer harmonic of degree n 2 N0 and order m 2 Z, n  m  n, at x  y can be expanded in terms of inner and outer harmonics as follows: 0

1 n X X

On;m .x  y/ D

In0 ;m0 .y/OnCn0 ;mCm0 .x/

(4.8.112)

In0 n;m0 m .y/On0 ;m0 .x/:

(4.8.113)

n0 D0 m0 Dn0 0

1 n X X

D

n0 Dn m0 Dn0

Note that by convention In;m D 0 if jmj > n which occurs in the second formulation. Exercise 4.8.44. Verify the translation theorem for inner harmonics: Let x; y 2 R3 . Then, the inner harmonic of degree n 2 N0 and order m 2 Z with n  m  n at x  y can be expanded in a finite sum of inner harmonics 0

In;m .x  y/ D

n n X X

0

.1/n In0 ;m0 .y/Inn0 ;mm0 .x/:

(4.8.114)

n0 D0 m0 Dn0

Note again that by convention In;m D 0 if jmj > n. Consider the translation of an outer harmonics expansion such as F .x/ D

n 1 X X

Fx^;O .n; m/On;m .x  x0 /; 0

(4.8.115)

nD0 mDn

which is uniformly convergent for x 2 R3 n B3r0 .x0 /, where the radius of the ball around x0 is r0 > 0. The expansion coefficients Fx^;O .n; m/ 2 C are called 0 multipole coefficients and denote the complex coefficients corresponding to the outer harmonic of degree n and order m with the center x0 in the expansion. They usually result from an expansion of a kernel like, e.g., the single pole in (4.8.104) (for more details see, e.g., Beatson and Greengard 1997; Epton and Dembart 1995; Greengard and Rokhlin 1997; Gutting 2008). Exercise 4.8.45. Show that the expansion of the function F of (4.8.115) can be shifted to 0

F .x/ D

1 n X X n0 D0 m0 Dn0

Fx^;O .n0 ; m0 /On0 ;m0 .x  x1 /; 1

(4.8.116)

4 Scalar Spherical Harmonics in R3

190

where x 2 R3 n B3r1 .x1 /  R3 n B3r0 .x0 / and the new expansion coefficients Fx^;O .n0 ; m0 / can be computed from the old ones via 1 0

Fx^;O .n0 ; m0 / 1

D

n n X X

Fx^;O .n; m/In0 n;m0 m .x0  x1 /: 0

(4.8.117)

nD0 mDn

Exercise 4.8.46. Prove that the expansion of the function F of (4.8.116) can be translated to 0

F .x/ D

1 n X X n0 D0 m0 Dn0

Fx^;I .n0 ; m0 /In0 ;m0 .x  x2 /; 2

(4.8.118)

where x 2 B3r2 .x2 / and B3r1 .x1 / \ B3r2 .x2 / D ;. The new expansion coefficients Fx^;I .n0 ; m0 / are related to (4.8.117) by 2 Fx^;I .n0 ; m0 / 2

D

n 1 X X

0

Fx^;O .n; m/.1/n Cm OnCn0 ;m0 m .x2  x1 /: 1

(4.8.119)

nD0 mDn

Remark 4.8.47. Since we are now dealing with an expansion in inner harmonics, the notation Fx^;I .n; m/ is used for the coefficient corresponding to the inner harmonic 2 of degree n and order m of the expansion of F with center x2 to distinguish the coefficients of (4.8.119) from those of (4.8.117). The coefficients Fx^;I .n; m/ are 2 called local coefficients corresponding to the inner harmonics expansion (a.k.a. local expansion) around x2 . Exercise 4.8.48. Show that the expansion of the function F of (4.8.118) can be translated to 0

F .x/ D

1 n X X n0 D0 m0 Dn0

Fx^;I .n0 ; m0 /In0 ;m0 .x  x3 /; 3

(4.8.120)

where x 2 B3r3 .x3 /  B3r2 .x2 / and the new expansion coefficients Fx^;I .n0 ; m0 / are 3 given by .n0 ; m0 / D Fx^;I 3

1 X n X nDn0

Fx^;I .n; m/Inn0 ;mm0 .x3  x2 /: 2

(4.8.121)

mDn

Remark 4.8.49. Note that we require the geometrical situation that B3r0 .x0 /  B3r1 .x1 / for Exercise 4.8.45. Moreover, we need that B3r1 .x1 / \ B3r2 .x2 / D ; in Exercise 4.8.46 as well as B3r2 .x2 / B3r3 .x3 / in Exercise 4.8.48.

4.8 Exercises

191

Remark 4.8.50. The three translation steps of Exercises 4.8.45–4.8.48 are the key ingredients of the fast multipole algorithm. Together they allow the fast interaction for points that are not located close to each other. The fast multipole method goes back to Greengard (1988), Greengard and Rokhlin (1987) and Rokhlin (1985) and has found many applications, also in geomathematics, e.g., for the solution of the oblique boundary value problem (see Gutting 2008, 2012). An overview can be found in, e.g., Beatson and Greengard (1997), Cheng et al. (1999), Greengard and Rokhlin (1997) and Gutting (2008). For further details on the translations we refer to Epton and Dembart (1995) and the references therein.

Wigner-D-Matrices Any two unit vectors ;  0 2 S2 are related by a rotation which can be described by a matrix R D R.˛; ˇ; / depending on the three Euler angles ˛, ˇ, and , i.e.,  D R 0 ;

or  0 D RT ;

(4.8.122)

where the matrix is given by R.˛; ˇ; / D 2 cos ˛ cos ˇ cos  sin ˛ sin 6 6sin ˛ cos ˇ cos C cos ˛ sin 4  sin ˇ cos

 cos ˛ cos ˇ sin  sin ˛ cos  sin ˛ cos ˇ sin C cos ˛ cos sin ˇ sin

(4.8.123) 3 cos ˛ sin ˇ 7 sin ˛ sin ˇ 7 5: cos ˇ

The Euler angles for a rotation of the coordinate system fx1 ; x2 ; x3 g ! fx10 ; x20 ; x30 g are defined as follows: (a) A rotation about the x3 -axis through an angle ˛ 2 Œ0; 2/, (b) A rotation about the new xQ 2 -axis through an angle ˇ 2 Œ0; , (c) A rotation about the new x30 -axis through an angle 2 Œ0; 2/. Alternatively, we find the same rotation as the following combination (using the same angles): (a) A rotation about the x3 -axis through an angle 2 Œ0; 2/, (b) A rotation about the initial x2 -axis through an angle ˇ 2 Œ0; , (c) A rotation about the initial x3 -axis through an angle ˛ 2 Œ0; 2/. Exercise 4.8.51. Prove that the inverse rotation is given by the Euler angles  , ˇ, ˛ and that it is also given by the Euler angles   , ˇ,   ˛.

4 Scalar Spherical Harmonics in R3

192

If we want to rotate the coordinate system in such a way that a point  whose spherical coordinates are # and ' lies on the "3 -axis afterwards, we use R D R.'; #; 0/. A spherical harmonic of degree n and order m evaluated at  0 with the spherical coordinates # 0 and ' 0 can be represented by a linear combination of spherical harmonics of the same degree at the point  with the spherical coordinates # and '; Yn;m . 0 / D

n X

n Dk;m .˛; ˇ; /Yn;k ./;

(4.8.124)

kDn

where the rotation is described in terms of the Euler angles ˛, ˇ, . Exercise 4.8.52. Let R1 , R2 be the rotations such that RT1  D "3 D RT2  0 , where ;  0 2 S2 with the corresponding spherical coordinates #; ' and # 0 ; ' 0 . Show that the rotation R with  0 D R can be described using the Euler angles ˛ D ' 0 , ˇ D # 0  # and D '. n .˛; ˇ; / 2 C of the linear combination (4.8.124) are the matrix The factors Dk;m entries of the complex Wigner rotation matrix to degree n. They can be decomposed into n n Dk;m .˛; ˇ; / D eik˛ dk;m .ˇ/eim :

(4.8.125)

This corresponds to a decomposition of the rotation into three rotations. The first n and the last of these three are obviously diagonal. The coefficients dk;m .ˇ/ 2 R that provide the matrix for the rotation around the x2 -axis have the following explicit representation (see, e.g., Edmonds 1964; Var˘salovi˘c et al. 1988) 1

n .ˇ/ D .1/nm ..n C m/Š.n  m/Š.n C k/Š.n  k/Š/ 2 dk;m

X  .1/s s

 2nmk2s  mCkC2s sin ˇ2 cos ˇ2 sŠ.n  m  s/Š.n  k  s/Š.m C k C s/Š

;

(4.8.126)

where the summation index s is extended over all integer values for which the factorials are non-negative. n Exercise 4.8.53. Show the following symmetries of the coefficients dk;m .ˇ/: n n n n .ˇ/ D .1/km dk;m .ˇ/ D .1/km dm;k .ˇ/ D dm;k .ˇ/; dk;m

(4.8.127)

n n n .ˇ/ D .1/km dk;m .ˇ/ D dm;k .ˇ/: dk;m

(4.8.128)

(See Var˘salovi˘c et al. 1988 for a complete overview of all symmetry relations.) n .ˇ/ and Exercise 4.8.54. Prove the following relation between the coefficients dk;m the Jacobi polynomials (see Sect. 3.3):

4.8 Exercises

193

 n dk;m .ˇ/ D k;m

sŠ.s C C /Š .s C /Š.s C /Š (

where k;m D

1=2        cos ˇ2 sin ˇ2 Ps. ;/ .cos.ˇ//; (4.8.129)

1 .1/

mk

;

m  k;

;

m < k;

(4.8.130)

and D jk  mj,  D jk C mj and s D n  12 . C /. Note that for the case that both k  m  0 and k C m  0 hold, we find  n .ˇ/ D .1/mk dk;m

.n C k/Š.n  k/Š .n C m/Š.n  m/Š

1=2 (4.8.131)

  kCm   km .km;kCm/ sin ˇ2  cos ˇ2 Pnk .cos.ˇ//: Remark 4.8.55. These coefficients are usually not calculated directly, but with the help of a recursion scheme (see, e.g., Biedenharn and Louck 1981; Edmonds 1964; Var˘salovi˘c et al. 1988; Zare 1988). We choose to go along with Choi et al. (1999) where a stable algorithm is developed that computes all the unitarian Wigner rotation matrices D n 2 C.2nC1/.2nC1/ for n D 1 up to a prescribed maximal degree from the rotation matrix of the coordinate system R.˛; ˇ; / or the corresponding Euler angles. The point of departure is the matrix for the first degree, i.e., n D 1, 2 1Ccos ˇ 2

ei.˛C /

sin ˇ i˛ p e 2

6 6 sin ˇ cos ˇ D 1 .˛; ˇ; / D 6  p2 ei 4 ˇ 1cos ˇ i. ˛/ p ei˛ e  sin 2 2

1cos ˇ i.˛ / e 2

3 7

7 sin ˇ i p e 7: 2 5 1Ccos ˇ i.˛C / e 2

(4.8.132)

n Note that the orders, i.e., the indices k and m of Dk;m , run from n to n. In our matrix notation k corresponds to the row index and m to the column index. Thus, 1 the upper left corner of D 1 contains the element D1;1 .

