Approximation with positive linear operators and linear combinations 978-3-319-58795-0, 3319587951, 978-3-319-58794-3

Table of contents :
Front Matter ....Pages i-xiii
Moments and Combinations of Positive Linear Operators (Vijay Gupta, Gancho Tachev)....Pages 1-14
Direct Estimates for Approximation by Linear Combinations (Vijay Gupta, Gancho Tachev)....Pages 15-23
Inverse Estimates and Saturation Results for Linear Combinations (Vijay Gupta, Gancho Tachev)....Pages 25-36
Voronovskaja-Type Estimates (Vijay Gupta, Gancho Tachev)....Pages 37-65
Pointwise Estimates for Linear Combinations (Vijay Gupta, Gancho Tachev)....Pages 67-87
Voronovskaja’s Theorem in Terms of Weighted Modulus of Continuity (Vijay Gupta, Gancho Tachev)....Pages 89-116
Direct Estimates for Some New Operators (Vijay Gupta, Gancho Tachev)....Pages 117-155
Convergence for Operators Based on Pǎltǎnea Basis (Vijay Gupta, Gancho Tachev)....Pages 157-173
Back Matter ....Pages 175-186

Citation preview

Developments in Mathematics

Vijay Gupta Gancho Tachev

Approximation with Positive Linear Operators and Linear Combinations

Developments in Mathematics Volume 50

Series editors Krishnaswami Alladi, Gainesville, USA Hershel M. Farkas, Jerusalem, Israel

More information about this series at http://www.springer.com/series/5834

Vijay Gupta • Gancho Tachev

Approximation with Positive Linear Operators and Linear Combinations

123

Vijay Gupta Department of Mathematics Netaji Subhas Institute of Technology New Delhi, India

Gancho Tachev Department of Mathematics University of Architecture Civil Engineering and Geodesy Sofia, Bulgaria

ISSN 1389-2177 ISSN 2197-795X (electronic) Developments in Mathematics ISBN 978-3-319-58794-3 ISBN 978-3-319-58795-0 (eBook) DOI 10.1007/978-3-319-58795-0 Library of Congress Control Number: 2017940878 Mathematics Subject Classification: 41A25, 41A30, 30E05, 30E10 © Springer International Publishing AG 2017 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. 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Approximation of functions by positive linear operators is an important branch of the approximation theory. To increase the order of approximation a useful tool is the method of linear combinations of positive linear operators (p.l.o.). The most known example of p.l.o. is the famous Bernstein operators introduced by S. Bernstein [26, 27], which for f 2 CŒ0; 1 is given by Bn .f ; x/ D

n X kD0

pn;k .x/f .k=n/;

! n k pn;k .x/ D x .1  x/nk : k

In 1932 Elena Voronovskaja [193]—a doctoral student of S. Bernstein—proved that if f is bounded on Œ0; 1, differentiable in some neighbourhood of x and has second derivative f 00 for some x 2 Œ0; 1 then lim nŒBn .f ; x/  f .x/ D

n!1

x.1  x/ 00 f .x/: 2

If f 2 C2 Œ0; 1 the convergence is uniform. This result shows that the rate of convergence Bn .f ; x/  f .x/ ! 0 as n ! 1 is of order not better than 1=n if f 00 .x/ ¤ 0: In this sense the theorem of E. Voronovskaja is a first example of saturation, i.e. the optimal rate of convergence is 1=n: To increase the rate of convergence, it was P. L. Butzer [34] who introduced in 1953 the following linear combinations: .2r  1/Bn;r .f ; x/ D 2r B2n;r1 .f ; x/  Bn;r1 .f ; x/; Bn;0 .f ; x/  Bn .f ; x/: Butzer showed that, for smooth functions f ; Bn;r .f ; x/  f .x/ tends to 0 faster than Bn .f ; x/  f .x/: More general combinations are considered by Rathore [163] and

v

vi

Preface

C. P. May in their PhD thesis and in the year 1976 in [145]. The kth order linear combinations Ln .f ; k; x/ of the operators Ldj n .f ; x/; discussed by [145], are given as Ln .f ; k; x/ D

k X

C.j; k/Ldj n .f ; x/;

jD0

where C.j; k/ D

k Y iD0 i¤j

dj ; k ¤ 0I C.0; 0/ D 1 dj  di

and d0 ; d1 ; :::::; dk are k C 1 distinct arbitrary and fixed positive integers. Z. Ditzian, the scientific advisor to C. P. May, in the famous book Moduli of Smoothness [50] written jointly with V. Totik in 1987 generalized the known methods of linear combinations (see Chapter 9 in the book). The linear combinations Ln;r for r 2 N of the operators Lni are given by Ln;r .f ; x/ D

r1 X

˛i .n/Lni .f ; x/

iD0

where the numbers ni and coefficients ˛i .n/ satisfy the following four conditions: (a) (b) (c) (d)

an D n0 < n1 <    nr1  An; Pr1 iD0 j˛i .n/j < C; Pr1 iD0 ˛i .n/ D 1; Pr1  D 0;  D 1; 2;    r  1: iD0 ˛i .n/ni

The last two conditions represent a linear system for the coefficient ˛i .n/ with unique solution ˛i .n/ D

r1 Y kD0;k¤i

ni : ni  nk

Note that Ln;0 D Ln (for a D 1). In our book we follow this more general framework of linear combinations. In Chapter 9 in [50], Ditzian and Totik among others proved the following equivalence result (see Theorem 9.3.2) jjLn;r f  f jjB D O.n˛=2 / , !'2r .f ; h/B D O.h˛ /; 0 < ˛ < 2r; where the space B and weight function 'pare defined as follows: for D x.1  x/I for Ln D Sn — Ln D Bn —Bernstein operator, B D CŒ0; 1; '.x/ p Szász–Mirakjan operator B D CŒ0; 1/; '.x/ D xI for Ln D Vn —Baskakov

Preface

vii

p operator B D CŒ0; 1/; '.x/ D x.1 C x/ and lastly for their Kantorovich variants the weight functions remain the same, where B D Lp Œ0; 1 for BOn and B D Lp .0; 1/; 1  p < 1 for SOn ; VOn and Ln;r are the linear combinations of these classical p.l.o.—Bernstein, Szász–Mirakjan and Baskakov operators and their Kantorovich [126] modifications. The case of Post–Widder operators was also considered. Since then in the last three decades, hundreds of papers have appeared and considering different problems connected with the methods of linear combinations of p.l.o. We only mention the dissertation of M. Heilmann [115] published in 1992 which may be considered as a second systematical study of linear combinations attached to Durrmeyer modifications of three classical operators mentioned above. It is hardly possible to mention all results on this topic. Together with known results in the past we include also the new results obtained very recently in our joint papers and also results obtained by many other mathematicians in the past 5–10 years. Some of the results are formulated and the reader may find the proofs in the references given at the end of the book. The book consists of eight chapters. In the first two chapters, we give the known results about the closed expressions (when it is possible) of the moments and the central moments of the operators Ln , two expressions which are crucial tools for further investigation of approximation by linear combinations. Direct and inverse estimates for a broad class of p.l.o. are considered in the next chapters. The cases of finite and unbounded intervals of the real-valued and complex-valued functions are considered. We list also the results for approximation by linear combinations in a pointwise form, obtained very recently. The known strong converse inequalities of type A in the terminology of Ditzian–Ivanov [51] for linear combinations of Bernstein and Bernstein–Kantorovich operators are also included. We represent also various Voronovskaja-type estimates for some linear combinations. Some open problems are also outlined, concerning the approximation by linear combinations of p.l.o. Quantitative estimates for the sequences of p.l.o. play an important role not merely in approximating the functions, but also in finding the error of approximation. One of the most important convergence results in the theory of approximation is the Voronovskaja-type theorem, which describes the rate of pointwise convergence. The quantitative version of the Voronovskaja theorem for any p.l.o. acting on compact intervals was obtained in [80]. Also Acar–Aral–Rasa in [7] established quantitative results for weighted modulus of continuity in the recent years. Pˇaltˇanea in [156, 157] introduced the weighted modulus of continuity. Here we discuss some of the results appeared in the recent years on such problems. Also in the last 3 years some papers on new hybrid operators appeared; we also discuss some of them. In the recent years R. Pˇaltˇanea in [155] proposed the generalization of Phillips operators based on certain parameter  > 0, which has a link to the well-known Szász–Mirakjan operators in limiting case. After that also many such operators have been appropriately modified so that they depend on certain parameters and in the limiting case they reduce to the well-known operators available in the literature. We also discuss some of the papers in this direction.

viii

Preface

It is our goal in this book to describe the most interesting features connected with approximation by linear combinations of p.l.o. We hope our book may not only be considered as a systematic overview but also be served as a basis for future study and development of this method. New Delhi, India Sofia, Bulgaria

Vijay Gupta Gancho Tachev

Some Words

The first author works as a professor in the Department of Mathematics, NSIT, New Delhi. His area of research is Approximation Theory, especially on linear positive operators and application of q-calculus in approximation theory. He has collaborated jointly with many researchers globally. The first author is inspired by the work of many researchers specially of Prof. Zeev Ditzian, Prof. Ulrich Abel, Prof. Margareta Heilmann, Prof. Mircea Ivan, Prof. Heiner Gonska, Prof. Gradimir V. Milovanovi´c, Prof. Octavian Agratini, Prof. Radu Pˇaltˇanea, Prof. Sorin Gal, Prof. Purshottam Narain Agrawal, Prof. Th. M. Rassias and Prof. Ali Aral. He got a chance to meet some of them personally during his visits to Jaen University, Spain; Indian Institute of Technology Roorkee, India; Lucian Blaga University, Romania, and Kirikkale University, Turkey. The second author works in the Department of Mathematics at the University of Architecture, Civil Engineering and Geodesy, Sofia, Bulgaria. His attention to the theory of positive linear operators, especially on linear combinations, was brought to him during his visits to the Universities of Duisburg-Essen and Wuppertal in the period 2000–2012 and collaboration with Prof. Heiner Gonska and Prof. Margareta Heilmann. The participation at Romanian-German Seminars on Approximation Theory in the last 15 years has given the opportunity to establish scientific contacts and collaboration with Prof. Radu Pˇaltˇanea, Prof. Ioan Gavrea, Prof. Sorin Gal, Prof. Daniela Kacso and others. In the last 4 years as a result of the joint work of both the authors several papers have appeared, dealing with approximation by linear combinations and direct estimates of p.l.o. The authors believe that the book will motivate other mathematicians to obtain new exciting results on this topic. They are thankful to their collaborative researchers, friends and also students for valuable suggestions. The authors thank the Springer-Verlag team for publishing the book timely. They are grateful to the reviewers and Ms. Razia Amzad for valuable suggestions leading to overall improvements of the manuscript.

ix

Contents

1 Moments and Combinations of Positive Linear Operators . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Some Kind of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Discretely Defined Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Kantorovich Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Durrmeyer Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Integral Type Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Phillips Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Szász–Mirakjan–Baskakov Operators . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Szász–Mirakjan–Laguerre Operators . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 2 5 6 9 9 10 12

2

Direct Estimates for Approximation by Linear Combinations . . . . . . . . . 2.1 Direct Estimates in Lp -Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Direct Results for Linear Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Post–Widder Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Durrmeyer Type Operators: Bn ; Sn ; V n . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Phillips Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Szász–Mirakjan–Baskakov Operators . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 18 18 20 21 22

3

Inverse Estimates and Saturation Results for Linear Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Inverse Estimates for Linear Combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Operators of Type: Bn ; Sn ; Vn ; b Bn ; b Sn ; b Vn . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Durrmeyer Type Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Phillips Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Saturation Results for Linear Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 25 26 29 31

Voronovskaja-Type Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 New Quantitative Estimates of Voronovskaja Type . . . . . . . . . . . . . . . . . . . 4.2 Voronovskaja-Type Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 In Terms of !' .f 00 ; ı/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 In Terms of !'  .f 00 ; ı/; 0    1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 43 43 45

4

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Contents

4.2.3 For f 2 C3 Œ0; 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 For Higher Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Asymptotic Expansion for Some Positive Linear Operators . . . . . . . . . . 4.4 Voronovskaja-Type Estimate for Schoenberg Operators . . . . . . . . . . . . . . 4.5 Voronovskaja-Type Estimates for Linear Combinations . . . . . . . . . . . . . . 4.5.1 Phillips Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Durrmeyer Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Simultaneous Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46 48 52 54 57 57 62 63

5

Pointwise Estimates for Linear Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Approximation by Linear Combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Discrete Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Szász–Mirakjan–Durrmeyer Operators . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Bernstein Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Simultaneous Pointwise Approximation by Bn;r . . . . . . . . . . . . . . . . . . . . . . . 5.3 Pointwise Estimates for Combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Bernstein–Kantorovich Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Phillips Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Linear Combinations of Complex Phillips Operators . . . . . . . . . . . . . . . . .