Exercise 4.8.56. Show the following recurrence relations for the computation of the matrices of higher degrees: n n 1 n1 n 1 n1 n 1 n1 D ak;m D0;0 Dk;m C bk;m D1;0 Dk1;m C bk;m D1;0 DkC1;m ; Dk;m

(4.8.133)

n n 1 n1 n 1 n1 n 1 n1 D ck;m D0;1 Dk;mC1 C dk;m D1;1 Dk1;mC1 C dk;m D1;1 DkC1;mC1 ; Dk;m (4.8.134) n n 1 n1 n 1 n1 n 1 n1 D ck;m D0;1 Dk;m1 C dk;m D1;1 Dk1;m1 C dk;m D1;1 DkC1;m1 ; Dk;m

(4.8.135)

4 Scalar Spherical Harmonics in R3

194

where (4.8.133) holds for n C 1  m  n  1, (4.8.134) for n  m  n  2 and (4.8.135) for n C 2  m  n. All three recurrence relations (4.8.133)– (4.8.135) hold for all k D n; : : : ; n. The factors ak;m , bk;m , ck;m , and dk;m are set to  n ak;m D

 n bk;m

D

.n C k/.n  k/ .n C m/.n  m/

 12

.n C k/.n C k  1/ 2.n C m/.n  m/

n ak;m D 0 for k D ˙n;

;  12

(4.8.136)

n bk;m D 0 for k D n or k D n C 1;

;

(4.8.137)  n D ck;m

 n D dk;m

2.n C k/.n  k/ .n C m/.n C m  1/ .n C k/.n C k  1/ .n C m/.n C m  1/

 12 ;

n ck;m D 0 for k D ˙n;

;

n dk;m D 0 for k D n or k D n C 1:

 12

(4.8.138)

(4.8.139) n n Note that the factors dk;m have to be distinguished from dk;m .ˇ/ of (4.8.126).

Remark 4.8.57. The recursions (4.8.134) and (4.8.135) are only applied for the cases m D n and m D n respectively. The repeated calculation of the square roots that form these factors can be avoided via a look-up table. For more details the reader is referred to Choi et al. (1999). Exercise 4.8.58. Prove the following symmetries of the matrix entries: n n Dk;m .˛; ˇ; / D .1/mk ei2k˛i2m Dk;m .˛; ˇ; /;

(4.8.140)

n n Dk;m .˛; ˇ; / D .1/mk Dm;k . ; ˇ; ˛/ n D .1/mk Dk;m .˛; ˇ; /

(4.8.141)

n . ; ˇ; ˛/; D Dm;k

and the following relations for the inverse rotation:  1 n n n D .˛; ˇ; / k;m D Dm;k .˛; ˇ; / D Dk;m . ; ˇ; ˛/:

(4.8.142)

Now, we apply these rotations to the inner and outer harmonics of Definitions 4.8.37 and 4.8.38. Their equivalents for the inner and outer harmonics, i.e.,   n;I D n;I .˛; ˇ; / D Dk;m .˛; ˇ; /

k;mDn;:::;n

  n;O .˛; ˇ; / D n;O .˛; ˇ; / D Dk;m

k;mDn;:::;n

;

(4.8.143)

;

(4.8.144)

4.8 Exercises

195

respectively, are related by p .n C m/Š.n  m/Š n;O n n n .˛; ˇ; / D Dk;m .˛; ˇ; / p .˛; ˇ; /Ok;m ; Dk;m D Dk;m .n C k/Š.n  k/Š (4.8.145) p .n C k/Š.n  k/Š n;I n n n Dk;m .˛; ˇ; / D Dk;m .˛; ˇ; / p .˛; ˇ; /Ik;m : D Dk;m .n C m/Š.n  m/Š (4.8.146) n;O n;I For the computation of the matrices Dk;m .˛; ˇ; / and Dk;m .˛; ˇ; / the factorials n n of Ok;m and Ik;m have to be incorporated in the recurrences. n W Exercise 4.8.59. Show the following recursive properties of the factors Ok;m

s

n Ok;m

.n C m/.n  m/ n1 D Ok1;m D .n C k/.n  k/ s .n C m/.n  m/ n1 D OkC1;m .n  k/.n  k  1/

s

n1 Ok;m

.n C m/.n  m/ .n C k/.n C k  1/ (4.8.147)

n and analogously for Ik;m W

s

n Ik;m

.n C k/.n  k/ n1 n1 D Ik1;m D Ik;m .n C m/.n  m/ s .n  k/.n  k  1/ n1 D IkC1;m : .n C m/.n  m/

s

.n C k/.n C k  1/ .n C m/.n  m/ (4.8.148)

Exercise 4.8.60. Calculate the initial matrices D 1;O .˛; ˇ; / and D 1;I .˛; ˇ; / corresponding to the inner and outer harmonics from (4.8.132). Exercise 4.8.61. Show the following recurrence relations which correspond to the recursions (4.8.133)–(4.8.135) of Exercise 4.8.56 for the case of the outer harmonics: n;O 1;O n1;O 1;O n1;O Dk;m DD0;0 Dk;m .1  ık;n /.1  ık;n / C D1;0 Dk1;m .1  ık;n /.1  ık;nC1 / 1;O n1;O C D1;0 DkC1;m .1  ık;n /.1  ık;n1 /;

(4.8.149)

n;O 1;O n1;O Dk;m DD0;1 Dk;mC1 .1  ık;n /.1  ık;n / 1;O n1;O C D1;1 Dk1;mC1 .1  ık;n /.1  ık;nC1 / 1;O n1;O C D1;1 DkC1;mC1 .1  ık;n /.1  ık;n1 /;

(4.8.150)

4 Scalar Spherical Harmonics in R3

196 n;O 1;O n1;O Dk;m DD0;1 Dk;m1 .1  ık;n /.1  ık;n /

1;O n1;O C D1;1 Dk1;m1 .1  ık;n /.1  ık;nC1 / 1;O n1;O C D1;1 DkC1;m1 .1  ık;n /.1  ık;n1 /;

(4.8.151)

where (4.8.149) holds for n C 1  m  n  1, (4.8.150) for n  m  n  2 and (4.8.151) for n C 2  m  n. All three recurrence relations (4.8.149)– (4.8.151) hold for all k D n; : : : ; n. Exercise 4.8.62. Show the following recurrence relations which correspond to the recursions (4.8.133)–(4.8.135) of Exercise 4.8.56 for the case of the inner harmonics: n;I 1;I n1;I Dk;m DD0;0 Dk;m

1 1;I n1;I .n C k/.n C k  1/ .n C k/.n  k/ C D1;0 Dk1;m .n C m/.n  m/ 2 .n C m/.n  m/

1 1;I n1;I .n  k/.n  k  1/ ; DkC1;m C D1;0 2 .n C m/.n  m/ n;I 1;I n1;I Dk;m D2D0;1 Dk;mC1

.n C k/.n  k/ .n  m/.n  m  1/

1;I n1;I C D1;1 Dk1;mC1

.n C k/.n C k  1/ .n  m/.n  m  1/

1;I n1;I DkC1;mC1 C D1;1 n;I 1;I n1;I Dk;m D2D0;1 Dk;m1

(4.8.152)

.n  k/.n  k  1/ ; .n  m/.n  m  1/

(4.8.153)

.n C k/.n  k/ .n C k/.n C k  1/ 1;I n1;I C D1;1 Dk1;m1 .n C m/.n C m  1/ .n C m/.n C m  1/

1;I n1;I DkC1;m1 C D1;1

.n  k/.n  k  1/ ; .n C m/.n C m  1/

(4.8.154)

where (4.8.152) holds for n C 1  m  n  1, (4.8.153) for n  m  n  2 and (4.8.154) for n C 2  m  n. All three recurrence relations (4.8.152)– (4.8.154) hold for all k D n; : : : ; n. Now, we apply these results to the translations of the fast multipole method. Suppose F is given as an expansion in terms of outer harmonics with expansion center x0 such as (4.8.115). The coefficients Fx^;O .n; m/ are to be shifted to the new expansion 0 center x1 as in Exercise 4.8.45. However, the shift is supposed to take place along the x3 -axis, i.e, the expansion needs to be rotated, shifted, and rotated back again. Exercise 4.8.63. Prove that the rotated expansion takes the form F .x/ D

n 1 X X nD0 kDn

e x^;O .n; k/On;k .R.x  x1 /  jx0  x1 j"3 /; F 0

(4.8.155)

4.8 Exercises

197

where the rotated coefficients fulfill n X

e x^;O .n; k/ D F 0

n;O Fx^;O .n; m/Dk;m .˛; ˇ; / 0

(4.8.156)

mDn

for k D n; : : : ; n. The rotation is R D R.˛; ˇ; / and for the Euler angles it holds that .˛; ˇ; / D .0; #; '/; (4.8.157) where .'; #/ denote the spherical coordinates of

x0 x1 jx0 x1 j .

Exercise 4.8.64. Show that the shift along the x3 -axis leads to the expansion 0

F .x/ D

1 n X X n0 D0 m0 Dn0

e x^;O .n0 ; m0 /On0 ;m0 .R.x  x1 // F 1

(4.8.158)

with the shifted coefficients 0

e x^;O .n0 ; m0 / D F 1

n X nDjm0 j

n0 n

e x^;O .n; m0 / jx0  x1 j F 0 .n0  n/Š

(4.8.159)

for m0 D n0 ; : : : ; n0 . Remark 4.8.65. The application of the inverse rotation gives 0

F .x/ D

1 n X X n0 D0 k 0 Dn0

Fx^;O .n0 ; k 0 /On0 ;k 0 .x  x1 /; 1

(4.8.160)

where the coefficients corresponding to degree n0 2 N0 are rotated back via 0

.n0 ; k 0 / D Fx^;O 1

n X m0 Dn0

e x^;O .n0 ; m0 /D n00 ;O0 . ; ˇ; ˛/; F k ;m 1

(4.8.161)

for k 0 D n0 ; : : : ; n0 using the Euler angles of Exercise 4.8.63. Since the rotation matrix is unitary the inverse rotation is already given by Exercise 4.8.63. Now, suppose that F is given as an expansion in outer harmonics with expansion center x1 such as (4.8.116) or (4.8.160). The multipole expansion is to be translated into an expansion in terms of inner harmonics around the new expansion center x2 as in Exercise 4.8.46. However, the shift is supposed to take place along the x3 -axis, i.e, the expansion needs to be rotated, shifted, and rotated back again. Analogously to Exercise 4.8.63 the rotated expansion takes the form

4 Scalar Spherical Harmonics in R3

198

F .x/ D

n 1 X X

e x^;O .n; k/On;k .R.x  x2 / C jx2  x1 j"3 /; F 1

(4.8.162)

nD0 kDn

where the rotated coefficients fulfill e x^;O .n; k/ D F 1

n X

n;O Fx^;O .n; m/Dk;m .˛; ˇ; / 1

(4.8.163)

mDn

for k D n; : : : ; n. The rotation is R D R.˛; ˇ; / and the Euler angles satisfy .˛; ˇ; / D .0; #; '/; where .'; #/ denote the spherical coordinates of

(4.8.164)

x2 x1 . jx2 x1 j

Exercise 4.8.66. Show that the shift along the x3 -axis leads to the expansion 0

F .x/ D

1 n X X n0 D0 m0 Dn0

e x^;I .n0 ; m0 /In0 ;m0 .R.x  x2 //; F 2

(4.8.165)

.n C n0 /Š ; jx2  x1 jnCn0 C1

(4.8.166)

with the shifted coefficients e x^;I .n0 ; m0 / D F 2

1 X nDjm0 j

e x^;O .n; m0 / F 1

for m0 D n0 ; : : : ; n0 . Remark 4.8.67. Then, the application of the inverse rotation gives 0

F .x/ D

1 n X X n0 D0 k 0 Dn0

Fx^;I .n0 ; k 0 /In0 ;k 0 .x  x2 /; 2

(4.8.167)

where the coefficients corresponding to degree n0 2 N0 are rotated back via 0

.n0 ; k 0 / Fx^;I 2

D

n X m0 Dn0

0

e x^;I .n0 ; m0 /D n0 ;I 0 . ; ˇ; ˛/; F k ;m 2

(4.8.168)

for k 0 D n0 ; : : : ; n0 using the Euler angles of (4.8.164). Finally, suppose that F is given as an inner harmonics expansion with expansion .n; m/ are to be shifted center x2 such as (4.8.118) or (4.8.167). The coefficients Fx^;I 2 to the new expansion center x3 as in Exercise 4.8.48. However, the shift is supposed to take place along the x3 -axis, i.e, the expansion needs to be rotated, shifted, and rotated back again.

4.8 Exercises

199

Exercise 4.8.68. Prove that the rotated expansion takes the form F .x/ D

n 1 X X

e x^;I .n; k/In;k .R.x  x3 / C jx3  x2 j"3 /; F 2

(4.8.169)

nD0 kDn

where the rotated coefficients fulfill n X

e x^;I .n; k/ D F 2

n;I Fx^;I .n; m/Dk;m .˛; ˇ; /; 2

(4.8.170)

mDn

for k D n; : : : ; n. The rotation is R D R.˛; ˇ; / and for the Euler angles it holds that .˛; ˇ; / D .0; #; '/; (4.8.171) where .'; #/ denote the spherical coordinates of

x3 x2 . jx3 x2 j

Exercise 4.8.69. Show that the shift along the x3 -axis leads to the expansion 0

F .x/ D

1 n X X n0 D0 m0 Dn0

e x^;I .n0 ; m0 /In0 ;m0 .R.x  x3 //; F 3

(4.8.172)

with the shifted coefficients e x^;I .n0 ; m0 / D F 3

1 X nDn

nn0

e x^;I .n; m0 / jx3  x2 j ; F 2 .n  n0 /Š 0

(4.8.173)

for m0 D n0 ; : : : ; n0 . Remark 4.8.70. The application of the inverse rotation yields 0

F .x/ D

1 n X X n0 D0 k 0 Dn0

Fx^;I .n0 ; k 0 /In0 ;k 0 .x  x3 /; 3

(4.8.174)

where the coefficients corresponding to degree n0 2 N0 are rotated back by 0

.n0 ; k 0 / D Fx^;I 3

n X m0 Dn0

0

e x^;I .n0 ; m0 /D n0 ;I 0 . ; ˇ; ˛/; F k ;m 3

(4.8.175)

for k 0 D n0 ; : : : ; n0 using the Euler angles of Exercise 4.8.68. Since the rotation matrix is unitary the inverse rotation is already given by Exercise 4.8.68. Remark 4.8.71. The rotations and the Wigner matrices can be used to accelerate the translations of the fast multipole algorithm and reduce the effort of a translation from p 4 to 3p 3 , where p denotes the truncation degree of the expansion (for more details we refer to Cheng et al. 1999; Greengard and Rokhlin 1997; Gutting 2008; White and Head-Gordon 1996). In Gutting (2008, 2012) this method is used to

4 Scalar Spherical Harmonics in R3

200

determine the Earth’s gravitational potential from known (oblique) gravity vectors on the actual Earth’s surface.

Spherical Harmonics in Quaternionic Representation Let R4 be the four-dimensional Euclidean vector space. Suppose that ei 1 D "i , i D 1; : : : ; 4, is the canonical orthonormal system in R4 . Hence, any vector a 2 R4 can be written as 3 X ai ei D a0 e0 C a; O a0 2 R; (4.8.176) aD i D0

where aO D a1 e1 C a2 e2 C a3 e3 2 R3 ;

ai 2 R; i D 1; 2; 3:

(4.8.177)

Let b 2 R4 be another vector such that a multiplication law is given by O 0 C aO ^ bO C a0 bO C ab ab D .a0 b0  aO  b/e O 0;

(4.8.178)

where aO  bO and aO ^ bO are the scalar product and the vector product in the Euclidean space R3 , respectively. Obviously, this product is not commutative. With the help of this multiplication the vector space R4 is furnished with the algebraic structure of a ring (see, e.g., G¨urlebeck and Spr¨oßig 1989), i.e., it is an algebra which is named real quaternionic algebra. In the honour of its discoverer (Hamilton 1866) it is designated by H. Exercise 4.8.72. Show that the members of H, i.e., the quaternions a 2 R4 , may be identified with matrices of R44 of the form 2

a0 6a1 aD6 4a2 a3

a1 a0 a3 a2

a2 a3 a0 a1

3 a3 a2 7 7 a1 5

(4.8.179)

a0

and the product of two quaternions corresponds to the matrix product of their matrix representations. Associated to the representation (4.8.179) the basis ei , i D 0; : : : ; 3, is given by 2

10 60 1 e0 D 6 40 0 00

3 2 00 0 1 61 0 0 07 7 ; e1 D 6 40 0 1 05 01 0 0

3 2 0 0 0 0 1 60 0 0 0 07 7 ; e2 D 6 41 0 0 0 15 1 0 0 1 0

3 2 0 00 60 0 17 7 ; e3 D 6 40 1 05 0 10

3 0 1 1 0 7 7: 0 05 0 0 (4.8.180)

4.8 Exercises

201

Remark 4.8.73. In the classical nomenclature (cf. Clifford 1878, Hamilton 1866) the basis elements are represented by 1; i; j; k. This is done in analogy to the approach known from the complex space C. The conjugate quaternions are defined by a D a0 e0 

3 X

a 2 H:

ai ei ;

(4.8.181)

i D1

From the law of multiplication (4.8.178) it immediately follows that aa D aa D

3 X

ai2 e0 ;

(4.8.182)

i D0

such p that by setting e0 D 1 (see Remark 4.8.73) the (Euclidean) norm jaj is equal to aa. The real part and the imaginary part is understood to be Re.a/ D a0 e0 D 12 .a C a/; Im.a/ D aO D

1 2 .a

(4.8.183)

 a/:

The inverse of the quaternion a 2 H n f0g is given by a1 D

(4.8.184) a . jaj2

Exercise 4.8.74. Verify that, for a; b 2 H and  2 R the following rules hold: a C b D a C b;

(4.8.185)

ab D ba;

(4.8.186)

a D a; p jaj D aa D jaj D j  aj;

(4.8.187)

jaj D jjjaj;

(4.8.189)

jabj D jajjbj;

(4.8.190)

Re.ab/ D Re.ba/;

(4.8.191)

jjaj  jbjj  ja  bj  jaj C jbj:

(4.8.188)

(4.8.192)

Exercise 4.8.75. Show that the basis quaternions satisfy the following relations: e02 D e0 ; ei2

D e0 ;

ei ej C ej ei D 0;

(4.8.193) i D 1; 2; 3; i ¤ j; i; j D 1; 2; 3;

(4.8.194) (4.8.195)

4 Scalar Spherical Harmonics in R3

202

e0 ei D ei e0 D ei ;

i D 0; 1; 2; 3;

(4.8.196)

e1 e2 D e3 ;

(4.8.197)

e2 e3 D e1 ;

(4.8.198)

e3 e1 D e2 :

(4.8.199)

Definition 4.8.76. Let R0;q stand for the real vector space Rq , q 2 N, provided with a non-degenerate symmetric bilinear form B such that, for any vectors a; b 2 R0;q ; B.a; b/ D a  b;

(4.8.200)

where a  b denotes the Euclidean inner product of a and b. In particular, B.ei ; ej / D ıi;j ;

i; j D 1; : : : ; q:

(4.8.201)

Looking at the associated quadratic form we have Q.a/ D B.a; a/ D jaj2 ;

(4.8.202)

where jaj stands for the Euclidean norm of a. Of course, Q.ei / D B.ei ; ei / D 1;

i D 1; : : : ; q:

(4.8.203)

The real Clifford algebra Cl.q/ constructed over R0;q is a real, linear associative algebra with identity e0 D "1 , of dimension 2q , containing R and R0;q as subspaces and in which for each vector a 2 R0;q ; a2 D Q.a/ D jaj2 :

(4.8.204)

such that an Henceforth we take the orthonormal basis e1 ; : : : ; eq for Pgranted q arbitrary element a 2 R0;q is representable in the form a D j D1 aj ej : A basis of Cl.q/ consists of the elements feA gAf1;:::;qg , where A is an element of the power set P .f1; : : : ; qg/ of f1; : : : ; qg such that, for A D fi1 ; : : : ; il g with 1  i1 < : : : < il  q, eA D ei1    eil and e; D e0 is the identity element. In such a way any element a 2 Cl.q/ may be written as aD

X

aA eA ;

aA 2 R:

(4.8.205)

A

Exercise 4.8.77. Verify the following statements: (a) Cl.1/ is isomorphic to C, i.e., Cl.1/ Š C. Moreover, dim.Cl.1// D 2, where e0 D e; , e1 D ef1g with e12 D 1 is a basis for Cl.1/ (in the classical Gaussian nomenclature of the complex space C, e0 D 1 and e1 D i).