67 67 67 68 70 75 76 76 77 82

6

Voronovskaja’s Theorem in Terms of Weighted Modulus of Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Quantitative Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 General Form of Voronovskaja’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Applications of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Szász–Mirakjan Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Baskakov Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Phillips Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Voronovskaja’s Estimate for Functions with Exponential Growth . . . 6.5 Applications of Voronovskaja’s Estimate for Functions with Exponential Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Baskakov Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Szász–Mirakjan Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Phillips Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 Lupa¸s–Szász Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Voronovskaja-Type Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Derivatives of Szász–Mirakjan Operators . . . . . . . . . . . . . . . . . . . . . 6.6.2 For Linear Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

102 102 103 104 105 108 108 109

Direct Estimates for Some New Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Lupa¸s–Durrmeyer Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Direct Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Rate of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Genuine Integral Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Complex Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Modified Baskakov–Durrmeyer Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 117 118 122 123 124 134

7

89 89 90 94 94 96 98 99

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7.3 Jain Generalization of Szász–Mirakjan–Durrmeyer Operators . . . . . . . 7.3.1 Moments Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Direct Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Quantitative Asymptotic Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Generalized Baskakov–Szász Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Baskakov–Jain Type Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Baskakov–Szász operators Based on IPED. . . . . . . . . . . . . . . . . . . . 7.5 Lupa¸s Type Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137 138 143 143 145 145 148 152

Convergence for Operators Based on Pˇaltˇanea Basis . . . . . . . . . . . . . . . . . . . . 8.1 Genuine Durrmeyer Type Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Genuine Bernstein–Durrmeyer Operators . . . . . . . . . . . . . . . . . . . . . 8.1.2 Genuine Hybrid Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Modified Phillips Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Some New Hybrid Durrmeyer Type Operators . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 General Lupa¸s Type Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Modified Baskakov–Szász Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Jakimovski–Leviatan–Pˇaltˇanea Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Some Other Linking Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 157 157 160 161 164 164 165 167 169

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Chapter 1

Moments and Combinations of Positive Linear Operators

1.1 Introduction The convergence of a sequence of positive linear operators (abbrev. p.l.o.) is one of the important areas of researchers related to approximation theory. Apart from the earlier known examples several new sequences of p.l.o. were introduced and their approximation properties have been discussed in the last few decades. There are several books in approximation theory, which deal with the linear and nonlinear operators of different kind. We mention here some of the books available in the related area, which are due to DeVore [42], DeVore–Lorentz [43], Ditzian–Totik [50] and Pˇaltˇanea [153]. Gal in [63] and [65] studied the quantitative overconvergence estimates in complex approximation by various operators. Mastroianni–Milovanovic in [144] and Szabados–Vértesi in [177] devoted only to the interpolation processes in their books. Aral–Gupta–Agarwal [18] discussed applications of q calculus in approximation theory. Also Gupta–Agarwal in [98] mainly studied the convergence estimates of p.l.o. The important book on Bernstein polynomials is due to Lorentz [134], actually Bernstein polynomials and Bézier curves are of fundamental importance in computer aided geometric design. The theoretical aspects of Bézier variants of some operators were discussed by many researchers, some of the results were compiled in [98]. A fundamental book for systematical study of Bézier curves and surfaces is due to Farin [55]. Also, for the application in CAGD of splines the well-known books are of de Boor [32] and Schumaker [168]. In the recent edited book by Rassias–Gupta [162] the achievements in pure mathematical analysis and approximation theory, as well as some of their applications have been highlighted. The book presents a collection of papers, which have been written by experts from the international mathematical community. The order of approximation for a sequence of p.l.o. Ln f is at best O.n1 /; even for smooth function, in this context C. P. May [145] (see also R. K. S. Rathore [163]), © Springer International Publishing AG 2017 V. Gupta, G. Tachev, Approximation with Positive Linear Operators and Linear Combinations, Developments in Mathematics 50, DOI 10.1007/978-3-319-58795-0_1

1

2

1 Moments and Combinations of Positive Linear Operators

considered the kth order linear combinations Ln .f ; k; x/ of the operators Ldj n .f ; x/; as follows: Ln .f ; k; x/ D

k X

C.j; k/Ldj n .f ; x/;

jD0

where C.j; k/ D

k Y iD0 i¤j

dj ; k ¤ 0I C.0; 0/ D 1 dj  di

and d0 ; d1 ; :::::; dk are k C 1 distinct arbitrary and fixed positive integers. Ditzian and Totik [50] considered the more general linear combinations Ln;r for r 2 N of the operators Lni which are defined by Ln;r .f ; x/ D

r1 X

˛i .n/Lni .f ; x/

iD0

where the numbers ni and coefficients ˛i .n/ satisfy the following four conditions: (a) an D n0 < n1 <    nr1  An; r1 X (b) j˛i .n/j < C; iD0

(c) (d)

r1 X iD0 r1 X

˛i .n/ D 1; 

˛i .n/ni

D 0;  D 1; 2;    r  1:

iD0

Also, there is another approach to improve the order of approximation, C. A. Micchelli [148] considered the iterates of the Bernstein polynomials and he was able to achieve the better order of approximation. In the present book we will not discuss these combinations, as only local results are available in literature for iterative combinations.

1.2 Some Kind of Operators 1.2.1 Discretely Defined Operators One of the most studied and popular linear positive operators is the famous Bernstein polynomials introduced by S. Bernstein in [26] and defined for f 2 CŒ0; 1

1.2 Some Kind of Operators

3

by ! n k Bn .f ; x/ D pn;k .x/f .k=n/; pn;k .x/ D x .1  x/nk : k kD0 n X

(1.2.1)

For f 2 CŒ0; 1/ the Szász–Mirakjan operators introduced in [178] are defined as Sn .f ; x/ D

1 X

sn;k .x/f .k=n/; sn;k .x/ D enx .nx/k =kŠ:

(1.2.2)

kD0

Also for f 2 CŒ0; 1/ the Baskakov operators introduced in [21] are defined as ! nCk1 xk vn;k .x/f .k=n/; vn;k .x/ D : Vn .f ; x/ D .1 C x/nCk k kD0 1 X

(1.2.3)

Mastroianni operators Mn (see [12, 143]) are defined as .Mn f /.x/ D

C1 X kD0

.1/k k .k/ x n .x/ f kŠ

  k ; n 2 N; n

where .n /n1 is a sequence of real-valued functions defined on RC D Œ0; C1/; which are infinitely differentiable on RC and satisfy the following conditions: .k/ n .0/ D 1 for every n 2 NI .1/k n .x/  0 for every n 2 N; x 2 RC and k 2 N [ f0g for each .n; k/ 2 N  N [ f0g; there exist a number p.n; k/ 2 N and a .iCk/ .i/ function ˛n;k 2 RRC ; such that n .x/ D .1/k p.n;k/ .x/ ˛n;k .x/; i 2 N[f0g; x 2 RC and lim

n n!C1 p.n;k/

D lim

n!C1

˛n;k .x/ nk

D 1: For n .x/  enx these operators provide

Szász–Mirakjan operators, for n .x/ D .1Cx/n , these provide Baskakov operators and for n .x/ D .1  x/n , one may get Bernstein polynomials. Further we adopt that CŒ0; 1/  L1 Œ0; 1/, i.e. we consider bounded functions on Œ0; 1/ also denoted by CB Œ0; 1/: All these operators defined above may be unified in a common definition, as it was given by M. Heilmann in her doctoral dissertationp [113, (2.14)], the weight function '.x/, associated with p these operators p is '.x/ D x.1  x/ for Bn , '.x/ D x for S and '.x/ D n    x.1 C x/ for Vn : Throughout this section En D An ; 1  An for Bn and En D An ; 1 for Sn ; Vn with A > 0 being fixed positive number. The central moment of order m for p.l.o Ln is defined by Ln;m .x/ D Ln ..t  x/m ; x/; m; n 2 N:

(1.2.4)

The absolute central moment be defined as L .x/ D Ln .jt  xjm ; x/; m; n 2 N: Mn;m

(1.2.5)

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1 Moments and Combinations of Positive Linear Operators

In the book of Ditzian–Totik [50, Lemma 9.4.3], the following recurrence relation was proved. Lemma 1.1 If Ln is one of the operators Bn ; Sn ; Vn and Am;n is given by Am;n .x/ D nm Ln;m .x/; then the recurrence relation AmC1;n D m:n:' 2 .x/Am1;n .x/ C ' 2 .x/A0m;n .x/;

(1.2.6)

A0;n .x/ D 1 and A1;n .x/ D 0 holds with '.x/ defined above for each of the three operators. Here we point out that similar recurrence relation holds true for almost all known positive linear operators as we will see later. From (1.2.6) we can recursively compute the moments Ln;m for example: Ln ..t  x/2 ; x/ D

' 2 .x/ ; n

(1.2.7)

where Ln is one of the operators Bn ; Sn ; Vn : The second moment Ln;2 plays an important role to establish direct and inverse estimates for the rate of approximation. We point out that it is hardly possible to represent the moment Ln;m for an arbitrary m 2 N exactly in a simple compact form as in (1.2.7) for m D 2: Even for the classical Bernstein polynomials, very recently J. Adell, J. Bustamante and J. Quesada in two papers (see for more details [9, 10]) in 2015 have proven sharp n and M Bn in a pointwise and C-norm. In [50, upper and lower bounds for Bn;m n;m Lemma 9.4.4] of the book by Ditzian–Totik, the following upper estimate was proved: Lemma 1.2 If Am;n as defined in Lemma 1.1, we have A2m;n .x/  C:nm :' 2m .x/

(1.2.8)

for x 2 En : The following representations for the central moments for the three operators Bn ; Sn ; Vn are given in [50, Lemma 9.5.5], which we formulate as: Lemma 1.3 For Ln —one of the operators Bn ; Sn ; Vn and the related ', we have Ln ..t  x/2j ; x/ D

j1 X ' 2.jm/ .x/ qm .x/ njCm mD0

Ln ..t  x/2jC1 ; x/ D

j1 X ' 2.jm/ .x/ pm .x/ njCmC1 mD0

(1.2.9)

(1.2.10)

1.2 Some Kind of Operators

5

where pm and qm are constants for Ln D Sn , fixed bounded polynomials for Ln D Bn ; and fixed bounded polynomials of degree 2m and 2m C 1, respectively, for Ln D Vn : The proof follows easily from the recurrence relation (1.2.6) and by induction. Now we arrive at the estimates for the moments Ln;r ..t  x/i ; x/ of linear combinations Ln;r defined in previous section. Lemma 1.4 ([50]) We have Ln;r ..t  x/i ; x/ D 0 for all 0 < i  r

(1.2.11)

and Ln one of Bn ; Sn ; Vn : Ln;r ..t  x/2ri ; x/ D ' 2r2i .x/  O.nr / for i  r and x 2 En :

(1.2.12)

To get information about the images of the monomials ek W t ! tk ; t > 0 we may proceed in the following way: ! k X k k k ek D t D ..t  x/ C x/ D .t  x/i :xki : i iD0 Now we apply Ln for the linear combinations Ln;r to the both sides and have ! k X k ki L Ln .ek ; x/ D x n;i .x/; i iD0 where Ln;i .x/ is as defined in (1.2.4). Another way to calculate Ln .ek ; x/ is directly to apply the definition of the operator Ln with f D ek :

1.2.2 Kantorovich Variants The classical Bernstein operator is appropriate only for continuous functions, to approximate integrable function f 2 Lp Œ0; 1; 1  p  1 L. Kantorovich has introduced the following modification of Bernstein operators as BO n .f ; x/ D

n X

Z

.kC1/=.nC1/

pn;k .x/:.n C 1/

f .t/dt;

(1.2.13)

k=.nC1/

kD0

 where pn;k .x/ D nk xk .1  x/nk : Similarly the Kantorovich variants of Szász– Mirakjan operator Sn and Baskakov operator Vn are defined by SO n .f ; x/ D

1 X kD0

Z

.kC1/=n

sn;k .x/:n

f .t/dt; k=n

(1.2.14)

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1 Moments and Combinations of Positive Linear Operators

where sn;k .x/ D enx .nx/k =kŠ and VO n .f ; x/ D

1 X

Z

.kC1/=.nC1/

vn;k .x/:n

f .t/dt;

(1.2.15)

k=.nC1/

kD0

 k  where vn;k .x/ D nCk1 x =.1 C x/nCk : k For the linear combinations Ln;r where Ln is one of the Kantorovich modifications BO n ; SO n ; VO n we have similar estimates for the moments like in Lemma 1.4, namely: Lemma 1.5 (See [50, Lemma 9.5.1]) We have Ln;r ..t  x/i ; x/ D 0 for all 0 < i  r

(1.2.16)

and Ln one of BO n ; SO n ; VO n : Ln;r ..t  x/2ri ; x/ D ' 2r2i .x/:O.nr / for i  r and x 2 En

(1.2.17)

    where En D An ; 1  An for BO n and En D An ; 1 for SO n ; VO n with A > 0 being fixed positive number.

1.2.3 Durrmeyer Variants To approximate integrable functions f 2 L1 Œ0; 1 it was J. L. Durrmeyer [52], who has introduced in 1967 the modification of Bn as Bn .f ; x/ D .n C 1/

n X

Z

1

pn;k .x/

pn;k .t/f .t/dt; f 2 L1 Œ0; 1

(1.2.18)

0

kD0

 where pn;k .x/ D nk xk .1  x/nk : Similarly the Durrmeyer variants of Szász– Mirakjan operator Sn and Baskakov operator Vn are defined by Sn .f ; x/ D n

1 X

Z

1

sn;k .x/

sn;k .t/f .t/dt;

(1.2.19)

0

kD0

where sn;k .x/ D enx .nx/k =kŠ; V n .f ; x/ D .n  1/

1 X

Z

kD0

where vn;k .x/ D

1

vn;k .x/

nCk1 k x =.1 C x/nCk : k

vn;k .t/f .t/dt; 0

(1.2.20)

1.2 Some Kind of Operators

7

In 1992 M. Heilmann [115] published in German language her thesis (habilitation work) devoted to approximation by the Durrmeyer modifications B; S; V: We cite here some of the most important results from this book. It was the idea of M. Heilmann that the three definitions of Bn ; Sn ; V n may be unified as follows: For n > c we define Mnc .f ; x/ D .n  c/

1 X

pcn;k .x/

kD0

Z

1

pcn;k .t/f .t/dt;

0

(1.2.21)

where pcn;k .x/ D .1/k

xk .k/  .x/ kŠ n;c

.k/

with n;c .x/ D .1  x/n for c D 1; we get the Bernstein basis function pn;k .x/ .k/ also n;c .x/ D enx for c D 0 we get the Szász–Mirakjan basis function sn;k .x/ and .k/ finally n;c .x/ D .1 C x/n for c D 1; we get Baskakov basis function vn;k .x/: We define the coefficients ˛ i .n/; 0  i  r  1 as follows: ˛ i .n/ D

r1 Y

.ni  nj /1

r Y

.ni  lc/:

(1.2.22)

lD2

jD0 j¤i

In this way f˛ i .n/gr1 iD0 with r  2 are the unique solution of the system r1 X iD0

˛ i .n/ D 1; and

r1 X iD0

˛ i .n/

mC1 Y

.ni  lc/1 D 0 m D 1; 2;    r  1:

(1.2.23)

lD2

We consider the following linear combinations: c

M n;r D

r1 X

˛ i .n/Mnci ;

iD0

with n D n0 < n1 <    < nr1  An and r1 X iD0

j˛ i .n/j  B:

(1.2.24)

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1 Moments and Combinations of Positive Linear Operators