4.8 Exercises

203

(b) Cl.2/ is isomorphic to H, i.e., Cl.2/ Š H. Moreover, dim.Cl.2// D 4, where e0 D e; , e1 D ef1g , e2 D ef2g , e3 D ef1;2g D e1 e2 is a basis for Cl.2/ (in the classical Hamiltonian nomenclature e0 D 1, e1 D i, e2 D j, and e3 D k). Hint: Apply (4.8.200) to the basis vectors ei , i D 0; 1; 2; 3. For more details on Clifford algebras the reader is referred to, e.g., Brackx et al. (1982), Delanghe (2001), G¨urlebeck et al. (2008) and Qian et al. (2004) and the references therein. Next, we deal with functions f defined on regions G and/or their boundaries @G in H. The so-called H-valued functions can be represented by f D

3 X

Fi ei ;

(4.8.206)

i D0

where the scalar functions Fi are real-valued. Properties such as continuity, differentiability, integrability and so on which are ascribed to f can be defined via the components Fi , i D 0; 1; 2; 3; in the standard way. In this manner, spaces of .k/ .p/ .k/ .p/ H-valued functions cH .G /, lH .G /, cH .@G /, lH .@G / are denoted accordingly. As for vector-valued functions in Chap. 5 we use lower case letters for these function spaces. 4 P3As already pointed out, any vector a 2 R corresponds to a quaternion a D i D0 ai ei 2 H. In formal analogy, the vector 

@ @ @ @ @x0 ; @x1 ; @x2 ; @x3

 (4.8.207)

corresponds to an operator @ given by @D

3 X i D0

@ e: @xi i

(4.8.208)

Remark 4.8.78. In the framework of C, the operator @ in (4.8.208) corresponds to 2 @z@ D

@ @x

@ C i @y ;

(4.8.209)

while the conjugate operator @D

@ @x0 e0



3 X i D1

@ @xi ei

(4.8.210)

can be regarded as an extension to 2 @z@ D

@ @x

@  i @y :

(4.8.211)

4 Scalar Spherical Harmonics in R3

204

Definition 4.8.79. The operator @ defined by (4.8.208) is called the Cauchy– Riemann operator, while D D @  @x@0 e0 (4.8.212) is named the Dirac operator. According to its definition the operator DD

3 X i D1

@ @xi ei

(4.8.213)

only operates on the variables x1 , x2 , x3 . Clearly, D D D ; where  is the Laplacian in R3 . The action of the Dirac operator D on an H-valued function f of the form (4.8.206) from the left and right is denoted by D f and f D, respectively. The Dirac operator D has important applications in physics, from which we mention the pre-Maxwell equations. Exercise 4.8.80. Use Exercise 4.8.75 to show that D f may be written in the form D f D r  fO C rF0 C r ^ fO;

(4.8.214)

where rF0 D

3 X i D1

r  fO D

3 X i D1

r ^ fO D



@ @xi

F0 ei ;

(4.8.215)

@ @xi

Fi ;

(4.8.216)

@ @x2 F3



@ @x3 F2



e1 C



@ @x3 F1



@ @x1 F3



e2 C



@ @x1 F2



@ @x2 F1

 e3 :

(4.8.217) In consequence, if f is H-continuously differentiable with D f D 0, we are led to the differential system r  fO D 0;

(4.8.218)

rF0 C r ^ fO D 0:

(4.8.219)

In particular, where F0 is constant, the resulting pre-Maxwell equations describe an irrotational fluid without sources. Equations (4.8.218) and (4.8.219) have been used by Moisil and Teodorescu (1931) as the point of departure for the development of the hypercomplex function theory (see also G¨urlebeck et al. 2008 for more details). Exercise 4.8.81. Let f W H ! H be given by x D .x0 ; x1 ; x2 ; x3 /T 7! f .x/ D f .x0 ; x1 ; x2 ; x3 / D x1 e1  x2 e2 :

(4.8.220)

4.8 Exercises

205

Show that the following statements are valid: (a) D f D 0, (b) D.f f / D 2x1 e1  2x2 e2 ¤ 0, (c) For x ¤ 0 we have f .x/ ¤ 0 and (in contrast to complex analysis) the function f 1 given by x1 e1 C x2 e2 (4.8.221) f 1 .x/ D x12 C x22 satisfies D f 1 .x/ D

2.x12  x22 / 4x1 x2 e0 C 2 e3 ¤ 0: .x12 C x22 /2 .x1 C x22 /2

(4.8.222)

Exercise 4.8.81 shows that the quaternionic product ff of a function f with Df D 0 does not satisfy D.ff / D 0: The trouble in Exercise 4.8.81 consists in the noncommutativity of the H-multiplication law. An idea to overcome this difficulty is due to Delanghe (1970). Now, we turn to the concept of H-holomorphic functions. Definition 4.8.82. Let f be an H-valued continuously differentiable function in a region G  H. Then f is called H-right holomorphic (or H-left holomorphic) in G , if for all x 2 G there exist quaternions wk such that f .x C h/ D f .x/ C

3 X

wk .hk  h0 ek / C o.jhj/;

h ! 0;

(4.8.223)

3 X .hk  h0 ek /wk C o.jhj/;

h ! 0;

(4.8.224)

kD1

f .x C h/ D f .x/ C

kD1

respectively, where hk are the coordinates of h. Exercise 4.8.83 (Cauchy–Riemann Differential Equations). Suppose that the Hvalued function f is continuously differentiable in G  H. Show that f is H-right holomorphic if and only if f @ D 0 in G (f is H-left holomorphic if and only if @f D 0 in G ). Remark 4.8.84. Throughout our work, we agree upon the fact that H-left holomorphic functions are just called H-holomorphic. The set of all H-holomorphic functions on G is denoted by holH .G /. The Laplace operator (in R4 ) is given by  D @@ D @@ D .2/

3 X @2 : @xi2 i D0

(4.8.225)

Obviously, if f 2 cH .G /, then the equation f D 0 follows from @f D 0 in G , i.e., each component function of an H-holomorphic function is a harmonic function.

4 Scalar Spherical Harmonics in R3

206

In this way, the theory of H-holomorphic functions refines the theory of harmonic functions (see, e.g. Brackx and Delanghe 2003 for a more general approach in terms of Clifford algebras). Assume that a real function F W G ! R mapping x D .x0 ; x1 ; x2 /T 7! F .x0 ; x1 ; x2 / with .x0 ; x1 ; x2 /T 2 G  R3 , is twice continuously differentiable and harmonic, i.e., 2 X @2 F .x/ D 0 (4.8.226) @xi2 i D0 for all x 2 G . Following our nomenclature (and considering F W G ! R with G embedded in H) we therefore have @@F D

3 2 X X @2 @2 F .x/ D F .x/ D 0 @xi2 @xi2 i D0 i D0

(4.8.227)

in G . Note that F does not depend on x3 . Identity (4.8.227) leads us to the following observation: Applying the operator @ from the left to the harmonic function F provides us with an H-holomorphic function. This strategy is used to derive Hspherical harmonics ˇ ˇ @UnC1;0 ˇ 2 ; S

ˇ ˇ @UnC1;m ˇ 2 ; S

ˇ ˇ @VnC1;m ˇ

S2

(4.8.228)

from the standard inner harmonics given by UnC1;m .x/ D r nC1 PnC1;m .cos.#// cos.m'/;

(4.8.229)

UnC1;0 .x/ D r

(4.8.230)

nC1

PnC1 .cos.#// cos.m'/;

VnC1;m .x/ D r nC1 PnC1;m .cos.#// sin.m'/;

(4.8.231)

where n 2 N0 , m D 1; : : : ; n C 1, PnC1;m are the associated Legendre functions of (4.3.108) and PnC1 is the Legendre polynomial of degree nC1. The point x 2 R3 is represented in terms of polar coordinates .r; #; '/ as introduced in (4.1.2). Exercise 4.8.85. Show that for n 2 N0 , fyn;m ; yn;0 ; zn;m gmD1;:::;nC1 given by ˇ ˇ yn;m .#; '/ D@ .UnC1;m .x// ˇ

rD1

(4.8.232)

DAn;m cos.m'/e0 C .Bn;m cos.'/ cos m'  Cn;m sin.'/ sin.m'// e1 C .Bn;m sin.'/ cos m' C Cn;m cos.'/ sin.m'// e2 ; ˇ ˇ yn;0 .#; '/ D@ .UnC1;0 .x// ˇ rD1

DAn;0 e0 C Bn;0 cos.'/e1 C Bn;0 sin.'/e2 ;

(4.8.233)

4.8 Exercises

207

ˇ ˇ zn;m .#; '/ D@ .UnC1;m .x// ˇ

(4.8.234)

rD1

DAn;m sin.m'/e0 C .Bn;m cos.'/ sin m' C Cn;m sin.'/ cos.m'// e1 C .Bn;m sin.'/ sin m'  Cn;m cos.'/ cos.m'// e2 ; with An;m

Bn;m

  ˇ 1 ˇ 2 d sin .#/ dt PnC1;m .t/ˇ D C .n C 1/ cos.#/PnC1;m .cos.#// ; t Dcos.#/ 2 (4.8.235)   ˇ 1 ˇ d sin.#/ cos.#/ dt PnC1;m .t/ˇ D  .n C 1/ sin.#/PnC1;m .cos.#// ; t Dcos.#/ 2 (4.8.236)

Cn;m D

m 1 PnC1;m .cos.#//; 2 sin.#/

(4.8.237)

represents a system of 2n C 3 H-spherical harmonics. Of course, the system of 2nC3 H-spherical harmonics developed in Exercise 4.8.85 is not closed in l2H .S2 /, since all occurring functions do not contain an e3 -coordinate different from zero. Following an approach due to G¨urlebeck et al. (2008) we consider the extended system fyn;0 ; yn;m ; zn;m ; yn;0 e3 ; yn;l e3 ; zn;l e3 gmD1;:::;nC1I lD1;:::;nC1

(4.8.238)

that consists of 4n C 6 functions. However, it is easily seen that zn;nC1 e3 D yn;nC1 ;

(4.8.239)

yn;nC1 e3 D zn;nC1 :

(4.8.240)

Therefore, we continue with the system of 4n C 4 H-spherical harmonics fyn;0 ; yn;m ; zn;m ; yn;0;3 ; yn;l;3 ; zn;l;3 gmD1;:::;nC1I lD1;:::;n

(4.8.241)

where we have introduced the abbreviations yn;0;3 D yn;0 e3 ;

yn;l;3 D yn;l e3 ;

zn;l;3 D zn;l e3 :