It was proved by M. Heilmann in 1987 in her doctoral dissertation [113] that c

c

M n;r .e0 ; x/ D 1; M n;r ..t  x/ ; x/ D 0;  D 1; 2;    r  1;

(1.2.25)

which follows from the fact that the linear combinations M n;r preserve all polynomials up to degree r  1: For the images of the monomials er we have the following explicit representation (see [115, Satz 4.1]). Lemma 1.6 We have ! j1  X Y Š  Mnc .e ; x/ D .n  cl1 /1 .n C cl2 / xj jŠ j 1 D2 jD0 l D0 C1 Y 1

(1.2.26)

2

c

According to the estimates of central moments M n;r ..t  x/ ; x/, for   r we cite here the following results (see [115, Lemma 4.5, Korollar 4.6]). Lemma 1.7 For C  0, we have c

jM n;r ..t  x/ ; x/j  C

 X kDr

nk

 X

xj ;

(1.2.27)

jDk

 1 c jM n;r ..t  x/ ; x/j  Cn ; x 2 0; ; n  1 c  r r jM n;r ..t  x/ ; x/j  Cn x ; x 2 ;1 ; n c

jM n;r ..t  x/ ; x/j  Cnr x ; x 2 Œ1; 1 :

(1.2.28) (1.2.29) (1.2.30)

The Durrmeyer modification Mnc satisfy the following recurrence relations for the central moments (see [113, Lemma 4.10]). Lemma 1.8 If n > . C 1/jcj and Tn; .x/ D Mnc ..t  x/ ; x/;  2 N, '.x/ D p x.1 C cx/ and x 2 Œ0; 1/, then Tn;0 .x/ D 1; Tn;1 .x/ D

1 C 2cx n  2c

Œn  c. C 2/Tn;C1 .x/ 0 D ' 2 .x/Œ2Tn;1 .x/ C Tn; .x/ C . C 1/.1 C 2cx/Tn; .x/:

(1.2.31)

Lemma 1.9 (See [115, Korollar 4.11]) The following upper bound holds true: Tn;2 .x/  Cn Œ' 2 .x/ C n1  :

(1.2.32)

1.3 Integral Type Operators

9

Recently V. Gupta, Th. M. Rassias and J. Sinha [111] presented in their paper a list of many Durrmeyer variants of well-known operators introduced in the last five decades.

1.3 Integral Type Operators 1.3.1 Phillips Operators The Durrmeyer variants Bn ; Sn ; V n of the original Bernstein, Szász–Mirakjan and Baskakov operators satisfy two important properties in contrast with their Kantorovich modifications, namely commutativity, i.e. Mn .Mm / D Mm .Mn / and also commutativity with an appropriate differential operator. These two properties will be considered and applied later in the book. The Phillips operator e Sn known also as genuine Szász–Mirakjan–Durrmeyer operator was introduced by R. S. Phillips [161] in 1954, is defined for integrable functions as follows: Z 1 1 X e Sn .f ; x/ D n sn;k .x/ sn;k1 .t/f .t/dt C sn;0 .x/f .0/; x 2 Œ0; 1/ (1.3.1) 0

kD1

Sn also with Sn preserves linear where sn;k .x/ is defined as in (1.2.2). The operator e functions and interpolates f at 0: On the other hand, e Sn like Sn is commutative and commutes with the differential operator e D2 f D ' 2 D2 f ; where ' 2 .x/ D x and D2 denotes the usual second derivative. The last two properties were proved recently in 2011 in a joint paper of G. Tachev and M. Heilmann [119]. The following moments and central moments were estimated in [119, Lemma 2.1]. Lemma 1.10 The following holds true: !  X   1 Š j j e Sn .e0 ; x/ D 1; e n x ;  2 N; Sn .e ; x/ D jŠ j  1 jD1 e Sn ..t  x/0 ; x/ D 1; e Sn ..t  x/; x/ D 0; ! X   j  1 Š e nj xj ;   2: Sn ..t  x/ ; x/ D jŠ j  1 jD1 Œ=2



In the second joint paper of Tachev with M. Heilmann [119] published in 2012, the following linear combinations were considered: e Sn;r .f ; x/ D

r X iD0

˛i .n/e Sni

(1.3.2)

10

1 Moments and Combinations of Positive Linear Operators

where ˛i .n/ D

r Y kD0 k¤i

ni : ni  nk

It was shown that e Sn;r preserves all polynomials up to degree r C 1 which implies that the central moments of e Sn;r of order  r C 1 are equal to zero. For central moments of higher order we have (see [119, Lemma 5.2]). Lemma 1.11 For   r C 2; we have   8  if x 2 0; 1n ; 0 and r  0, we have Dn .tr ; x/ D

.n  r  1/.r C 1/ 1 F1 .rI 1I nx/: .n  1/

(1.3.5)

Further, we have Dn .tr ; x/ D

.n  r C 1/.r C 1/ Lr .nx/; .n  1/

where Lr .nx/ are the Laguerre polynomials represented by ! k X k uj j .1/ Lk .u/ D : k  j jŠ jD0 Remark 1.1 By definition of the operators Dn .1; x/ D 1, using Lemma 1.12, we have Dn .t; x/ D

1 C nx n2 x2 C 4nx C 2 ; Dn .t2 ; x/ D : n2 .n  2/.n  3/

The higher order moments can be obtained easily by Lemma 1.12. For fixed x 2 I  Œ0; 1/, define the function x by x .t/ D t  x. The central moments for the operators Sn are given by .Dn

0 x /.x/

D 1; .Dn

1 x /.x/

D

1 C 2x ; .Dn n2

2 x /.x/

D

.n C 6/x2 C 2.n C 3/x C 2 : .n  2/.n  3/

Moreover, let x 2 I be fixed. For r D 0; 1; 2; : : : and n 2 N, the central moments for the operators Dn satisfy .Dn

r x /.x/

D O.nŒ.rC1/=2 /:

In view of above, an application of the Schwarz inequality, for r D 0; 1; 2; : : :, yields q .Dn j xr j/.x/  .Dn x2r /.x/ D O.nr=2 /: (1.3.6) Also it was shown in [105] that ! k1 .n  k  1/ k k .n  k  1/kŠ X k nj xj n x C : Dn .t ; x/ D .n  1/ .n  1/ k  j jŠ jD0 k

The linear combinations take the form: Dn;r D

r X iD0

˛i .n/  Dni ;

12

1 Moments and Combinations of Positive Linear Operators

where ni ; i D 0; 1; : : : ; r—are different positive integers. Determine ˛i .n/ such that Dn;r p D p for all p 2 Pr . This seems to be natural as the operators Dn don’t preserve linear functions. The requirement that each polynomial of degree at most r should be reproduced leads to a linear system of equations: Dn;r .tk ; x/ D xk ; 0  k  r:

(1.3.7)

Therefore the system is given by: ˛ C ˛1 C    C ˛r D 1 P0 r .ni k1/ k iD0 ˛i  .ni 1/  ni D 1; 1  k  r:

(1.3.8)

The unique solution of this system is ˛i D

r Y 1 .ni  1/  ; 0  i  r: .ni  r  1/ jD0 .ni  nj / j¤i

To obtain a direct estimate for approximation by linear combinations Dn;r one needs two additional assumptions: n D n0 < n1 <    < nr  A  n; .A D A.r//; r X

j˛i .n/j  C:

(1.3.9)

iD0

The first of these conditions guarantees that  Dn;r j

rC1 j x



rC1  .x/ D O n 2 ; n ! 1;

(1.3.10)

which follows from (1.3.6).

1.3.3 Szász–Mirakjan–Laguerre Operators For p  1 we denote by Lwp Œ0; 1/ the linear space of p-absolutely integrable functions on Œ0; 1/ with respect to the weight function w˛ .t/ D t˛ :et ; t  0; ˛ > 1, supplied with norm Z jjf jjLwp D

0

1

jf .t/jp w˛ .t/dt

1=p :

1.3 Integral Type Operators

13

Let us denote by .; /˛ the Laguerre product Z

1

.f ; g/˛ D

f .t/g.t/w˛ .t/dt: 0

Recently in [38] A. Delgado and T. Pérez introduced for n 2 N the Szász–Mirakjan– Laguerre operators of order n as follows: Sn.˛/ .f ; x/ D

1 X .f ; sn;k /˛ :sn;k .x/; x 2 Œ0; 1/ .1; sn;k /˛ kD0

(1.3.11)

where sm;k are the Szász–Mirakjan basis functions. For the images of monomials by .˛/ Sn it was proved in [38, Lemma 2.3] that: Lemma 1.13 Sn.˛/ .em ; x/ D



n nC1

m

:xm C lower degree terms:

(1.3.12)

Naturally as in the previous paragraphs we can consider the linear combinations .˛/ .f ; x/ D Sn;r

r1 X

˛i .n/Sn˛i .f ; x/

(1.3.13)

iD0

where the coefficients ˛i .n/ satisfy the four conditions, described in Section 1.1 of .˛/ book. Therefore from Lemma 1.13, it follows that Sn;r f reproduce all polynomials .˛/ of degree  r  1: In [38] it was established the eigen structure of the operators Sn , namely (see [38, Theorem 3.1]). Lemma 1.14 Classical Laguerre orthogonal polynomials with respect to the weight w.t/ are the eigen-functions of the Szász–Mirakjan–Laguerre operators .˛/ .˛/ Sn.˛/ Lm .x/ D nm :Lm .x/

(1.3.14)

with nm

n D nC1

!m :

The following convergence results are among the results obtained by A. Degaldo and T. Pérez (see [38, Theorem 4.1, Theorem 5.2, Theorem 5.3, Corollary 6.2]. Lemma 1.15

(i) For f 2 Lwp Œ0; 1/ jjSn.˛/ f  f jjLwp ! 0

(1.3.15)

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1 Moments and Combinations of Positive Linear Operators

T .˛/ (ii) For f 2 CŒ0; 1/ Lwp Œ0; 1/ we have uniform convergence of Sn f to f on each compact subset of Œ0; 1/: i.e. jjSn.˛/ f  f jjCŒa;b ! 0; n ! 1 for Œa; b  Œ0; 1/: (iii) Let f 2 CŒ0; 1/ with limx!1 jf .x/j < 1: Then we have uniform convergence .˛/ on the whole interval Œ0; 1/, i.e. jjSn f  f jjCŒ0;1/ ! 0; n ! 1: m (iv) We have also that for f 2 C Œ0; 1/; m  1 such that f .r/ 2 Lwp Œ0; 1/; 0  r  .˛/ m then Sn f 2 Cm Œ0; 1/ and .Sn.˛/ f /.r/ .x/ ! f .r/ .x/ uniformly on each compact subset of Œ0; 1/:

(1.3.16)

Chapter 2

Direct Estimates for Approximation by Linear Combinations

2.1 Direct Estimates in Lp -Norm The aim of this chapter is to collect the known results for the error of approximation by linear combinations Ln;r , measured in different norms Lp .B/ and usually in terms of Ditzian–Totik moduli of smoothness !'r .f ; t/p , or the ordinary moduli of continuity ! r .f ; t/p . We will see the importance of the information about central moments of the p.l.o. (respectively of Ln;r ). We recall some well-known definitions from the book of Ditzian–Totik. The classical modulus of continuity of order r is defined by ! r .f ; t/p D sup jj rh f jjLp .B/ ;

(2.1.1)

0 ˛ and all k; n  n0 , we have .n/  M1 n˛ for all n  n0 :

(3.1.3)

© Springer International Publishing AG 2017 V. Gupta, G. Tachev, Approximation with Positive Linear Operators and Linear Combinations, Developments in Mathematics 50, DOI 10.1007/978-3-319-58795-0_3

25

26

3 Inverse Estimates and Saturation Results for Linear Combinations

The proofs of Lemma 3.1 and Lemma 3.2 are given in [50, Section 9.4] and [50, Section 9.7], respectively. The proof of Lemma 3.3 can be found in Berens–Lorentz [25, p. 969]. Using the three lemmas, formulated above Ditzian–Totik [50] proved the following inverse result: (see [50, Theorem 9.3.2 ]). Theorem 3.1 For Ln —one of the operators Bn ; Sn ; Vnb Sn ; b V n with D and ' Bn ; b described above and 1  p  1, we have  r k K2r;' .f ; n //D  jjLk;r f  f jjD C M: K2r;' .f ; kr /D ; n

(3.1.4)

jjLn;r f  f jjD D O.n˛=2 / , !'2r .f ; h/D D O.h˛ /; 0 < ˛ < 2r:

(3.1.5)

r

Proof For the sake of completeness we give the proof as made in the book due to Ditzian–Totik [50]. We write f D f  Lk;r f C Lk;r f and hence .2r/

K2r;' .f ; nr /D  jjf  Lk;r f jjD C nr jj' 2r Lk;r jjD h i .2r/ .2r/  jjf  Lk;r f jjD C nr jj' 2r Lk;r .f  g/jjD C jj' 2r Lk;r gjjD    jjf  Lk;r jjD C nr M:kr jjf  gjjD C Mkrr jj' 2r g.2r/ jjD  r  k  jjf  gjjD C kr jj' 2r g.2r/ jjD :  jjf  Lk;r jjD C M n Now we take the infimum over all g W g.2r1/ 2 A:C:loc and get the proof of (3.1.4). The equivalence (3.1.5) follows from direct estimate Theorem 2.1, the inverse estimate (3.1.4) and the Berens–Lorentz Lemma 3.3. Thus the proof is completed. 

3.1.2 Durrmeyer Type Operators The Durrmeyer modifications of the three classical p.l.o. Bernstein, Szász–Mirakjan and Baskakov operators are considered in a unified manner by M. Heilmann [115, Chapter 7] and by E. Berdysheva  [24], by the following way:  Let c 2 R: Put Ic D 0;  1c for c < 0 and Ic D Œ0; 1/ for c  0: For n > 0; k 2 N0 and x 2 Ic , we define pcn;k .x/

D .1/

k

! n=c n .cx/k .1 C cx/ c k ; c ¤ 0: k

p0n;k .x/ D lim pcn;k .x/ D c!0

.nx/k nx e ; c D 0: kŠ

(3.1.6)

3.1 Inverse Estimates for Linear Combinations

27

Then for n > c if c  0 or n D c:l with some l, if c < 0 define Mnc .f /.x/

D

1 X kD0

R

Œc

Ic

R

pn;k .t/f .t/dt Œc Ic pn;k .t/dt

Œc

pn;k .x/;

(3.1.7)

for f 2 Lp .Ic /; 1  p  1. This definition is equivalent to (1.2.21). For c D 1 we get well-known Bernstein–Durrmeyer operator Bn , introduced by Durrmeyer [52] and independently by A. Lupa¸s in [136] and further studied by Derriennic in [41], Bn is well defined on Lp Œ0; 1 and on CŒ0; 1: The operator Mn1 D Bn reproduces constant and is contractive, i.e. the inequality jjMnc .f /jjp  jjf jjp ;

(3.1.8)

holds true with c D 1; 1  p  1: For f 2 Lp Œ0; 1; 1  p < 1, or f 2 CŒ0; 1 the convergence jjMn1 .f /  f jjp ! 0; n ! 1 takes place. For c D 0 the operator Mn0 D Sn was introduced by Mazhar and Totik in [147] and also independently by Kasana et al. in [128]. The operator Mnc with c > 0 was introduced by Heilmann [114] and for c D 1 Baskakov–Durrmeyer operators V n independently by Sahai and Prasad in [166]. The operators Mnc with c  0 were studied by M. Heilmann [114], Heilmann and Müller [116]. It is not difficult to show that the operator Mnc is well defined in f 2 L1 .Ic / or f 2 L1 .Ic / and (3.1.8) holds for p D 1 and p D 1: Consequently Mnc is well defined for all f 2 L1 .Ic / and L1 .Ic /, i.e. for all f 2 Lp .Ic /; 1  p  1: By the Riesz–Thorin interpolation theorem (3.1.8) holds for all 1  p  1: The operators Mnc reproduce constants. It was shown in [114, Theorem 3.2] that jjMnc f  f jjp ! 1; n ! 1; f 2 Lp Œ0; 1/; 1  p < 1. In fact the right-hand side of (3.1.7) is also well defined for functions which in some cases do not belong to Lp -spaces, for example for polynomials. Let us consider the c .f / defined by the same way as in (1.2.24) with coefficients linear combinations Mn;r ˛i .n/; 0  i  r  1; satisfying the conditions (1.2.23). We see that the same inverse results and correspondent equivalence to (3.1.5) hold true in this case too. Theorem 3.2 (See [114, Satz. 6.4]) If f 2 Lp .I/; 1  p < 1; 0 < ˛ < r, we have that c jjMn;r f  f jjp D O.n˛ // , !'2r .f ; t/p D O.t2˛ /;

(3.1.9)

p with weight function '.x/ D x.1 C cx/: The needed inequalities of Bernstein type to prove (3.1.9) are already established by Ditzian and Ivanov [48] in Theorem 7.2 for c D 1, and by Heilmann for c  0 in [114]. Actually in [48] Ditzian and Ivanov considered a slightly different type of

28

3 Inverse Estimates and Saturation Results for Linear Combinations

linear combinations with 2r coefficients instead of r coefficients, which is the usual case. They considered the following linear combinations: Qn;r .f ; x/ D

2r1 X

˛i .n/Mn1 .f ; x/; i

(3.1.10)

iD0

n0 D n < n1 < n2    < n2r1  An; where A is independent of n and ˛i .n/ satisfy the conditions Qn;r .1; x/ D 1; Qn;r ..t P x/m ; x/ D 0 for 1  m  2r  1; 2r1 iD0 j˛i .n/j  B: The equivalence result in [51] states the following (see [48, Corollary 7.3]): Theorem 3.3 For Qn;r given by (3.1.10) and ˛ < r we have that jjQn;r f  f jjp D O.n˛ / , !'2r .f ; t/p D O.t2˛ /;

(3.1.11)

with 1  p  1: Even stronger result was proved by Ditzian and Ivanov in [48, Theorem 7.2]: Theorem 3.4 Suppose for k < 2r  1 Qn;k f D

k X

˛i .n/Mn1 f; i

iD0

n0 D n < n1 < n2    < nk  An;

k X

j˛i .n/j  B:

iD0

Then jjQn;k f  f jjp D O.n˛ / ) !'2r .f ; t/p D O.t2˛ /; ˛ < r:

(3.1.12)

Note that here k < 2r1 is an arbitrary number and the condition Qn;r ..tx/m ; x/ D 0; 1  m  2r  1 is missing here. In concrete example, Ditzian and Ivanov showed in [48] that such linear combinations Qn;r defined by (3.1.10) exist. In [115] Heilmann considered also the weighted simultaneous approximation by the linear combinations of Szász–Mirakjan–Durrmeyer operators and relevant inverse and saturation results. Theorem 3.5 (See Satz. 7.1 in [115]) For the linear combinations Sn;r .f ; x/ D

r1 X iD0

˛i .n/Sni ;

3.1 Inverse Estimates for Linear Combinations

where the coefficients ˛i .n/ satisfy (1.2.23), '.x/ D Lp Œ0; 1/; 1  p < 1 and 0 < ˛  s < r, we have that

29

p x; s 2 N0 ; ' 2s f .2s/ 2

jj' 2s .Sn;r f  f /.2s/ jjp D O.ns˛ /

(3.1.13)

!'2rC2s .f ; t/p D O.t2˛ /:

(3.1.14)

implies

The following saturation result also holds true: Theorem 3.6 (See [115, Satz 7.5]) By the same conditions as in Theorem 3.5, we have that (3.1.14) is equivalent to (3.1.13) and ' 2s f .2s/ 2 Lp Œ0; 1/: Theorem 3.7 (See [115, Satz. 7.7]) By the same conditions as in Theorem 3.5, the following four relations are equivalent: (i) (ii) (iii) (iv)

jj' 2s .Sn;r f  f /.2s/ jjp D O.ns˛ / and ' 2s f .2s/ 2 Lp Œ0; 1/; !'2rC2s .f ; t/p D O.t2˛ /; jj' 2sC2r .Sn f /2sC2r jjp D O.nsCr˛ / jjSn;rCs f  f jjp D O.n˛ /:

3.1.3 Phillips Operators The aim of this section is to consider the known inverse results for approximation by linear combinations of Phillips operators e Sn;r as defined by (1.3.2). As it was pointed out by May [146] the Phillips operator does not satisfy the differential equation for the so-called exponential type operators, like Bn ; Sn , etc. This fact causes some technical difficulties in the analogous estimates of e Sn . Although the main steps to prove inverse result for e Sn;r are similar to that for Ln;r in Theorem 3.1 or by Mn;c;r in Theorem 3.2, their proofs are different. For example, if we want to establish a global inverse theorem for approximation by e Sn;r of bounded continuous functions in CB Œ0; 1/, we must follow quite different approach from that for Theorem 3.2 which is valid only for f 2 Lp Œ0; 1/; 1  p < 1; because the proof of the last statement essentially uses the Hardy’s inequality. Let us briefly list the known results for inverse statements for e Sn;r . The first known result was proved by May [146], namely Theorem 3.8 Let f 2 CN Œ0; 1/ D ff W jf .x/j  MeNx for all x 2 Œ0; 1/; N > 0g, i.e. functions with exponential growth. Then if 0 < a < a1 < b1 < b < 1, we have nrC1 jje Sn;r f  f jjCŒa;b D O.1/ is fulfilled whenever f .2rC2/ 2 L1 Œa; b:

(3.1.15)

30

3 Inverse Estimates and Saturation Results for Linear Combinations

We point out that this equivalence has a local character, i.e. does not cover the whole interval Œ0; 1/: Also the case of bounded continuous functions f 2 CB Œ0; 1/ is not covered. Also May considered e Sn;r only for a specific choice of coefficients ˛i .n/. The next inverse result, which extends the previous result of May, was proved by Gupta and Sahai in [103]. They considered more general linear combinations containing as a partial case the combinations of May. The following inverse estimate was proved in [103]: Theorem 3.9 Let f 2 CN Œ0; 1/; 0 < a1 < a2 < b2 < b1 < 1 and 0 < ˛ < 2, then jje Sn;r f  f jjCŒa2 ;b2  D O.n˛.rC1/=2 /

(3.1.16)

f 2 Liz.˛; r C 1; a2 ; b2 /

(3.1.17)

is equivalent to

equivalent to (a) For m < ˛.r C 1/ < m C 1; m D 0; 1; 2;    ; 2r C 1; f .m/ exists and f .m/ 2 Lip.˛.r C 1/  m; a2 ; b2 /:

(3.1.18)

(b) For ˛.r C 1/ D m C 1; m D 0; 1;    ; 2rI f .m/ exists and f .m/ 2 Lip .1; a2 ; b2 /; where Liz.˛; k; a; b/ denotes the class of functions for which !2k .f ; h; a; b/  M:h˛k I If k D 1, Liz.˛; 1/ reduces to the Zygmund class Lip ˛: The proof of the last theorem relies on the approach of Steklov means. This result has again a local character and is in terms of the moduli !2r .f ; h/: In [14], Agrawal and Gupta considered iterative combinations of Micchelli type for Phillips operators and proved direct results in sup and Lp -norms, respectively, but all results have a local character. Very recently in 2015 Tachev proved in [184] a global inverse theorem for approximation of bounded continuous functions f 2 CB Œ0; 1/ in sup-norm by linear combinations e Sn;r : We formulate his result as: Theorem 3.10 Let f 2 CB Œ0; 1/; r 2 N0 ; k; n 2 N: Then we have for ˛ < 2r C 2 K'2rC2 .f ; nr1 /

 jje Sn;r f  f jjCB Œ0;1/ C M

 rC1 k K'2rC2 .f ; kr1 /; n

jje Sn;r f  f jjCB Œ0;1/ D O.n˛=2 / , !'2rC2 .f ; h/1 D O.h˛ /:

(3.1.19) (3.1.20)

3.2 Saturation Results for Linear Combinations

31

Proof The crucial steps for the proof are the Bernstein type inequalities: .2r/ f jjCB Œ0;1/  C.r/nr jjf jjCB Œ0;1/ ; jj' 2re Sn;r

Sn;r f /jjCB Œ0;1/  C.r/jj' 2r f .2r/ jjCB Œ0;1/ ; jj' 2r D2r .e

(3.1.21) (3.1.22)

which are corollaries in [184, Theorems 2.7 and 2.8]. Let g 2 CB Œ0; 1/ be the function such that ' 2rC2 g.2rC2/ 2 CB Œ0; 1/. Then to prove (3.1.19) we write f D Sk;r f and hence f e Sk;r f C e .2rC2/ Sk;r f  f jjCB Œ0;1/ C nr1 jj' 2rC2e K'2rC2 .f ; nr1 /  jje Sk;r f jjCB Œ0;1/ :

To estimate the second term from above we use an auxiliary function g: .2rC2/

Sk;r f jjCB Œ0;1/ jjnr1 ' 2rC2e h i .2rC2/  nr1 jj' 2rC2e Sk;r .f  g/jjCB Œ0;1/ C jj' 2rC2e Sk;r gjjCB Œ0;1/    C.r/nr1 krC1 jjf  gjjCB Œ0;1/ C krC1 :kr1 jj' 2rC2 g.2rC2/ jjCB Œ0;1/ : In the last upper bounds we used (3.1.21) and (3.1.22). Taking the infimum over all auxiliary functions g, we complete the proof of (3.1.19). The implication ( in (3.1.20) follows from direct estimate (2.2.7) in Theorem 2.13. The inverse direction ) now follows from (3.1.19) and the Berens–Lorentz Lemma 3.3–(3.1.3). Thus the proof is completed.  Let us point out that the proof of (3.1.21) and (3.1.22) essentially lies on the important and specific properties of the Phillips operator e Sn like commutativity and commutativity with the appropriate differential operator [183] and don’t use the Hardy inequality. We hope that the ideas of the proof of Theorem 3.10 may be applied in CŒ0; 1/-norm for other Durrmeyer modifications like Bn ; Sn ; V n , etc.

3.2 Saturation Results for Linear Combinations We differ between local and global saturation results for approximation by linear combinations. Some local results are shown in the previous section, see Theorems 3.8 and 3.9. One of the first local saturation results for Bernstein polynomials was given by H. Berens and G. G. Lorentz in [25]. Their statement says that jBn .f ; x/  f .x/j  M.x.1  x//˛=2 if and only if f 2 Lip ˛, i.e. if and only if jj 2h f jjCŒh;1h D jjf .x C h/  2f .x/ C f .x  h/jjCŒh;1h D O.h˛ /:

(3.2.1)

32

3 Inverse Estimates and Saturation Results for Linear Combinations

We recall the first linear combinations of Bn f introduced by Butzer [34]: .2r  1/Bn .f ; r; x/ D 2r B2n .f ; r  1; x/  Bn .f ; r  1; x/;

(3.2.2)

with Bn .f ; 0; x/ D Bn .f ; x/: The local inverse theorem of Bn .f ; k; x/  f .x/ was investigated by C. P. May in [145]. In [49] the local saturation of Bn .f ; k; x/  f .x/ was also investigated by Ditzian and May. The saturation result for combinations of Bernstein polynomials and Szász operators discussed in [49] are the following theorems: Theorem 3.11 For f 2 CŒ0; 1; 0 < a < a1 < b1 < b < 1 and fni g contains n0 2i I.f I ni ; k; a; b/  nikC1 jjBni .f ; k; /  f ./jjCŒa;b  M implies f .2kC1/ .x/ 2 A:C:.a; b/ and f .2kC2/ .x/ 2 L1 Œa; b;

(3.2.3)

and (3.2.3) implies I.f I n; k; a1 ; b1 /  M1 I I.f I n; k; a; b/ D o.1/; n ! 1 implies

2kC2 X

qi .t/f .i/ .t/ D 0

iD1

P .i/ for t 2 .a; b/; and 2kC2 iD1 qi .t/f .t/ in .a; b/ implies I.f I n; k; a1 ; b1 / D o.1/ n ! 1; where qi .t/ are fixed polynomials that depend on k: Theorem 3.12 For f 2 CŒ0; 1/; jf .t/j  KeLt for some K and L; 0 < a < a1 < b1 < b < 1 and m D 0 2m we have: J.f I i ; k; a; b/ D ik1 jjSi .f ; k; /  f ./jjCŒa;b  M implies (3.2.3) and (3.2.3) implies J.f I ; k; a1 ; b1 /  M1 I J.f I i ; k; a; b/ D o.1/ i ! 0 C implies