(4.8.242)

Exercise 4.8.86. Show that the system (4.8.241) satisfies the following properties: (a) For every n 2 N0 , the subsystem fyn;0 ; yn;m ; zn;m gmD1;:::;nC1 is an orthogonal system in the l2H .S2 /-sense, where the l2H .S2 /-norms are given by

4 Scalar Spherical Harmonics in R3

208

p .n C 1/;

kyn;0 kl2 .S2 / D H

(4.8.243) s

kyn;m kl2 .S2 / D kzn;m kl2 .S2 / D H

H

.n C 1 C m/Š  .n C 1/ 2 .n C 1  m/Š

(4.8.244)

with m D 1; : : : ; n C 1. (b) For every n 2 N0 , the subsystem fyn;0;3 ; yn;l;3 ; zn;l;3 glD1;:::;n is an orthogonal system in the l2H .S2 /-sense, where the l2H .S2 /-norms are given by kyn;0;3 kl2 .S2 / D H

p .n C 1/;

(4.8.245) s

kyn;l;3 kl2 .S2 / D kzn;l;3 kl2 .S2 / D H

H

.n C 1 C l/Š  .n C 1/ 2 .n C 1  l/Š

(4.8.246)

with l D 1; : : : ; n. (c) For every n 2 N0 and m D 1; : : : ; n C 1, l D 1; : : : ; n, we have that hyn;0 ; yn;0;3 il2 .S2 / D hyn;0 ; yn;l;3 il2 .S2 / D hyn;0 ; zn;l;3 il2 .S2 / D 0;

(4.8.247)

hyn;m ; yn;0;3 il2 .S2 / D hyn;m ; yn;l;3 il2 .S2 / D 0;

(4.8.248)

hzn;m ; yn;0;3 il2 .S2 / D hzn;m ; zn;l;3 il2 .S2 / D 0;

(4.8.249)

H

H

H

H

H

H

H

and ( hyn;m ; zn;l;3 il2 .S2 / D hzn;m ; yn;l;3 il2 .S2 / D H

H

0

; m ¤ l;

.nCmC1/Š  2 m .nmC1/Š

; m D l: (4.8.250)

In consequence, only a few functions out of the different systems are nonorthogonal. For a total orthonormalization process we first start with the normalized polynomials corresponding to (4.8.241): fyQn;0 ; yQn;m ; zQn;m ; yQn;0;3 ; yQn;l;3 ; zQn;l;3 gmD1;:::;nC1I lD1;:::;n :

(4.8.251)

Hence, from our preceeding considerations, it is clear that the two systems fyQn;0 ; yQn;m ; zQn;m gn2N0 ImD1;:::;nC1 ;

(4.8.252)

fyQn;0;3 ; yQn;l;3 ; zQn;l;3 gn2N0 I lD1;:::;n

(4.8.253)

are orthonormal systems, respectively. In addition, for every n 2 N0 , all H-spherical harmonics of the systems (4.8.252) and (4.8.253) are orthogonal, however, with the only exception.

4.8 Exercises

209

Exercise 4.8.87. Show that, for n 2 N0 and l D 1; : : : ; n, hyQn;l ; zQn;l;3 il2 .S2 / D hQzn;l ; yQn;l;3 il2 .S2 / D H

H

1 : nC1

(4.8.254)

Exercise 4.8.88. Verify that the system ˚

ON ON ON ON ON yn;0 ; yn;m ; zn;m ; yn;0;3 ; yn;l;3 ; zON n;l;3

n2N0 I mD1;:::;nC1I lD1;:::;n

(4.8.255)

with ON yn;0 D yQn;0 ;

(4.8.256)

ON D yQn;m ; yn;m

(4.8.257)

zON Qn;m ; n;m D z

(4.8.258)

ON D yQn;0;3 ; yn;0;3

(4.8.259)

1 ON yn;l;3 D p ..n C 1/yQn;l;3 C l zQn;l / ; .n C 1/2  l 2

(4.8.260)

1 ..n C 1/Qzn;l;3  l yQn;l / zON n;l;3 D p .n C 1/2  l 2

(4.8.261)

represents a fully l2H .S2 /-orthonormalized system of 4n C 4 H-spherical harmonics. The proof of Exercise 4.8.88 is straightforward by observing that the first 2n C 3 functions in (4.8.255) are already orthonormal. In the next step it should be realized that zQn;l;3 

nC1 X l ON ON ON ON yQn;l DQzn;l;3  yn;0 hyn;0 ; zQn;l;3 il2 .S2 /  yn;m hyn;m ; zQn;l;3 il2 .S2 / H H nC1 mD1



nC1 X mD1

ON zON Qn;l;3 il2 .S2 / : n;m hzn;m ; z

(4.8.262)

H

Next, we go over to the H-holomorphic inner harmonics defined by ˚ n ON n ON n ON n ON n ON n ON r yn;0 ; r yn;m ; r zn;m ; r yn;0;3 ; r yn;l;3 ; r zn;l;3 n2N

: (4.8.263)

0 I mD1;:::;nC1I lD1;:::;n

Exercise 4.8.89. Show that, for every n 2 N0 , the system of n C 1 H-holomorphic inner harmonics o n ON n ON ; r yn;2 ; r n zON (4.8.264) r n yn;0 n;2 ;3 D1;:::;b.nC1/=2cI D1;:::;bn=2c

4 Scalar Spherical Harmonics in R3

210

forms an orthogonal basis of the space of all H-holomorphic homogeneous polynomials of degree n. Exercise 4.8.90. Show that, for every n 2 N0 , the system of n C 1 H-holomorphic inner harmonics o n ON ; r n zON (4.8.265) r n yn;2C1 n;2 1;3 D0;:::;bn=2cI D1;:::;b.nC1/=2c

forms an orthogonal basis of the space of all H-holomorphic homogeneous polynomials of degree n. In order to establish orthonormality we need the following statement. Exercise 4.8.91. Let fHn;l g be an l2H .S2 /-orthonormal system of H-spherical harmonics. Prove that hHn;l ; Hk;j il2 .B3 / D H

1

1 hHn;l ; Hk;j il2 .S2 / : H nCkC3

(4.8.266)

Indeed, by virtue of Exercise 4.8.91, G¨up rlebeck et al. (2008) are able to guarantee that each of the two systems (with kn D 2n C 3 ) n n

ON ON kn r n yn;0 ; kn r n yn;2 ; kn r n zON n;2 ;3 ON kn r n yn;2C1 ; kn r n zON n;2 1;3

o

o n2N0 I D1;:::;b.nC1/=2cI D1;:::;bn=2c

n2N0 I D0;:::;bn=2cI D1;:::;b.nC1/=2c

(4.8.267) (4.8.268)

represents a closed l2H .B31 /-orthonormal system in the space l2H .B31 / \ holH .B31 / D l2H .B31 / \ ker.@/:

(4.8.269)

Remark 4.8.92. For a discussion of spherical operators and differential equations in Clifford algebras the reader is referred to, e.g., Liu and Ryan (2002) and the references therein. Homogeneous harmonic polynomials in Clifford algebras are discussed by Brackx and Delanghe (2003) and many others. In particular, the results presented in this work for quaternionic spherical harmonics can be formulated in Clifford algebras, too. In this respect it should be noted that (De Bie and Sommen 2007; De Bie et al. 2009) extended essential results known from classical spherical harmonics theory such as the Funk–Hecke formula, decomposition theorems, eigenvalue expansions to the so-called superspace using techniques of Clifford analysis.

Chapter 5

Vectorial Spherical Harmonics in R3

Various applications imply different formulations of vector spherical harmonics, putting the accent on different issues (see, e.g., Sects. 1.3 and 1.4). One important aspect in our understanding is the easy transition from scalar spherical harmonics to the vectorial ones. A simple approach is to formulate the vectorial problem in terms of Cartesian components. However, this procedure leads back to anisotropic scalar component equations, so that the physical relevance usually is difficult to realize, the mathematical formulation is lengthy, and modeling often becomes complicated. For many physically motivated applications, where the underlying differential equations are separable in spherical coordinates, it is advantageous to separate the vector fields into a normal and a tangential part (see, e.g., Sect. 1.3). Our approach to vector spherical harmonics takes this into account. These geophysically motivated aspects are guaranteed adequately within a vectorial framework, transforming scalar functions into vector fields by use of certain operators o.i / ; i D 1; 2; 3, that are defined in Sect. 5.2. The operator o.1/ separates the normal part of a vector field from the tangential part, o.2/ defines a (tangential) surface gradient field that is curl-free, while o.3/ yields a (tangential) surface curl gradient field that is divergence-free. In doing so, we are led to definitions that are independent of any particular choice of spherical harmonics and do not relate to any particular choice of a spherical coordinate system. Moreover, the rotational symmetry, i.e., the isotropy can be reflected in suitable (vectorial) manner. Further material can be found, e.g., in Blatt and Weisskopf (1952), Freeden et al. (1998), Freeden and Schreiner (2009), Gervens (1989), Hill (1954), and Morse and Feshbach (1953), for applications we refer to Backus et al. (1996), Gerhards (2011), Maier (2003), Mayer (2003) (geomagnetism), Freeden (1999), Nutz (2002), Rummel and van Gelderen (1992), Schreiner (1994), Freeden and Nutz (2011) (satellite geodesy), or Fengler (2005), Fengler and Freeden (2005), Freeden and Schreiner (2009) (Navier–Stokes equation on the sphere), Abeyratne et al. (2003), Bauch (1981), Freeden (1990), Freeden and Michel (2004, 2005), Freeden et al. (1990), Grafarend (1986), Gervens (1989), Gurtin (1972), Jaswon and Symm (1977), Lurje (1963), Knops and Payne (1971),

W. Freeden and M. Gutting, Special Functions of Mathematical (Geo-)Physics, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-0348-0563-6 5, © Springer Basel 2013

211

5 Vectorial Spherical Harmonics in R3

212

and Mochizuki (1988) (elasticity). For applications in quantum mechanics using another approach we just mention Edmonds (1964), James (1976), and Jones (1985). The layout of this chapter on vector spherical harmonics is as follows: Sect. 5.1 is concerned with the necessary notation for spherical vector fields, in particular the occurring differential operators. It also summarizes several helpful results from vector analysis. In Sect. 5.2, we introduce the vector spherical harmonics based on the properties of the operators o.i / , i D 1; 2; 3. Section 5.3 is dedicated to the Helmholtz decomposition formula for spherical vector fields by use of the Green function with respect to the Beltrami operator that has been introduced in Sect. 4.6. Section 5.4 shows the closure and completeness of vector spherical harmonics intrinsically on the sphere based on Bernstein summability. The interrelations between vector spherical harmonics and homogeneous harmonic vector polynomials are investigated in more detail in Sect. 5.5. Section 5.6 shows us that the vector spherical harmonics can be regarded as eigenfunctions of a vectorial analog of the Beltrami operator. Section 5.7 presents the formulation of the addition theorem in terms of vector spherical harmonics thereby introducing the Legendre rank-2 tensor kernels as the counterpart of the Legendre polynomials in the scalar case. In Sect. 5.8, we prove vectorial versions of the Funk–Hecke formula and deal with orthogonal invariance. Vectorial counterparts of the Legendre polynomial are investigated in detail in Sect. 5.9. For an application we go back to the Cauchy– Navier equation of Sect. 1.4 and consider vector polynomials for its solution in Sect. 5.10. We develop a system of “Cauchy–Navier polynomials” such that the vector spherical harmonics of Sect. 5.2 act as corresponding boundary values in the Cauchy–Navier boundary value problem. Finally, we provide the reader with some exercises in Sect. 5.11 that are concerned with Heisenberg’s uncertainty principle, its application to classify zonal functions, as well as the treatment of the Navier–Stokes equation on the sphere of Sect. 1.3.