2kC2 X

Qi .t/f .i/ .t/ D 0

iD1

for t 2 .a; b/ where Qi .t/ are fixed polynomials that depend on k and i and P 2kC2 .i/ iD1 Qi .t/f .t/ in .a; b/ implies J.f I ; k; a1 b1 / D o.1/: These results related jjBn .f ; k; :/f jjCŒa;b to smoothness in .˛; ˇ/ where Œ˛; ˇ  .a; b/ and Œa; b  .0; 1/: In [46] Ditzian using the technique of space interpolation obtained a global result and thus overcame the difficulty, caused by the singularity at 0 and 1: Let us denote jjf jj D sup0x1 jf .x/j, jjf jj2r D sup0x1 jf .2r/ .x/xr .1  x/r j and A2r D ff W jjf jj2r < 1 and f .2r1/ 2 A:C:loc g: The Peetre’s K-functional is given by K.t2r ; f / D inf fjjf  gjj C t2r jjgjj2r g: g2A2r

3.2 Saturation Results for Linear Combinations

33

The intermediate space .C; A2r /ˇ for some 0 < ˇ < 2r is the collection of all f for which the norm supt>0 tˇ :K.t2r ; f / is finite. The main global results of Ditzian in [46] state the following: Theorem 3.13 For f 2 CŒ0; 1; 0 < ˇ < 2r the following are equivalent: (i) jjBn .f ; r  1; x/  f jj D O.nˇ=2 /; n ! 1I (ii) f 2 .C; A2r /ˇ I (iii) suphr 0, we have e ! .f ; :/  .1 C /!.f ; / and the last cannot be improved for each  > 0 and D 1. Hence we have !.f ; /  e ! .f ; /  2!.f ; /:

4.1 New Quantitative Estimates of Voronovskaja Type

39

Therefore the estimate (4.1.4) is an improvement of (4.1.3). Theorem 4.3 is a corollary from more general quantitative statements for a broad class of p.l.o. preserving linear functions (see [80, Theorem 6.2]), which are formulated here as: Theorem 4.4 Let L W CŒ0; 1 ! CŒ0; 1 be a p.l.o. such that Lei D ei for i D 0; 1. If f 2 C2 Œ0; 1; and x 2 Œ0; 1, then ˇ ˇ ˇ ˇ ˇL.f ; x/  f .x/  1 f 00 .x/L..e1  x/2 ; x/ˇ ˇ ˇ 2 0 1 s 4 ; x/ 1 L..e  x/ 1 1 A:  L..e1  x/2 ; x/e ! @f 00 ; 2 3 L..e1  x/2 ; x/

(4.1.7)

The proof of this theorem relies on the precise estimation of the remainder in Taylor formula in terms of least concave majorant e ! . We give the proof of Theorem 4.4 after the proof of the following crucial step, namely (see [80, Theorem 3.2]): Theorem 4.5 For n 2 N0 , let f 2 Cn Œa; b; and x; x0 2 Œa; b. If Rn .f I x0 ; x/ represents the remainder in Taylor’s formula in Lagrange form: Rn .f I x0 ; x/ WD f .x/  then it holds:

n X 1 .k/ f .x0 /.x  x0 /k ; kŠ kD0

  jx  x0 jn .n/ jx  x0 j e ! f ; : jRn .f I x0 ; x/j  nŠ nC1

(4.1.8)

Proof It is clear that following the Lagrange’s form of the remainder in Taylor’s formula, there exists a point x between x and x0 , such that Rn .f I x0 ; x/ D

.x  x0 /n .n/ .f .x /  f .n/ .x0 //: nŠ

Hence  jx  x0 jn  .n/ ! f I jx  x0 j W Œa; b nŠ jx  x0 jn .n/ jjf jjCŒa;b : 2 nŠ

jRn .f I x0 ; x/j 

If now g 2 CnC1 Œa; b again using Lagrange form of the remainder, we may write jRn .gI x0 ; x/j 

jx  x0 jnC1 .nC1/ jx  x0 jnC1 .nC1/ :jg jjg .x /j  jjCŒa;b ; .n C 1/Š .n C 1/Š

for some x between x and x0 :

40

4 Voronovskaja-Type Estimates

Keeping f fixed and letting g be arbitrary in CnC1 Œa; b, we have jRn .f I x0 ; x/j D jRn .f  g C gI x0 ; x/j  jRn .f  gI x0 ; x/j C jRn .gI x0 ; x/j :  2jx  x0 jn jx  x0 j .nC1/ .n/ jjg jj.f  g/ jjCŒa;b C  jjCŒa;b : n 2.n C 1/ We take the infimum over all g 2 CnC1 Œa; b to get using (4.1.5)   2jx  x0 jn jx  x0 j .n/ 1 K ; f I CŒa; b; C Œa; b jRn .f I x0 ; x/j  nŠ 2.n C 1/   jx  x0 jn jx  x0 j : D e ! f .n/ ; nŠ nC1 This completes the proof of the theorem.



Proof of Theorem 4.4 If the p.l.o. reproduces polynomials up to degree n1, using Theorem 4.5, we arrive at ˇ ˇ ˇ ˇ ˇL.f ; x/  f .x/  1 f .n/ .x/L..e1  x/n ; x/ˇ ˇ ˇ nŠ ˇ  ˇ   ˇ jt  xjn jt  xj ˇˇ (4.1.9) D ˇˇL e ! f .n/ ; ˇ: nŠ nC1 If L is positive operator, n D 2 by (4.1.5), we get

 D  

ˇ ˇ ˇ ˇ ˇL.f ; x/  f .x/  1 f 00 .x/L..e1  x/2 ; x/ˇ ˇ ˇ 2     .e1  x/2 00 je1  xj L ;x e ! f ; 2 3     je1  xj 00 0 L .e1  x/2 K ; f I C Œ0; 1; C1 Œ0; 1 ; x 6     je1  xj 2 00 000 ; jjf jj ; x L .e1  x/ jj.f  g/ jj C 6   1 L.je1  xj3 ; x/ 000 2 00 jjf jj : L..e1  x/ ; x/ jj.f  g/ jj C 6 L..e1  x/2 ; x/

Using the Cauchy–Schwarz inequality for p.l. functionals, we obtain that L.je1  xj3 ; x/ 

p

p L..e1  x/4 ; x/ L..e1  x/2 ; x/:

(4.1.10)

4.1 New Quantitative Estimates of Voronovskaja Type

41

Consequently taking infimum over all g 2 C3 Œ0; 1, we get ˇ ˇ ˇ ˇ ˇL.f ; x/  f .x/  1 f 00 .x/L..e1  x/2 ; x/ˇ ˇ ˇ 2 ! p 1 L..e1  x/4 ; x/ 2 00 1  L..e1  x/ ; x/e : ! f p 2 3 L..e1  x/2 ; x/  We point out that last quantitative results are based on estimates for the remainder in Taylor’s formula, represented in Lagrange form. There are other methods, based on integral representation of the remainder in Taylor’s formula and upper bounds in terms of other moduli and K-functionals, which are considered later. Now we return to the representation (4.1.8). Let Ln;r .f ; x/ be the linear combinations defined in Section 1.1. The condition (d) for the coefficients ˛i .n/ in most cases, as we have already seen guarantees that Ln;r reproduces all polynomials up to degree r1. Then (4.1.8) implies for t; x 2 Œa; b Ln;r .f ; x/  f .x/ 

1 .r/ f .x/Ln;r ..e1  x/r ; x/ rŠ

D Ln;r .Rr .f I t; x/; x/ D

r1 X

˛i .n/Lni .Rr .f I t; x/; x/:

(4.1.11)

iD0

Due to the fact that Ln;r are not positive operators, using the conditions (a) and (b) for coefficients ˛i .n/ we may proceed as follows: ˇ ˇ ˇ ˇ ˇLn;r .f ; x/  f .x/  1 f .r/ .x/Ln;r ..e1  x/r ; x/ˇ ˇ ˇ rŠ 

r1 X

j˛i .n/j:Lni .jRr .f I t; x/j; x/

iD0

 C max :Lni .jRr .f I t; x/j; x/ 0ir1

 C.r/:Ln .jRr .f I t; x/j; x/     jt  xjr jt  xj :! f .r/ ; ;x :  C.r/Ln rŠ rC1 As in the proof of Theorem 4.5, we get that jRr .f ; t; x/j  2

jt  xjr .r/ jjf jjCŒa;b : rŠ

(4.1.12)

42

4 Voronovskaja-Type Estimates

Further for g 2 CrC1 Œa; b, in a similar way, we have jRr .gI t; x/j 

jt  xjrC1 .rC1/ jjg jjCŒa;b : .r C 1/Š

Now (4.1.12) and the last two estimates imply ˇ ˇ ˇ ˇ ˇLn;r .f ; x/  f .x/  1 f .r/ .x/Ln;r ..e1  x/r ; x/ˇ ˇ ˇ rŠ  C.r/ ŒLn .jRr .f  gI t; x/j; x/ C Ln .jRr .gI t; x/j; x/  2 1  C.r/ jj.f  g/.r/ jj:Ln .jt  xjr ; x/ C jjg.rC1/ jj:Ln .jt  xjrC1 ; x/ rŠ .r C 1/Š  1 1 Ln .jt  xjrC1 ; x/ .rC1/ 2 :jjg jj :  C.r/ Ln .jt  xjr ; x/ jj.f  g/.r/ jj C rŠ 2 .r C 1/ Ln .jt  xjr ; x/ We take infimum over all g W g.rC1/ 2 CŒa; b to get ˇ ˇ ˇ ˇ ˇLn;r .f ; x/  f .x/  1 f .r/ .x/Ln;r ..e1  x/r ; x/ˇ ˇ ˇ rŠ   2 1 1 Ln .jt  xjrC1 ; x/ .r/ 1  C.r/ Ln .jt  xjr ; x/K ; f I CŒa; b; C Œa; b rŠ 2 .r C 1/ Ln .jt  xjr ; x/   Ln .jt  xjrC1 ; x/ Ln .jt  xjr ; x/  C.r/ e ! f .r/ ; : : (4.1.13) rŠ .r C 1/Ln .jt  xjr ; x/ In the most of known p.l.o. Ln , we have that dn;r WD

Ln .jt  xjrC1 ; x/ ! 0; as n ! 1 .r C 1/Ln .jt  xjr ; x/

(4.1.14)

uniformly on x 2 Œa; b: So we arrive at the proof of the following quantitative form of Voronovskaja theorem for approximation of f 2 Cr Œa; b by linear combinations: Theorem 4.6 Let f 2 Cr Œa; b and Ln;r be the linear combinations defined in Section 1.1. Then, we have ˇ ˇ .r/ ˇ ˇ ˇLn;r .f ; x/  f .x/  f .x/ Ln;r ..e1  x/r ; x/ˇ ˇ ˇ rŠ  C.r/

 Ln .jt  xjr ; x/  .r/ e ! f ; dn;r .x/ : rŠ

(4.1.15) (4.1.16)

4.2 Voronovskaja-Type Estimates

43

4.2 Voronovskaja-Type Estimates 4.2.1 In Terms of !' .f 00 ; ı/ In this section we represent a new quantitative variant of Voronovskaja’s theorem for Bernstein operator. This estimate improves in some sense the estimate for Bn — (4.1.4), obtained in [80]. Starting again from the representation of the remainder in Taylor’s formula in Lagrange form, it is clear that ˇ ˇ ˇ ˇ ˇL.f ; x/  f .x/  1 f 00 .x/L..e1  x/2 ; x/ˇ ˇ ˇ 2    .:  x/2  00 L ! f ; xI j:  xjI x ; 2

(4.2.1)

where L is an l.p.o. reproducing linear functions and !.f ; xI ı.t; x// is the local moduli of smoothness: !.f ; xI ı.t; x// D supfjf .x C h/  f .x/j W jhj  ı.t; x/; x; x C h 2 Œ0; 1g h

and ı.t; x/ is a continuous positive function of x 2 Œ0; 1 for any t 2 .0; 1. The average moduli of smoothness  we shall use are given by  .f ; xI ı.t; x// WD jj!.f ; :I ı.t; ://jjCŒ0;1 The properties of  -moduli and their applications to many problems in approximation theory and numerical analysis can be found in the monograph [169]. The main result proved by Tachev in [179, Theorem 4] is the following: Theorem 4.7 For the Bernstein operator Bn ; n  1 and f 2 C2 Œ0; 1, we have ˇ ˇ ˇ ˇ ˇnŒBn .f ; x/  f .x/  1 f 00 .x/:x.1  x/ˇ ˇ ˇ 2  C:x.1  x/ .f 00 .x/; ı.n1=2 ; ://;

(4.2.2)

where 1 1 ı.n1=2 ; x/ D p x.1  x/ C ; n  1: n n The proof of this theorem relies on the use of auxiliary function G1;n .x/, which is the function Gk;n from [122, Theorem 3.1] for k D 1 and also we apply the Cauchy– Schwarz inequality and representations of the second and fourth moments of Bn . On the other hand, from [123, Corollary 5.1], it follows that

44

4 Voronovskaja-Type Estimates

C1  .f ; ı.t; ://  K1;' .f ; t/1  C2 .f ; ı.t; ://; where Kr;' .f ; tr /1 was defined in (2.1.4). But we know that K1;' .f ; t/1 !'1 .f ; t/1 : Hence we get: Theorem 4.8 For the Bernstein operator Bn ; n  1 and f 2 C2 Œ0; 1, we have ˇ ˇ ˇ ˇ ˇnŒBn .f ; x/  f .x/  1 f 00 .x/:x.1  x/ˇ ˇ ˇ 2  C0 :x.1  x/!'1 .f 00 ; n1=2 /;