5.1 Basic Notation Vector fields on the unit sphere f W S2 ! R3 can be written in the form f ./ D

3 X

Fi ./"i ;

 2 S2 ;

(5.1.1)

i D1

where the component functions Fi are given by Fi ./ D f ./  "i (as usual, vector fields will be denoted by lower case letters). First, we introduce function spaces for vector fields on the sphere. l2 .S2 / denotes the space of all square-integrable vector fields on S2 which is a Hilbert space in connection with the scalar product Z hf; gil2 .S2 / D

f ./  g./ dS./; S2

f; g 2 l2 .S2 /:

(5.1.2)

5.1 Basic Notation

213

The space c.p/ .S2 /, 0  p  1, consists of all p-times continuously differentiable vector fields on the sphere. c.S2 / D c.0/ .S2 / is the Banach space of continuous vector fields on the sphere with respect to the norm kf kc.S2 / D sup jf ./j;

f 2 c.S2 /:

(5.1.3)

2S2

We know that c.S2 /

kkl2 .S2 /

D l2 .S2 /:

(5.1.4)

As in the scalar case, there is the following norm estimate for f 2 c.S2 /: kf kl2 .S2 / 

p

4 kf kc.S2 / :

(5.1.5)

Any vector field f 2 c.S2 / can be decomposed into its normal and its tangential part: f ./ D fnor ./ C ftan ./;

(5.1.6)

where the normal and tangential field are given by  7! pnor f ./ D fnor ./ D .f ./  /;

 2 S2 ;

 7! ptan f ./ D ftan ./ D f ./  .f ./  /;

(5.1.7)  2 S2

(5.1.8)

with the projection operators pnor and ptan . For f; g 2 c.S2 /,  2 S2 , it holds that f ./  g./ D fnor ./  gnor ./ C ftan ./  gtan ./:

(5.1.9)

The spaces c.S2 / and l2 .S2 / can also be decomposed accordingly: cnor .S2 / D ff 2 c.S2 / W f D pnor f g;

(5.1.10)

ctan .S2 / D ff 2 c.S2 / W f D ptan f g;

(5.1.11)

c.S / D cnor .S / ˚ ctan .S /; 2

2

2

(5.1.12)

as well as l2nor .S2 / D cnor .S2 / l2tan .S2 / D ctan .S2 / l .S / D 2

2

l2nor .S2 /

kkl2 .S2 /

;

(5.1.13)

;

(5.1.14)

l2tan .S2 /:

(5.1.15)

kkl2 .S2 /

˚

5 Vectorial Spherical Harmonics in R3

214

From Sect. 4.1 we remember the surface gradient operator p 1 @ @ C "t 1  t 2 r  D "' p @t 1  t 2 @'

(5.1.16)

which contains the tangential derivatives of the gradient and the surface curl gradient operator (L F ./ D  ^ r F ./ for F 2 C.1/ .S2 /): p @ 1 @ : L D "' 1  t 2 C "t p 2 @t @' 1t

(5.1.17)

A simple calculation using the coordinate representation of r  (5.1.16) and L (5.1.17) as well as the definition of the surface divergence of (4.1.8) and the surface curl of (4.1.9) shows that L  r F ./ D 0

and r  L F ./ D 0:

(5.1.18)

The following results are known from surface differential geometry (see, e.g., Strubecker 1964). Lemma 5.1.1. The tangential field of f vanishes, i.e., ftan ./ D 0 for all  2 S2 , if and only if f ./   D 0 for every unit vector  that is perpendicular to . Proof. First, let ftan D 0. Then, for all  2 S2 , f ./   D .f ./  /   C .f ./  .f ./  //  

(5.1.19)

D 0 C ftan ./   D 0: On the other hand, assume now that ftan ./ D f ./  .f ./  / ¤ 0. Then,

ftan ./ jftan ./j

./ is a unit vector field that is perpendicular to . Hence, ftan ./ jfftan D jftan ./j D 0 tan ./j which is a contradiction. Thus, ftan ./ D 0. t u

Lemma 5.1.2. Let f 2 c.S2 / satisfy Z C

for every curve C  S2 . Then,

  f ./ d./ D 0

ftan ./ D 0

(5.1.20)

(5.1.21)

for all  2 S2 . Proof. Choose any point 0 2 S2 . Let 0 be a unit vector with 0  0 D 0. Then, there is a curve C  S2 which passes through 0 and has 0 as tangent vector at 0 . Let C0 be any subset of C containing 0 . Then,

5.1 Basic Notation

215

Z C0

  f ./ d./ D 0:

(5.1.22)

Now, we let the length of C0 tend to zero and find that 0  f .0 / D 0 in the limit. Then, Lemma 5.1.2 implies that ftan .0 / D 0. Since 0 has been chosen arbitrarily, we have ftan D 0. t u Lemma 5.1.3. Suppose F is of class C.1/ .S2 /. Then, (i) r F ./ D 0 holds for all  2 S2 if and only if F is constant. (ii) L F ./ D 0 holds for all  2 S2 if and only if F is constant. Proof. First, we treat the case with r  . Suppose F is continuously differentiable in an open set in R3 that contains S2 and C  S2 is any curve from 0 2 S2 to 1 2 S2 . Let  be the unit tangent vector at  on C pointing from 0 to 1 . Then, Z F .1 /  F .0 / D   r F ./ d./: (5.1.23) C

If r F ./ D 0, we obtain from (5.1.23) that F .1 / D F .0 / for all points 0 ; 1 2 S2 . If on the other hand F is constant, (5.1.23) gives us that the vector field f D R r  F fulfills C   f ./ d./ D 0 for all curves C  S2 . By Lemma 5.1.2 it holds that ftan D 0 on S2 and thus, ftan ./ D f ./  .f ./  /  D f ./ D r F ./ D 0

(5.1.24)

for all  2 S2 . Now, we consider the case with L . If L F ./ D 0, i.e.,  ^ r F ./ D 0 for all  2 S2 , then we have      ^  ^ r F ./ D   r F ./   r F ./.  / D r F ./ D 0 (5.1.25) for all  2 S2 . By part (i) of Lemma 5.1.3 it follows that F is constant. If on the other hand F is constant, then L F ./ D  ^ r F ./ D  ^ 0 D 0 for t u all  2 S2 . Lemma 5.1.4. Let f 2 c.S2 / be a tangential vector field with Z   f ./ d./ D 0

(5.1.26)

C

for every closed curve C  S2 . Then, there is a scalar field P on S2 such that f ./ D r P ./; where P 2 C.1/ .S2 / and P is unique up to a constant.

(5.1.27)

5 Vectorial Spherical Harmonics in R3

216

Proof. Take an arbitrary, but fixed 0 2 S2 . We set Z



  f ./ d./

P ./ D

(5.1.28)

0

with the integral along any curve that starts at 0 and ends at  2 S2 . Then, for 0 ; 1 2 S2 and any curve C  S2 starting at 0 and ending at 1 , we obtain Z

1

P .1 /  P .0 / D

  f ./ d./

(5.1.29)

  r P ./ d./;

(5.1.30)

0

and

Z P .1 /  P .0 / D

1 0

since the surface gradient acts like an ordinary gradient in R3 when integrating along lines on S2 . Therefore, we are able to combine the two equations to Z

1 0

  .f ./  r P .// d./ D 0

(5.1.31)

for any curve C  S2 connecting 0 and 1 . By Lemma 5.1.2 we find that f ./  r P ./ D 0;

 2 S2 :

(5.1.32)

The proof that P is continuously differentiable can be found in standard lectures on vector analysis (see, e.g., J¨anich 2004). One basic idea is to take P constant on each straight line passing through S2 in the normal direction. In order to prove uniqueness, we consider r P1 ./ D r P2 ./,  2 S2 , which implies that r .P1  P2 /./ D 0. Therefore, due to Lemma 5.1.3, P1  P2 D const. t u Theorem 5.1.5. Let f 2 c.1/ .S2 / be a tangential vector field. Then, L  f ./ D 0,  2 S2 , if and only if there is a scalar field P such that f ./ D r P ./;

 2 S2 :

(5.1.33)

P is called a potential function for f and is unique up to an additive constant. Similarly, r  f ./ D 0,  2 S2 , if and only if there is a scalar field S such that f ./ D L S./;

 2 S2 :

(5.1.34)

S is called a stream function for f and is unique up to an additive constant.

5.1 Basic Notation

217

Proof. The condition f D r  P implies that L  f D L  r  P D 0. f D L S implies that r   f D r   L S D 0. Conversely, assume that L  f ./ D 0,  2 S2 . Then, we can use the surface theorem of Stokes (Theorem 4.1.2) to obtain Z C

  f ./ d./ D 0

(5.1.35)

for every closed curve C  S2 . By Lemma 5.1.4 there exists a scalar field P such that f D r  P , where P is unique up to a constant. Suppose now that r  f ./ D 0,  2 S2 . For  2 S2 (note that f D ftan is a tangential field and that r   D 2): L  . ^ f .// D . ^ r /  . ^ f .//

(5.1.36)

D .  /.r  f .//  .  f .//.r   / D r  f ./  2.  ftan .// D r  f ./; i.e., L  . ^ f .// D 0, since r  f ./ D 0. Thus, as before, there exists a scalar field S (unique up to a constant) such that   ^ f ./ D r S./;

 2 S2 :

(5.1.37)

This is equivalent to   ^ . ^ f .// D . ^ r /S./;

 2 S2 ;

(5.1.38)

 ^ . ^ f .// D .  f .//  .  /f ./

(5.1.39)

or f D L S on S2 , because

D .  ftan .// C f ./ D f ./: t u

This yields the second part of Theorem 5.1.5.

Theorem 5.1.6. Let f 2 c.1/ .S2 / be a tangential vector field such that, for all  2 S2 ; r  f ./ D 0

and

L  f ./ D 0:

(5.1.40)

Then, f D 0 on S2 . Proof. Since L  f ./ D 0, Theorem 5.1.5 guarantees the existence of a scalar field P such that f ./ D r P ./ for all  2 S2 . Moreover, r  f ./ D 0, i.e., r  r P ./ D  P ./ D 0. Because of (4.1.19) in Theorem 4.1.4 we find that

5 Vectorial Spherical Harmonics in R3

218

Z  2 r P ./ dS./ D 0:

(5.1.41)

S2

Hence, f ./ D r P ./ D 0.

t u

5.2 Definition of Vector Spherical Harmonics Now, we define three operators mapping scalar functions to vectorial functions. We analyze the properties of these operators as well as their adjoints and use them to define vector spherical harmonics. In this form the introduction goes back to Freeden et al. (1998). Definition 5.2.1. For  2 S2 and F 2 C.0i / .S2 /, we define the operators o.i / W C.0i / .S2 / ! c.S2 /;

i D 1; 2; 3;

(5.2.1)

by .1/

o F ./ D F ./;

(5.2.2)

o F ./ D r F ./;

(5.2.3)

o F ./ D L F ./;

(5.2.4)

.2/ .3/

where we have used the abbreviation ( 0i D

0

if i D 1;

1

if i D 2; 3:

(5.2.5)

We also introduce the notation N0i , which is N0 if i D 1 and N if i D 2; 3. It is clear that o.1/ F 2 cnor .S2 / and o.2/ F; o.3/ F 2 ctan .S2 /. Remark 5.2.2. Let F 2 C.0i / .S2 / be an -zonal function, i.e., F ./ D F .  /. Then, .1/

o F ./ D F .  /;

(5.2.6)

o F ./ D r F .  / D F 0 .  /.  .  //;

(5.2.7)

o F ./ D L F .  / D F 0 .  /. ^ /:

(5.2.8)

.2/ .3/

By Green’s integral formulas we can introduce the adjoint operators.

5.2 Definition of Vector Spherical Harmonics

219

Definition 5.2.3. For f 2 c.0i / .S2 / and G 2 C.0i / .S2 / (i D 1; 2; 3), we have that ho.i /G; f il2 .S2 / D hG; O .i / f iL2 .S2 / :

(5.2.9)

Therefore, for f 2 c.0i / .S2 /,  2 S2 , we find the adjoint operators to o.1/ , o.2/ , and o.3/ : .1/

O f ./ D   pnor f ./;

(5.2.10)

O f ./ D r  ptan f ./;

(5.2.11)

O f ./ D L  ptan f ./:

(5.2.12)

.2/ .3/

Lemma 5.2.4. Let F 2 C.2/ .S2 /. .i / .j /

(i) If i ¤ j , i; j 2 f1; 2; 3g, then O o F ./ D 0,  2 S2 . ( F ./ ; i D 1; .i / .i / (ii) O o F ./ D   F ./ ; i D 2; 3: Proof. These are well-known properties of the differential operators r  and L . t u Definition 5.2.5. For any Yn 2 Harmn .S /, the vector field 2

yn.i / D o.i / Yn ;

n  0i ; i 2 f1; 2; 3g;

(5.2.13)

is called a vector spherical harmonic of degree n and type i . By harm.in / .S2 / we denote the space of all vector spherical harmonics of degree n and type i . We also set the spaces .1/

harm0 .S2 / D harm0 .S2 /;

harmn .S2 / D

3 M

harm.in / .S2 /;

n  1:

(5.2.14)

i D1 .1/

.2/

.3/

Remark 5.2.6. We have  ^ yn D 0,   yn D 0, and   yn D 0. Theorem 5.2.7. If fYn;k gn2N0 ;kDn;:::;n is an L2 .S2 /-orthonormal system of scalar spherical harmonics, then the system of .i /

yn;k ./ D q

1 .i / n

.i /

o Yn;k ./;

i D 1; 2; 3; n  0i ; k D n; : : : ; n;

(5.2.15)

with .in /

(  .i / .i /  1 D O o Yn;k L2 .S2 / D n.n C 1/

;

i D 1;

;

i D 2; 3;

(5.2.16)

5 Vectorial Spherical Harmonics in R3

220

forms an l2 .S2 /-orthonormal system of vector spherical harmonics, i.e., Z S2

.i /

.j /

yn;k ./  ym;l ./ dS./ D ın;m ık;l ıi;j :

(5.2.17)

Proof. This follows directly from the orthogonality of the scalar spherical harmont u ics and the properties of the operators o.i / , O .i / with i D 1; 2; 3. In Fig. 5.1 some orthonormal vector spherical harmonics of type 2 and of type 3 are depicted to give an impression of the different characteristic behavior.

5.3 The Helmholtz Decomposition Theorem In the following, we formulate and prove the decomposition theorem for spherical vector fields that motivates the choice of the three operators o.i / . Theorem 5.3.1 (Helmholtz Decomposition Theorem). Let f be of class c.1/ .S2 /. There exist uniquely determined functions F1 2 C.1/ .S2 / and F2 ; F3 2 C.2/ .S2 / satisfying Z S2

Fi ./ dS./ D 0;

i D 2; 3;

(5.3.1)

such that f ./ D

3 X

o.i /Fi ./ D F1 ./ C r F2 ./ C L F3 ./;

 2 S2 :

(5.3.2)

i D1

The functions Fi are given by .1/

F1 ./ D O f ./;  2 S2 ; Z G. I ; /O.2/ f ./ dS./; F2 ./ D  S2

Z F3 ./ D 

S2

G. I ; /O.3/ f ./ dS./;

(5.3.3)  2 S2 ;

(5.3.4)

 2 S2 ;

(5.3.5)

where G. I ; / is the Green function with respect to the Beltrami operator of Definition 4.6.1. Proof. Any vector field f 2 c.1/ .S2 / can be written as f D fnor C ftan with 2 fnor D pnor f 2 c.1/ nor .S /;

(5.3.6) .1/

ftan D ptan f 2 ctan .S2 /:

(5.3.7)

5.3 The Helmholtz Decomposition Theorem

221

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05 0 0.45

0.45 0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

.i/

Fig. 5.1 Vector spherical harmonics yn;k of type i D 2 (left column) and of type i D 3 (right column), the degree n D 8 in each case, the order k D 0 in the top row, k D 4 in the middle row, and k D 8 in the bottom row .1/

Clearly, fnor ./ D o.1/ F1 ./ with F1 ./ D O f ./ for  2 S2 . For the tangential field ftan , the Stokes theorem for S2 (Theorem 4.1.2 for a closed surface, i.e., there is no boundary) yields that Z S2

L  ftan ./ dS./ D 0:

(5.3.8)

5 Vectorial Spherical Harmonics in R3

222

Thus, L  ftan is suitable as a right-hand side of the Beltrami differential equation and by Theorem 4.6.9 we find F3 2 C.2/ .S2 / such that Z

 F3 D L  ftan

and S2

F3 ./ dS./ D 0:

(5.3.9)

In other words, L  L F3 D L  ftan

or

L  .ftan  L F3 / D 0:

(5.3.10)

Note that the difference ftan  L F3 is still a tangential vector field. Therefore, we can now use Theorem 5.1.5 which gives us a scalar field F2 2 C.2/ .S2 / with ftan  L F3 D r  F2

or ftan D r  F2 C L F3 :

(5.3.11)

R R 1 If S2 F2 ./ dS./ ¤ 0, we replace F2 by FQ2 D F2  4 S2 F2 ./ dS./. Then, it holds that Z r  F2 D r  FQ2 and FQ2 ./ dS./ D 0: (5.3.12) S2

Assume that there exists another triple Gi , i D 1; 2; 3, such that .1/

.2/

.3/

.1/

.2/

f ./ D o F1 ./ C o F2 ./ C o F3 ./; .3/

f ./ D o G1 ./ C o G2 ./ C o G3 ./:

(5.3.13) (5.3.14)

Then, it follows that F1 D O .1/ f D G1 , i.e., F1 is uniquely defined. Thus, .2/

.3/

.2/

.3/

o F2 ./ C o F3 ./ D o G2 ./ C o G3 ./:

(5.3.15)

We now apply O .2/ and O .3/ which yield  F2 D  G2 ;

 F3 D  G3 :

(5.3.16)

Hence, we find uniqueness up to a constant and the normalization conditions on F2 and F3 imply that F2 D G2 and F3 D G3 . The specific representation of F2 , F3 involving integrals with Green’s function G. I ; / follows directly from Theorem 4.6.9. t u Remark 5.3.2. A vector field of the form  7! F1 ./ C r F2 ./;

 2 S2 ;

(5.3.17)

is called spheroidal, one of the form  7! L F3 ./;

 2 S2 ;

(5.3.18)

5.4 Closure and Completeness of Vector Spherical Harmonics

223

is said to be toroidal (cf. Backus 1967, 1986). Thus, the Helmholtz decomposition theorem represents the decomposition of a c.1/ -vector field into its spheroidal and its toroidal parts. It also implies an orthogonal decomposition of the space c.1/ .S2 /, namely .1/

.1/

.1/

c.1/ .S2 / D c.1/ .S2 / ˚ c.2/ .S2 / ˚ c.3/ .S2 /;

(5.3.19)

2 c.1/ .S2 / D c.1/ nor .S /; ˚  .1/ c.2/ .S2 / D f 2 c.1/ .S2 / W O .1/ f D O .3/ f D 0 ; ˚  .1/ c.3/ .S2 / D f 2 c.1/ .S2 / W O .1/ f D O .2/ f D 0 :

(5.3.20)

where .1/

(5.3.21) (5.3.22)

These definitions can also be extended to c.k/ .S2 /, k 2 N, to c.S2 / and to l2 .S2 / by a density argument, i.e., kkc.S2 / ˚ c.i / .S2 / D o.i / F W F 2 C.1/ .S2 / ;

(5.3.23)

˚ kkl2 .S2 / l2.i / .S2 / D o.i / F W F 2 C.1/ .S2 / :

(5.3.24)

Therefore, we find the orthogonal decompositions l2 .S2 / D l2nor .S2 / ˚ l2tan .S2 /;

(5.3.25)

l2tan .S2 / D l2.2/ .S2 / ˚ l2.3/ .S2 /:

(5.3.26)

5.4 Closure and Completeness of Vector Spherical Harmonics Next, we prove the closure and completeness of vector spherical harmonics intrinsically on the sphere (note that a non-intrinsic proof is included in Sect. 5.5). For our purpose here, we use vectorial variants of the scalar zonal Bernstein kernels (cf. Freeden and Gutting 2008). The vector zonal Bernstein kernel approximations can be shown to guarantee the closure property in the space of continuous spherical normal as well as tangential vector fields. In consequence, they also assure closure and completeness in the Hilbert space of (Lebesgue-)square-integrable vector fields. Essential tools are the theory of the Green function with respect to the Beltrami operator of Sect. 4.6 and the Helmholtz decomposition theorem (Theorem 5.3.1) of Sect. 5.3.