(4.2.3)

where C0 is absolute positive constant. It is known that !'r .f ; t/ 4: We point out that Corollary 4.1 shows that the estimate in Theorem 4.11 allows to get the best possible order of approximation in Voronovskaja’s theorem. Corollary 4.3 (See [180, Corollary 3]) If f 2 C3 is such that f 000 is Lipschitz function of order 1 and if f is not a polynomial of degree  1, then the order of uniform approximation of f by Bn f is exactly n1 :

4.2.4 For Higher Order Derivatives The famous theorem of E. Voronovskaja—Theorem 4.1 has attracted the attention of many authors in the last 80 years. Inspired by the result of Voronovskaja, her scientific advisor S. Bernstein generalized Theorem 4.1, showing in [28] the asymptotic expansion of Bn f for f 2 Cq Œ0; 1 for q-even as follows: Theorem 4.12 If q 2 N is even, f 2 Cq Œ0; 1, then uniformly in x 2 Œ0; 1, we have " q=2

n

q X

f .r/ .x/ Bn ..e1  x/ ; x/ Bn .f ; x/  f .x/  rŠ rD1 r

# ! 0; n ! 1 (4.2.18)

In [142] Mamedov considered also the case f 2 Cq Œ0; 1, q-even namely: Theorem 4.13 Let q 2 N be even, f 2 Cq Œ0; 1, and Ln W CŒ0; 1 ! CŒ0; 1 be a sequence of p.l.o. such that Ln .e0 ; x/ D 1; x 2 Œ0; 1 Ln .e1  x/qC2 ; x/ D0 n!1 Ln ..e1  x/q ; x/ lim

for at least one q D 1; 2; 3;    . Then for n ! 1 " # q X 1 f .r/ .x/ r Ln ..e1  x/ ; x/ Ln .f ; x/  f .x/  ! 0: (4.2.19) Ln ..e1  x/q ; x/ rŠ rD1 A complete asymptotic expansion in quantitative form was given some 30 years ago by Sikkema and Van der Meer in [170]:

4.2 Voronovskaja-Type Estimates

49

Theorem 4.14 Let WCq Œ0; 1 denote the set of all functions on Œ0; 1 whose q-th derivative is piecewise continuous, q  0: Moreover, let .Ln / be a sequence of p.l.o. Ln W WCq Œ0; 1 ! CŒ0; 1, satisfying Ln .e0 ; x/ D 1: Then for all f 2 WCq Œ0; 1; q 2 N; x 2 Œ0; 1; n 2 N, and ı > 0 one has "

q X

f .r/ .x/ Ln .f ; x/  f .x/  Ln ..e1  x/ ; x/ rŠ rD1 r

where Cn;q .x; ı/ D ı q Ln .sq; if Ln ..e1  x/q ; x/ < 0, and

 e1 x ı

#  Cn;q .x; ı/!.f .q/ ; ı/; (4.2.20)

 ; x ,  D 1=2, if Ln ..e1  x/q ; x/  0;  D 1=2

   1 1 1 q q juj C :u C BqC1 .juj/  BqC1 .juj  Œjuj/ ; sq; .u/ D qŠ 2 .q C 1/Š BqC1 is the Bernoulli polynomial of degree q C 1 and Œt D maxfz 2 Z; z  tg: The function Cn;q .x; ı/ are the best possible for each f 2 Cq Œ0; 1; x 2 Œ0; 1; n 2 N; ı > 0: The expression for Cn;q .x; ı/ is very complicated and it is hardly possible to derive from (4.2.20) the extension of (4.2.18) to all q 2 N. Another quantitative result for f 2 Cq Œ0; 1; q 2 N was proved recently by H. Gonska [72, Theorem 3.2] and states the following: Theorem 4.15 Let q 2 N0 ; f 2 Cq Œ0; 1 and L W CŒ0; 1 ! CŒ0; 1 be a p.l.o. Then "

# f .r/ .x/ L.f ; x/  f .x/  Ln ..e1  x/ ; x/ rŠ rD1   L.je1  xjqC1 ; x/ L.je1  xjqC1 ; x/ .q/ : e ! f ;  qŠ .q C 1/L.je1  xjq ; x/ q X

r

(4.2.21)

Very recently in [181], Tachev extended the results of Mamedov—Theorem 4.13 and of S. Bernstein-Theorem 4.12 for q odd natural numbers. Let us formulate the main statements in [181]: Theorem 4.16 (Extension of Mamedov’s Result) Let q 2 N be odd, f 2 Cq Œ0; 1 and Ln W CŒ0; 1 ! CŒ0; 1 be sequence of p.l.o. such that Ln .e0 ; x/ D 1; x 2 Œ0; 1 Ln ..e1  x/lC2j ; x/ D0 n!1 Ln ..e1  x/l ; x/ lim

for l 2 N—any even number and for at least one j D 1; 2; ::: Then if n ! 1 we have that (4.2.19) holds true. Theorem 4.17 (Extension of Bernstein’s Result) If q 2 N is odd, f 2 Cq Œ0; 1, then uniformly in x 2 Œ0; 1, when n ! 1 we have that (4.2.18) holds true. Even more we have

50

4 Voronovskaja-Type Estimates

"

q X

f .r/ .x/ n Bn ..e1  x/ ; x/ Bn .f ; x/  f .x/  rŠ rD1 ! r p Aq x.1  x/ 1 .q/ e ! f ;  : qŠ .q C 1/ n q=2

#

r

(4.2.22)

for some constant Aq > 0, depending only on q: Theorem 4.18 (Optimal Rate of Convergence) Let q 2 N; f 2 Cq Œ0; 1: There exists no constant ˇ > 0, such that the uniform convergence over Œ0; 1 holds true: " q=2

n

q X

f .r/ .x/ Bn ..e1  x/ ; x/ Bn .f ; x/  f .x/  rŠ rD1 r

# ! 0;

(4.2.23)

when n ! 1. Proof of Theorem 4.17 The proof relies on the quantitative estimate of Gonska (4.2.21) in Theorem 4.15 and on the use of K-functional (4.2.6) equivalent to the least concave majorant e ! . What we are going to do is to get an upper bound for the argument of e ! in (4.2.21) if L D Bn —the Bernstein operator. Using the Cauchy– Schwarz inequality we get Bn .je1  xjq ; x/ 

p Bn .je1  xj2q ; x/:

It is known that (see the book of G.G. Lorentz [134, p. 15]) Bn ..e1  x/2q ; x/ 

Aq ; nq

for all x 2 Œ0; 1: The last two estimates imply Bn ..je1  xjq ; x/ 

p Aq nq=2 :

(4.2.24)

For arbitrary g 2 C1 Œ0; 1 we apply the definition of K-functional (4.2.6) and estimate the right-hand side of (4.2.21) as follows:   Bn ..je1  xjq ; x/ 1 Bn .je1  xjqC1 ; x/ :e ! f .q/ ; : qŠ q C 1 Bn .je1  xjq ; x/ Bn ..je1  xjq ; x/ Bn .je1  xjqC1 ; x/ 0 :2jjf .q/  gjj1 C jjg jj1 qŠ .q C 1/Š p p p Aq nq=2 Bn .je1  xj2q ; x/: Bn ..e1  x/2 ; x/ 0 .q/  2jjf  gjj1 C jjg jj1 qŠ ..q C 1/Š 

4.2 Voronovskaja-Type Estimates

51

" # r p Aq nq=2 x.1  x/ 0 1 .q/ 2 jjf  gjj1 C :jjg jj1  qŠ 2.q C 1/ n

(4.2.25)

In the last estimate we used the fact that Bn .je1  xj2 ; x/ D

x.1  x/ : n

Lastly we take the infimum over all g W g 2 C1 Œ0; 1 and using (4.1.5) in Lemma 4.1 we complete the proof of Theorem 4.17.  Another way to establish Theorem 4.17 without using the technique of Kfunctional is to prove the following conjecture, formulated in [181]. Prove that for all natural numbers q, the following holds true: Bn ..je1  xjqC1 ; x/ D 0: n!1 Bn ..je1  xjq ; x/ lim

(4.2.26)

At the time of publishing of [181] it was not clear how to verify (4.2.26). Some suggestions in this direction are made by Gonska in [72], but from his consideration it was not clear how to derive at the (4.2.26). Very recently I. Gavrea and M. Ivan solved this conjecture in [71] giving the positive answer for the case of Bernstein operator. Their main result [71, Theorem 14] states the following: Theorem 4.19 For any q  4 and x 2 .0; 1/ there exists a constant Aq such that Aq Bn ..je1  xjqC1 ; x/  p ; n  5: Bn ..je1  xjq ; x/ n

(4.2.27)

For q D 0; 1; 2; 3, (4.2.27) is easily verified. Even more in [71, Theorem 15] Gavrea and Ivan proved a converse form of inequality (4.2.27), namely ˇˇ ˇˇ ˇˇ Bn ..je1  xjqC1 ; x/ ˇˇ M.q/ ˇˇ ˇˇ  p ; ˇˇ B ..je  xjq ; x/ ˇˇ n n 1 CŒ0;1

(4.2.28)

for any q 2 N and M.q/ is a positive constant depending only on q: Let us finish this paragraph with the observation that the estimates of Gavrea and Ivan can be delivered from the sharp upper and lower bounds for the central moments of Bn , obtained very recently in 2015 by Adell, Bustamante, Quesada (see [9, 10]).

52

4 Voronovskaja-Type Estimates

4.3 Asymptotic Expansion for Some Positive Linear Operators The main contribution concerning complete asymptotic expansion is due to U. Abel and collaborators, who established many important results which are available in the literature (see, e.g., [2, 3, 6], etc). In the recent book some examples of complete asymptotic expansions were provided (see [98, Chapter 3]). Here we mention some more important examples: U. Abel and M. Ivan in [4] established the asymptotic expansion of the Jakimovski–Leviatan operators and their derivatives, which are much more general than the original Szász–Mirakjan operators. Jakimovski and Leviatan [125] gave a generalization of Szász operators by using 1 X the Appell polynomials. Let g.z/ D ak zk be an analytic function in the disc kD0

jzj < R; .R > 1/ and g.1/ ¤ 0. It is well known that the Appell polynomials pk .x/ are defined by the following generating functions: g.u/eux D

1 X

pk .x/uk :

(4.3.1)

kD0

Let E be the class of all functions of exponential type which satisfy the property jf .t/j  ceAt for some finite constants c; A > 0: Jakimovski–Leviatan [125] operators associate to each function f 2 E are defined as follows: 1

enx X pk .nx/f Pn .f ; x/ D g.1/ kD0

  k ; n

(4.3.2)

where pk .x/ are Appell polynomials given by (4.3.1). The operators (4.3.2) are well defined, for all sufficiently large n, since the infinite sum in (4.3.2) is convergent if n > A= log R. For the special case g.z/ D 1 the operators Pn reduced to Szász operators. Let q 2 N and x 2 Œ0; 1/ be fixed, let K Œq .x/ be the class of all functions f W E ! R such that f admits a derivative of order q at x: The following complete asymptotic expansion was obtained in [4]: Theorem 4.20 Let q 2 N; x  0 and f 2 K Œ2q .x/: The Jakimovski–Leviatan operators satisfy the asymptotic relation Pn .f ; x/ D f .x/ C

q X kD1

ck .f I x/nk C o.nq /;

4.3 Asymptotic Expansion for Some Positive Linear Operators

53

where the coefficient ck .f I x/ is given by ck .f I x/ D

k X

a.k; s/

sD0

xs .kCs/ .x/ f sŠ

with ! s k s 1 X k g.m/ .1/ X r sr .1/  rCk a.k; s/ D  .r C k; r C m/ kŠ mD0 m g.1/ rD0 rCm and .m; n/ is the Stirling number of second kind. Also, the following complete asymptotic expansion for derivatives of Jakimovski– Leviatan operators was obtained by U. Abel and M. Ivan in [4]: Theorem 4.21 Let q 2 N; x  0; l 2 N0 and f 2 K Œ2qC2l .x/: The derivatives of Jakimovski–Leviatan operators satisfy the asymptotic relation .l/ P.l/ n .f ; x/ D f .x/ C

q X

Œl

ck .f I x/nk C o.nq /;

kD1 Œl

where the coefficient ck .f I x/ is given by Œl



ck .f I x/ D

d dx

l

Œl

ck .f I x/.x/

and ck .f 0 x/ is defined in Theorem 4.20. Furthermore Œl ck .f I x/

! k k 1 X xs .lCkCs/ X k g.m/ .1/ f D .x/ kŠ sD0 sŠ m g.1/ mD0 s X r C r D 0s .1/sr  rCkCl  .r C k C l; r C m C l/: rCmCl

.˛;ˇ/

For f 2 L1w Œ0; 1 the n-th Bernstein–Durrmeyer operator with respect to the Jacobi weight is defined by (see [3]):

.Mn.˛;ˇ/ f /.x/

D

n X kD0

pn;k .x/

1

Z1

.˛;ˇ/

cn;k

0

pn;k .t/ w.˛;ˇ/ .t/ f .t/ dt;

54

4 Voronovskaja-Type Estimates

where pn;k .x/ is Bernstein basis given in (1.2.1), w.˛;ˇ/ .x/ D x˛ .1x/ˇ ; (˛; ˇ > 1) R1 .˛;ˇ/ .˛;ˇ/ is the Jacobi weight on .0; 1/I cn;k D pn;k .t/ w.˛;ˇ/ .t/ dt and Lpw Œ0; 1 is the 0

space of Lebesgue-measurable functions f on Œ0; 1; for which, the weighted Lp norm is finite. r For r 2 N; let L1 Œ0; 1 be the class of all functions f with f .r1/ absolutely continuous on Œ0; 1 and f .r/ 2 L1 Œ0; 1: The following complete asymptotic expansions were calculated in [3]. Theorem 4.22 Let q 2 N; x 2 .0; 1/ and f 2 K Œ2q .x/: Then we have .Mn.˛;ˇ/ f /.x/ D

1 w.˛;ˇ/ .x/

q X

1

kD0

kŠ.n C ˛ C ˇ C 2/k

 .˛Ck;ˇCk/ .k/ w .x/ f .k/ .x/ C o.nq /; where xl D

Ql1

vD0 .x

C v/:

Theorem 4.23 Let q 2 N; x 2 .0; 1/; r 2 N0 and f 2 K Œ2qCr .x/: Then we have .Mn.˛;ˇ/ f /.r/ .x/ D

q X

1

kD0

kŠ.n C ˛ C ˇ C 2/k



1 w.˛;ˇ/ .x/

 .˛Ck;ˇCk/ .k/ w .x/ f .k/ .x/

.r/

C o.nq /:

4.4 Voronovskaja-Type Estimate for Schoenberg Operators We recall the definition of the famous variation-diminishing operator Sn;k .f ; x/; introduced by I. Schoenberg in [167]. Consider the knot sequence n D fxi gnCk k ; n  1; k  1 with equidistant “interior knots”, namely n W xk D : : : D x0 D 0 < x1 < x2 < : : : < xn D : : : D xnCk D 1 and xi D ni for 0  i  n: For bounded real-valued function f defined over Œ0; 1; the Schoenberg spline operator Sn;k of degree k w.r.t. n is given by Sn;k .f ; x/ WD

n1 X

f .j;k / Nj;k .x/ for 0  x < 1

jDk

and Sn;k .f ; 1/ WD

lim Sn;k .f ; y/;

y!1; y 1; n > 1; the exact representation of the second moment of Sn;k ; i.e., Sn;k ..e1  x/2 ; x/ is still missing. The following upper bound for the second moment of Sn;k valid for all n; k  1 and x 2 Œ0; 1 was recently proved in [29]: ˚ min 2x.1  x/I nk Sn;k ..e1  x/ ; x/  1:  nCk1 2

(4.4.2)

Some lower bounds for the second moment can be found in [29, 30, 182]. We refer the reader to [29, 30], where many quantitative estimates are proved, connected with various properties of Schoenberg operator Sn;k : Here, we point out that the lack of suitable sharp upper and lower bounds for the central moments of Sn;k makes it very difficult to apply the method of linear combinations of Sn;k : Nonetheless, very recently, H. Gonska and coauthors proved in [72, 80] the quantitative forms of uniform convergence in Voronovskaja’s Theorem for the cases S1;k and Sn;1 : For example, for the case k D 1; n > 1—the linear interpolant at the knots xi —H. Gonska established in [72] the following result: Theorem 4.24 If zn .x/ D fnxg WD nx  Œnx is the fractional part of nx; then ˇ ˇ   ˇ n2 f 00 .x/ ˇˇ 1 00 1 ˇ  ŒS e ! f : .f ; x/  f .x/  I ˇ z .x/ .1  z .x// n;1 2 ˇ 2 3n n n

(4.4.3)

This is obtained via representation of the second and fourth moments as given by A. Lupa¸s in his Ph.D. thesis [137]: Sn;1 ..e1  x/2 ; x/ D

Sn;1 ..e1  x/4 ; x/ D

1 zn .x/ .1  zn .x//; n2

1 zn .x/ .1  zn .x// Œ1  3 zn .x/ .1  zn .x//: n4

Very recently, in 2012, the following two quantitative Voronokskaja’s type estimates are proved in [182, Theorems 2.1 and 3.1]:

56

4 Voronovskaja-Type Estimates

Theorem 4.25 For f 2 C2 Œ0; 1; we have ˇ ˇ ˇ ˇ ˇSn;k .f ; x/  f .x/  1 Sn;k ..e1  x/2 ; x/ f 00 .x/ˇ ˇ ˇ 2   1 1 kC1  Sn;k ..e1  x/2 ; x/ e : ! f 00 I 2 3 2n

(4.4.4)

Theorem 4.26 For f 2 C2 Œ0; 1; we have ˇ ˇ ˇ ˇ ˇSn;k .f ; x/  f .x/  1 Sn;k ..e1  x/2 ; x/ f 00 .x/ˇ ˇ ˇ 2   1 1p  Sn;k ..e1  x/2 ; x/ e ! f 00 I n;k .x/ ; 2 3

(4.4.5)

where n;k .x/ D 3

n k

   2 1 3 1 x.1  x/ C : k k

The estimate in Theorem 4.25 corresponds to the so-called spline case .k  n1/ and that in Theorem 4.26 corresponds to the so-called polynomial case .k  n/: Further, in the first case, we suppose nk ! 0 and in the second case nk ! 0: In the first case, we suppose that lim

n2 Sn;k ..e1  x/2 ; x/ D g.x/; k

where the convergence is uniform w.r.t. x 2 Œ0; 1: Now, from Theorem 4.25, we get the following Voronovskaja-type estimate: Theorem 4.27 If

k n

! 0; then for f 2 C2 Œ0; 1;

n2 1 ŒSn;k .f ; x/  f .x/ D f 00 .x/:g.x/: nCk!1 k 2 lim

The convergence is uniform w.r.t. x 2 Œ0; 1: Theorem 4.28 If f 2 C2 Œ0; 1; and limk!1 k: Sn;k ..e1 x/2 ; x/ D e.x/ and then lim k ŒSn;k .f ; x/  f .x/ D

nCk!1

The convergence is uniform w.r.t. x 2 Œ0; 1:

1 00 f .x/ e.x/: 2

k n

! 1,

4.5 Voronovskaja-Type Estimates for Linear Combinations

57

To generalize the last two Voronovskaja’s estimates for all other cases when n C k ! 1; it is sufficient to prove Sn;k ..e1  x/4 ; x/ D 0: nCk!1 Sn;k ..e1  x/2 ; x/

(4.4.6)

lim

The last remains still an open problem. Of course, it may be generalized for an arbitrary p.l.o. Ln (instead of Sn;k ) to find necessary and sufficient conditions, such that the relation (4.4.6) holds true.

4.5 Voronovskaja-Type Estimates for Linear Combinations 4.5.1 Phillips Operators In two joint papers [119] and [118] Heilmann and Tachev have studied the approximation properties of Phillips operators e Sn , defined in (1.3.1) and their linear combinations e Sn;r defined in (1.3.2). In order to derive a strong converse inequality of type A we need an appropriate strong Voronovskaja type result. In [118, Theorem 4.1], the following estimate was proved: p Theorem 4.29 Let g 2 CB Œ0; 1/; ' 2 g000 ; ' 3 g000 2 CB Œ0; 1/ with '.x/ D x and n > 0. Then ˇˇ ˇˇ ˇˇ 1 2 00 ˇˇˇˇ ˇˇe ˇˇS n g  g  n ' g ˇˇ ( ) r p 4p 6 1 1 C 2c 1 2 000 1 3 000  : max jj' g jj .1 C 2c/ p :jj' g jjI (4.5.1) 2 n 3 c n n where c denotes an arbitrary positive constant. Proof We sketch the proof and omit some details. Representing the remainder in Taylor expansion of g in integral form it follows g.t/ D g.x/ C g0 .x/.t  x/ C g00 .x/

1 .t  x/2 C 2 2

Z

t

g000 .u/.t  u/2 du:

x

We apply the operator e Sn to both sides and use Lemma 4.1 to derive ˇ ˇ ˇ ˇ 1 e2 ˇe ˇ S . D .g; x/  g.x/  g/.x/ ˇ n ˇ n ˇ  ˇZ t ˇ ˇ 1  :e Sn ˇˇ g000 .u/.t  u/2 duˇˇ ; x : 2 x We differ two cases, depending upon the position of x.

(4.5.2)

58

4 Voronovskaja-Type Estimates

Case 1. If x 

1 , cn

we have ˇ  ˇZ t ˇ ˇ 000 2 ˇ e Sn ˇ g .u/.t  u/ duˇˇ ; x x ˇ  ˇZ t ˇ .t  u/2 ˇˇ 3 000 e ˇ  jj' g jj:Sn ˇ duˇ ; x : '.u/3 x

As

jtuj u



jtxj , x

(4.5.3)

it follows ˇZ ˇ e Sn ˇˇ

t

x

ˇ  2 1 e .t  u/2 ˇˇ duˇ ; x  Sn .jt  xj3 ; x/: 3 '.u/ 3 x3=2

(4.5.4)

Using the Cauchy–Schwarz inequality, we get the estimate q q e Sn ..t  x/2 ; x/ e Sn ..t  x/4 ; x/ r s   2 2x 12x ; x C  n n2 n

e Sn .jt  xj3 ; x/ 

where we have used Lemma 1.10. For x  ˇZ ˇ e Sn ˇˇ

x

t

1 cn

the last two estimates imply

p ˇ  4 6p .t  u/2 ˇˇ duˇ ; x  1 C 2c:n3=2 : '.u/3 3

Therefore p ˇ  ˇZ t ˇ ˇ 4 6p 000 2 e Sn ˇˇ g .u/.t  u/ duˇˇ ; x  1 C 2c:n3=2 jj' 3 g000 jj: 3 x Case 2. x 

1 . cn

(4.5.5)

In a similar way as in Case 1, we get

r ˇ  ˇZ t p ˇ ˇ 1 C 2c 2 3 000 000 2 e :n jj' g jj: Sn ˇˇ g .u/.t  u/ duˇˇ ; x  6 c

(4.5.6)

x

Now substituting (4.5.5) and (4.5.6) into (4.5.2), we complete the proof.  The strong variant of Voronovskaja-type theorem is a crucial step for the proof of strong converse inequality of type A for Phillips operator e Sn (see [118, Theorem 5.2]):

4.5 Voronovskaja-Type Estimates for Linear Combinations

59

Theorem 4.30 For every f 2 CB Œ0; 1/ and n > 0, the following inequality holds true:   1  92:16:jje Sn f  f jjCB Œ0;1/ : K2;' f ; (4.5.7) n The proof of Theorem 4.30 relies on the following three Bernstein type inequalities (see [118, Lemmas 5.1–5.3].) jje D2 .e Sn f /jj  2n:jjf jj:

(4.5.8)

2 .'/ WD fg W jj' 2 g00 jj1 < 1g and n > 0 we have For every g 2 W1

p Sn g/jj  1:47 njje D2 gjj: jj' 3 D3 .e

(4.5.9)

2 .'/ we have For every g 2 W1

Sn g/jj  2njje D2 gjj: jj' 3 D3 .e

(4.5.10)

We point out that (4.5.8) improves upon the constant in [59, Lemma 6] and (4.5.9) gives an explicit value for the constant in [59, Lemma 7]. The proof of Theorem 4.30 follows the general framework developed by Z. Ditzian and K. Ivanov in [51] and is influenced also by the remarkable paper of Knoop and Zhou [129] where strong converse inequality of type A for Bn was established for the first time. By the standard way using again the integral representation of the remainder in Taylor formula we proved in [118, Theorem 5.1]: Theorem 4.31 For every f 2 CB Œ0; 1/ and n > 0, there holds   ˇˇ ˇˇ 1 ˇ ˇ ˇˇe : Sn f  f  2:K2;' f ; n

(4.5.11)

As a corollary from Theorems 4.30 and 4.31, we get the following equivalence:   ˇˇ ˇˇ ˇˇ 1 ˇˇˇˇe 1  92:16 ˇˇe Sn f  f ˇˇ ; Sn f  f ˇˇ  2:K2;' f ; 2 n   ˇˇ ˇˇ ˇˇ ˇˇ 1 2 ˇ ˇ ˇ ˇe C1 Sn f  f  !' f ; p  C2 ˇˇe Sn f  f ˇˇ : n

(4.5.12)

Now we continue with Voronovskaja-type estimate for linear combinations of Phillips operators e Sn;r : In [119], we proved the following result:

60

4 Voronovskaja-Type Estimates

Theorem 4.32 Let f 2 CB Œ0; 1/ be .2r C 2/-times differentiable at a fixed point x. Then with e D2.rC1/ D Dr ' 2.rC1/ DrC1 we have

n!1

r Y

  .1/r e2.rC1/  D Sn;r f  f D nk g e f .x/: .r C 1/Š kD0

lim f

(4.5.13)

Proof For the function f we use the Taylor expansion 2.rC1/

f .t/ D

X .t  x/ f ./ .x/ C .t  x/2.rC1/ R.t; x/ Š D0

WD e f .t/ C .t  x/2.rC1/ R.t; x/;

(4.5.14)

where jR.t; x/j < C for every t 2 Œ0; 1/ and limt!x R.t; x/ D 0: Now we use the following exact representations of the central moments of e Sn;r (see [119, Lemma 5.2]), if f;r D .t  x/ , then .e Sn;r f0;x /.x/ D 1I .e Sn;r f;x /.x/ D 0; 1    r C 1;

.e Sn;r f;x /.x/ D .1/r

r Y 1 k n kD0

(4.5.15)

8 .rC1/ !   X ˆ 1 1   j  1 Š j ˆ < x jr1 ; r C 2    2r C 2; ; ::: jŠ n0 nr j1 jD1

 ˆ ˆ : PŒ=2 j1 Š xj  1 ; ::: 1 ;   2r C 2 jD1

j1

jr1

jŠ

n0

nr

where j .x0 ; x1 ; : : : xm / denotes the complete symmetric function which is the sum of all products of x0 ; x1 ; : : : xm of total degree j for j 2 N and 0 .x0 ; x1 ; : : : xm / WD 1: Now from (4.5.16), we get 2.rC1/

e Sn;r .e f ; x/  f .x/ D

X f ./ e Sn;r .f;x ; x/ Š DrC2

D .1/r

2.rC1/ r Y 1 X ./ f .x/ nk DrC2 kD0

.rC1/

X jD1

!   1 1 j1 1 j x jr1 ;::: jŠ n0 nr j1

4.5 Voronovskaja-Type Estimates for Linear Combinations

61

 2rC1   r Y 1 1 X 1 D .1/ j.rC1/ ;::: nk jDrC1 n0 nr kD0 r

2.rC1/

X

DjC1

f

./

! j1 1 j x : .x/   j  1 .  j/Š

(4.5.16)

From [119, Lemma 5.1] and the assumption (a) in Section 1.1 it follows that we only have to consider the summand with j D r C 1 for the following limit. Thus we get ( r ) Y k lim n :.e f ; x/  f .x// Sn;r .e n!1

kD0 2.rC1/ r

D .1/

X

f

DrC2

D

./

! r 1 xr1 .x/   r  2 .  r  1/Š

.1/r r rC1 rC2 .1/r e2.rC1/ f .x/; D D Œx D f .x/ D .r C 1/Š .r C 1/Š

(4.5.17)

where we used Leibniz rule. For the remainder term, we have to show that ( r ) Y   k lim n : e (4.5.18) Sn;r ..t  x/2.rC1/ :R.t; x/; x/ D 0 n!1

kD0

For " > 0 let ı > 0 be a positive number, such that jR.t; x/j <  for jt  xj < ı. Thus for every t 2 Œ0; 1/ we have jR.t; x/j <  C C

.t  x/2 : ı 02

Due to the assumptions (a) and (b) for ˛i .n/ we have ˇ  ˇ ˇe Sn;r .t  x/2.rC1/ :R.t; x/; x ˇ    C:e Sn .t  x/2.rC1/ :jR.t; x/j; x    M   2.rC1/ 2.rC2/ e e  C:  Sn .t  x/ ; x C 02 Sn .t  x/ ;x ı From the upper bounds for the moments of order 2.r C 1/ and 2.r C 2/ in Lemma 1.11 (see [119, Corollary 5.3]), we get (4.5.18). The proof is completed.  We point out that the both Voronovskaja-type estimates in Theorems 4.29 and in 4.32 (for linear combinations) are proved by assumption that f 2 CB Œ0; 1/: In the next chapter we extend these results for unbounded function with polynomial and exponential growth.