5 Vectorial Spherical Harmonics in R3

224

We start our considerations by convolving Green’s function with respect to the Beltrami operator (see Definition 4.6.1 or (4.6.6) for the explicit representation) against the Bernstein kernel (4.4.28) of Definition 4.4.6,   BGn .  / D G. I  /  KB;n . / ./ D

Z S2

G. I   /KB;n .  / dS. /: (5.4.1)

Written in terms of a Legendre series, we find the finite sum BGn .  / D

n ^ X .k/ 2k C 1 KB;n kD1

k.k C 1/

4

Pk .  /:

(5.4.2)

Note that the Bernstein kernel is a polynomial and the Green function is of class L1 .Œ1; 1 /, hence, the existence of the convolution integral as a bandlimited Legendre expansion is obvious. Next, we are interested in the Bernstein summability of Fourier expansions in terms of vector spherical harmonics. To this end, we need some preparatory material (more precisely, Lemmas 5.4.1 and 5.4.2). An essential tool of our considerations is the Green function with respect to the Beltrami operator (cf. Freeden and Gutting 2008). Lemma 5.4.1. For i 2 f1; 2; 3g, .n/

lim kFi  Fi kC.S2 / D 0;

(5.4.3)

n!1

where Fi , i D 1; 2; 3, are the functions occurring in the Helmholtz decomposition theorem (Theorem 5.3.1) .1/

F1 ./ D O f ./; Z Fi ./ D  G. I   /O.i / f ./ dS./;

(5.4.4) i D 2; 3;

(5.4.5)

S2

.n/

and Fi , i D 1; 2; 3, are given by Z .n/

F1 ./ D .n/

S2

Fi ./ D 

KB;n .  /O.1/ f ./ dS./;

(5.4.6)

Z S2

BGn .  /O.i / f ./ dS./;

i D 2; 3:

(5.4.7)

Proof. Clearly, the case i D 1 of Lemma 5.4.1 is easy to handle. It follows immediately from the scalar theory. Thus, it remains to study the cases i D 2; 3. We start from the convolution integrals

5.4 Closure and Completeness of Vector Spherical Harmonics

  .n/ Fi ./ D  BGn  O .i / f ./ D 

225

Z S2

BGn .  /O.i / f ./ dS./;

(5.4.8)

i D 2; 3. It is not difficult to see that kFi  Fi kC.S2 / D kG. I /  O .i /f  BGn  O .i / f kC.S2 / .n/

 kO .i / f kC.S2 / kG. I /  BGn kL1 .Œ1;1 / :

(5.4.9)

Since both kernels G. I / and BGn are of class L2 .Œ1; 1 / and, for all k 2 N0 , the ^ Legendre coefficients of the Bernstein kernel KB;n .k/ converge to 1 for n tending to infinity, we are able to deduce that lim kG. I /  BGn kL2 .Œ1;1 / D 0:

n!1

(5.4.10)

.n/

Obviously, this implies L1 -convergence as well as kFi  Fi kC.S2 / ! 0 for i D 1; 2; 3 and n ! 1, as required. u t Considering the o.i /-derivatives, we have to verify the following lemma. Lemma 5.4.2. For i 2 f1; 2; 3g, ˇ ˇ ˇ ˇ .i / .i / .n/ lim sup ˇo Fi ./  o Fi ./ˇ D 0:

n!1

(5.4.11)

2S2

Proof. The case i D 1 is obvious and it is not difficult to see that, for i 2 f2; 3g, .i /

.i /

.n/

ko Fi ./  o Fi ./kc.S2 / ˇ Z Z ˇ .i / .i / .i / D sup ˇˇo G. I   /O f ./ dS./o S2

2S2

ˇZ ˇ D sup ˇˇ 2 2S

S2

o G. I   /O f ./ dS./ .i /

S2

.i /

Z S2

(5.4.12) ˇ ˇ .i / BGn .  /O f ./ dS./ˇˇ

ˇ ˇ .i / .i / o BGn .  /O f ./ dS./ˇˇ;

where it is clear that the operator o.i / can be drawn inside the two integrals. This leads us to the following estimate: ˇZ ˇ Z ˇ ˇ .i / .i /  .i / .i / ˇ sup ˇ o G. I   /O f ./ dS./  o BGn .  /O f ./ dS./ˇˇ 2 2 2

2S

S

S

Z ˇ ˇˇ ˇ ˇˇ ˇ ˇ .i / .i /  sup ˇo G. I   /  o BGn .  /ˇ ˇO.i / f ./ˇ dS./ 2S2

S2

 kO .i / f kC.S2 / sup 2S2

Z ˇ ˇ ˇ .i / ˇ .i / ˇo G. I   /  o BGn .  /ˇ dS./: (5.4.13) S2

5 Vectorial Spherical Harmonics in R3

226

We have to study the convergence of the last integral. In more detail, we are interested in proving that Z ˇ ˇ ˇ .i / ˇ .i / lim sup ˇo G. I   /  o BGn .  /ˇ dS./ D 0:

n!1

2S2

(5.4.14)

S2

.i /

For that purpose, we notice that the Bernstein kernels o BGn .  /; i D 2; 3; admit the (Legendre) series expansions .2/

o BGn .  / D

n ^ X .k/ 2k C 1 KB;n kD1

.3/

o BGn .  / D

4

k.k C 1/

n ^ X .k/ 2k C 1 KB;n kD1

4

k.k C 1/

Pk0 .  / .  .  // ;

(5.4.15)

Pk0 .  / . ^ / :

(5.4.16)

Moreover, an easy calculation shows that the application of the o.i / -operators, i D 2; 3, to the Green function with respect to the Beltrami operator leads us to the identities o G. I   / D  .2/

1   .  / 1  ^ .3/ ; o G. I   / D  : 4 1     4 1     (5.4.17)

Consequently, for i D 2; 3, our integral can be expressed in the form Z ˇ ˇ ˇ ˇ .i / .i / ˇo G. I   /  o BGn .  /ˇ dS./ S2

D

(5.4.18)

ˇ .i / Z ˇ n ˇ 1 o .  / 1 X ˇ 2k C 1 ^ .i / 0 ˇ ˇ dS./  K .k/P .  /o .  / B;n k  ˇ ˇ 4 k.k C 1/ S2 4 1     kD1

ˇ Z ˇ n ˇ ˇˇ X ˇ 1 1 2k C 1 ^ ˇˇ ˇ .i / 0 ˇ dS./ D  K .  / .k/P .  / ˇˇ ˇo B;n k ˇ 4 S2 1  k.k C 1/ kD1

D

1 2

Z

1 1

ˇ ˇ n X p ˇ 1 ˇ 2k C 1 ^ 0 2 ˇ  K .k/Pk .t/ˇˇ dt: 1t ˇ 1t k.k C 1/ B;n kD1

At this point, we use the recurrence relation (3.4.56) of Theorem 3.4.11 for the Legendre polynomials, i.e.,  D 12 in the version for ultraspherical polynomials. This gives us the identity

5.4 Closure and Completeness of Vector Spherical Harmonics

227

Z ˇ ˇ ˇ .i / ˇ .i / ˇo G. I   /  o BGn .  /ˇ dS./

(5.4.19)

S2

1 D 2 D

1 2

Z

1

r

ˇ ˇ n X 1 C t ˇˇ PkC1 .t/  Pk1 .t/ ˇˇ ^ 1  .1  t/ KB;n .k/ ˇ dt 1t ˇ t2  1

r

ˇ ˇ n ˇ 1 X ^ 1 C t ˇˇ 1C KB;n .k/ .PkC1 .t/  Pk1 .t// ˇˇ dt: ˇ 1t 1Ct

1

Z

1

1

kD1

kD1

For the occurring sum, it follows that n X

^ KB;n .k/ .PkC1 .t/  Pk1 .t//

kD1 ^ ^ ^ ^ DKB;n .n/PnC1 .t/ C KB;n .n  1/Pn .t/  KB;n .2/P1 .t/  KB;n .1/P0 .t/

C

n1 X  ^  ^ KB;n .k  1/  KB;n .k C 1/ Pk .t/;

(5.4.20)

kD2

where a simple calculation shows that ^ ^ ^ KB;n .k  1/  KB;n .k C 1/ D KB;nC1 .k/.2k C 1/

2 : .n C 2/

(5.4.21)

We put (5.4.21) into (5.4.20) getting the following result: n X

^ KB;n .k/ .PkC1 .t/  Pk1 .t//

(5.4.22)

kD1 ^ ^ ^ .n/PnC1 .t/ C KB;n .n  1/Pn .t/  KB;n .2/ P1 .t/ D KB;n

2 X ^ KB;nC1 .k/.2k C 1/Pk .t/ nC2 n1

^  KB;n .1/P0 .t/ C

kD2

D

2 nC2

nC1 X

^ KB;nC1 .k/.2k C 1/Pk .t/  .1 C t/

kD0

 1 C t nC1 2 .n C 1/  .1 C t/: D nC2 2 Keeping this result in mind, we return to the integral (5.4.19). As a matter of fact, the identity (5.4.19) can be rewritten in the form

5 Vectorial Spherical Harmonics in R3

228

r

ˇ ˇ n ˇ 1 X ^ 1 C t ˇˇ ˇ dt K .k/ .P .t/  P .t// 1 C kC1 k1 B;n ˇ ˇ 1t 1Ct 1 kD1 ˇ  ˇ Z r 1 1 1 C t ˇˇ n C 1 1 C t n ˇˇ D ˇ dt: 2 1 1  t ˇ n C 2 2

1 2

Z

1

(5.4.23)

Clearly, as the Bernstein kernel is non-negative, we are left with the integral expression  Z ˇ Z r ˇ 1Ct 1Ct n 1nC1 1 ˇ ˇ .i / .i /  G. I   /  o BG .  / dS./ D dt ˇ ˇo n  2 n C 2 1 1  t 2 S2 D

.n C 32 / . 12 / .n C 2/

;

(5.4.24)

which follows by induction. It is well-known that the value of our integral can be estimated as follows: .n C 32 / 1 2 p < :