62

4 Voronovskaja-Type Estimates

4.5.2 Durrmeyer Modification Agrawal and Gupta [13] considered combinations due to Rathore [163] and C.P. May [145] of Bernstein–Durrmeyer operators, and obtained the following asymptotic formula in simultaneous approximation: Theorem 4.33 Let f 2 LB .0; 1; the class of bounded and Lebesgue integrable functions on Œ0; 1, which admits a derivative of order 2kCr C2 at a point x 2 Œ0; 1, then .r/

lim nkC1 ŒBn .f ; k; x/  f .r/ .x/ D

n!1

2kCrC2 X

f .i/ .x/Q.i; k; r; x/

iDr

and .r/

lim nkC1 ŒBn .f ; k C 1; x/  f .r/ .x/ D 0

n!1

where Q.i; k; r; x/ are certain polynomials in x: Also, the above limits hold uniformly if f .2kCrC2/ 2 CŒ0; 1: In this direction Kasana [127] also considered such combinations and established some direct results including a Voronovskaja-type asymptotic formula in ordinary approximation for Szász–Durrmeyer operators. Kasana [127] considered the following class of functions: P˛ Œ0; 1/ D ff 2 PŒ0; 1/ W f .t/ D O.e˛t /; t ! 1; ˛ > 0 R1 with PŒ0; 1/ D ff W 0 ent f .t/dt < 1; n > n0 .f /g and by using the linear approximating method, viz. Steklov mean, he obtained the following asymptotic formula: Theorem 4.34 Let f 2 P˛ Œ0; 1/ and let f .2kC2/ exist at a point x 2 Œ0; 1/. Then lim nkC1 ŒSn .f ; k; x/  f .x/ D

n!1

2kC2 X

f .m/ .x/ Q.m; k; x/ mŠ mDkC1

and lim nkC1 ŒSn .f ; k C 1; x/  f .x/ D 0

n!1

where Q.m; k; x/ are certain polynomials in x of degree at most Œm=2 such that Q.2k C 1; k; x/ D

.1/k .2k C 2/Š k .1/k .2k C 2/Š kC1 x ; Q.2k C 2; k; x/ D x : Qk Q 2kŠ jD0 dj .k C 1/Š kjD0 dj

4.5 Voronovskaja-Type Estimates for Linear Combinations

63

Moreover, if f .2k C 2/ exists and is continuous on Œa1 ; b1 , then the above limits in the theorem hold uniformly on Œa1 ; b1 : c Here we consider the linear combinations Mn;r as defined in (1.2.25). We recall that for c D 1 we get combinations of Bni , for c D 0 combinations of Sni and c D 1 combinations of V ni . The following Voronovskaja-type theorem for approximation of f 2 CŒ0; 1; .c D 1/ and f 2 CB Œ0; 1/; .c D 0; 1/ was proved by M. Heilmann [115]. Theorem 4.35 Let f 2 CŒ0; 1; c D 1 and f 2 CŒ0; 1/ for c D 0; 1, '.x/ D p x.1 C cx/; r 2 N and f is 2r-times differentiable at a fixed point x. Then we have r1 Y c f  f /.x/ lim f .nl  c.r C 1/g.Mn;r

n!1

D

lD0

.1/r1 dr 2r f' .x/:f .r/ .x/g: rŠ dxr

(4.5.19)

The proof of Theorem 4.35 follows by the same way as Theorem 4.32 and relies c ..t  x/ ; x/, given in [115]. We on the explicit representations of the moments Mn;r omit the details.

4.5.3 Simultaneous Approximation It is well known that if f 2 Ck Œ0; 1 for any k  0, then lim .Bn f /.k/ .x/ D f .k/ .x/;

n!1

uniformly on Œ0; 1 (see, for example, the book of Lorentz [134]). Recently M. Floater [60] proved that Voronovskaja’s formula can be “differentiated”, namely Theorem 4.36 If f 2 CkC2 Œ0; 1 for some k  0, then lim n..Bn f /.k/ .x/  f .k/ .x// D

n!1

1 dk fx.1  x/f 00 .x/g 2 dxk

(4.5.20)

uniformly for x 2 Œ0; 1. Also for f 2 C2 Œ0; 1 the following error bound is well known (see [43]): j.Bn .f ; x/  f .x/j 

1 x.1  x/jjf 00 jj: 2n

(4.5.21)

64

4 Voronovskaja-Type Estimates

This estimate was generalized by Floater [60, Theorem 1] as follows: Theorem 4.37 If f 2 CkC2 Œ0; 1 for some k  0, then j.Bn f /.k/ .x/  f .k/ .x/j  1 k.k  1/:jjf .k/ jj C kj1  2xj:jjf .kC1/ jj  2n .kC2/ Cx.1  x/jjf jj :

(4.5.22)

The both results Theorems 4.36 and 4.37 are based on differentiation of remainder formula, studied by Stancu [173]. The case of simultaneous approximation by linear combinations of Bernstein operator Bn;r will be considered in the next chapter. Now we represent two Voronovskaja-type estimates for approximation by linear combinations of Szász– Mirakjan–Durrmeyer operators Sn;r and by the combinations of Phillips operators e Sn;r (i.e. combinations of genuine Szász–Mirakjan–Durrmeyer operators). The first result states the following (see [115, Satz 8.5]): p Theorem 4.38 If f .s/ 2 CB Œ0; 1/; s 2 N0 ; '.x/ D x and for r 2 N, f .s/ is 2rtimes differentiable at a point x 2 Œ0; 1/, then the following holds true: ( r1 ) Y .1/r1 drCs 2r nl .Sn;r f  f /.s/ .x/ D Œ' .x/f .r/ .x/: lim rCs n!1 rŠ dx lD0

(4.5.23)

The second result is for the linear combinations e Sn;r (see [119, Theorem 5.5]): Theorem 4.39 If f 2 CB Œ0; 1/ be .m C 2r C 2/-times differentiable at a fixed point x, then with e D2.rC1/ D Dr ' 2.rC1/ DrC2 , we have ( lim

n!1

r Y

) nk .Sn;r f  f /.m/ .x/ D

kD0

.1/r .Dm e D2.rC1/ f /.x/: .r C 1/Š

(4.5.24)

The proofs of both statements are similar to the proofs of Theorems 4.32 and 4.35 and rely on the explicit representations of the central moments of linear combinations. In [1] U. Abel considered the linear combinations of Bernstein–Durrmeyer operators as Bn;r .f ; x/ D

r1 X iD0

˛i .n/Bni .f ; x/;

4.5 Voronovskaja-Type Estimates for Linear Combinations

65

where the coefficient ˛i .n/ satisfies the condition (a) of linear combinations given in and ˛i .n/ D .ni C 2/r1

r1 X .ni  nj /1

(4.5.25)

jD0 k 0 where the rising factorial Pr1 n D n.nC1/    .nCk1/; n D 1. It was observed in [1] that the condition iD0 j˛i .n/j  B with certain constant B independent of n is not required. The condition (4.5.25) guarantees that this condition is valid if in addition niC1  n ; .i D 0; 1; 2;    r  1/ with some constant  > 1: The following local estimate was obtained by Abel in [1] for the combinations of Bernstein–Durrmeyer operators:

Theorem 4.40 Let r; q 2 N; m 2 N/ and x 2 Œ0; 1: Then for f 2 KŒm; 2.q C r/I x, (the class of all functions f 2 L1 Œ0; 1 which are m C 2q C 2r times differentiable at x), the linear combinations Bn;r .f ; x/ with the conditions as mentioned above satisfy .m/ Bn;r .f ; x/

Df

.m/

q X



' 2.kCr/ .x/f .kCr/ .x/ .x/ C S.k; rI n0 ;    ; nr1 / .k C r/Š kD0

mCkCr

Co.n.qCr/ /; as n ! 1, where '.x/ D

p

x.1  x/ and

! r1 k X .1/rC1 X k .1/v .ni C r C 1 C v/1 : S.k; rI n0 ;    ; nr1 / D kŠ v vD0 jD0 Moreover, we have S.k; rI n0 ;    ; nr1 / D Ø.n.kCr/ /; n ! 1: Also it was remarked in [1] that for q D 0, the following Voronovskaja type formula holds: .mCr/  2r r1 Y ' .x/f .r/ .x/ .nj C r C 1/ŒBn;r .f ; x/  f .x/.m/ D .1/rC1 : n!1 rŠ jD0 lim

The special case m D 0 was discussed by M. Heilmann in [115, Satz 8.4].

Chapter 5

Pointwise Estimates for Linear Combinations

5.1 Approximation by Linear Combinations 5.1.1 Discrete Operators It was pointed out by Feilong and Zongben [56] that the Baskokov operators with the weight function ' 2 .x/ D x.1 C x/ are non-concave on Œ0; 1/: By using the Ditzian– Totik modulus !'r  .f ; t/ of r-th order with r 2 N; 0   < 1 they established the following four main results for the classical Baskakov operators: Theorem 5.1 Let 0    1; 0 < ˛ < r; r 2 N and f 2 CŒ0; 1/; we have ˛ jVn;r .f ; x/  f .x/j D O.n1=2 A1 n .x//

if and only if !'r  .f ; t/ D O.t˛ /; p p where An .x/ D '.x/ C 1= n  max.'.x/ C 1= n/: Theorem 5.2 If 0    1; 0 < ˛ < r; r 2 N; then for f 2 CŒ0; 1/ and !'r  .f ; t/ D O.tˇ / with certain ˇ > 0; we have   ' r .x/ jVn.r/ .f ; x/j D O min n2

n .x/ ' 2 .1  /

.r˛/=2

if and only if !'r  .f ; t/ D O.t˛ /: © Springer International Publishing AG 2017 V. Gupta, G. Tachev, Approximation with Positive Linear Operators and Linear Combinations, Developments in Mathematics 50, DOI 10.1007/978-3-319-58795-0_5

67

68

5 Pointwise Estimates for Linear Combinations

Theorem 5.3 Let 0    1; r 2 N and f 2 CŒ0; 1/; we have jVn;r .f ; x/j  C !'r  .f ; n1=2 A1 n .x//: Theorem 5.4 Let f 2 CŒ0; 1/; r 2 N; 0 < ˛ < r; 0    1; we have .x/ !'r  .f ; n1=2 A1 j' r .x/ Vn.r/ .f ; x/j  C nr=2 Ar.1/ n n /:

5.1.2 Szász–Mirakjan–Durrmeyer Operators As far as we know the first characterization of pointwise approximation of bounded continuous functions f 2 CB Œ0; 1/ by linear combinations of Szász–Mirakjan– Durrmeyer operators (abbrev. SMD operators) was given by D.X. Zhou in 1993 in [205]. This result is in terms of r-th order moduli of smoothness ! r .f ; t/ defined by ! r .f ; t/ D sup jj rh f .:/jjCŒ0;1/ 0 I f xC k 2 2 k

and rh f .x/ D 0, otherwise. Zhou extended the results of Mazhar and Totik considering the linear combinations Ln;r defined in Section 1.1 with Ln D Sn the Durrmeyer modification of Szász–Mirakjan operator. The equivalent K-functional to ! r .f ; t/ is given by Kr .f ; tr / D inf fjjf  gjjCŒ0;1/ C tr jjgjjDr g; g2Dr

where Dr D fg 2 CB Œ0; 1/ W f .r1/ 2 ACloc ; g.r/ 2 L1 Œ0; 1/g and jjgjjDr D jjgjjCŒ0;1/ C jjg.r/ jj1 :

5.1 Approximation by Linear Combinations

69

The direct result states the following (see [205, Theorem 1]): Theorem 5.5 Let f 2 CB Œ0; 1/; r 2 N: Then we have  ! 1 r=2 x C 2 f; n n ! r 1 x C 2 ; f; n n 

jLn;r .f ; x/  f .x/j  M:Kr  M0! r

(5.1.1)

where M; M 0 are constants depending on f ; n 2 N and x  0: Using some ideas from [23, 206] Zhou established the following inverse result in a pointwise form (see [205, Theorem 2]): Theorem 5.6 Let f 2 CB Œ0; 1/; r 2 N; 0 < ˛ < r: Then we have  jLn;r .f ; x/  f .x/j  C

1 x C 2 n n

˛=2 (5.1.2)

with a constant independent of x and n, if and only if ! r .f ; h/ D O.h˛ /:

(5.1.3)

Also the following connections between derivatives and smoothness were proved in [205, Theorem 3]: Theorem 5.7 Let f 2 CB Œ0; 1/; r 2 N; 0 < ˛ < r: Then we have

n .r˛/=2 jLn.r/ .f ; x/j  M min n2 ; , ! r .f ; h/ D O.h˛ /: x

(5.1.4)

The results of Zhou were extended in 1998 by S. Guo and coauthors in [87] in terms of r-th order Ditzian–Totik moduli of smoothness with parameter . The main result in [87] states the following: Theorem 5.8 Let f 2 CB Œ0; 1/; r 2 N; 0 < ˛ < r; 0    1: Then the following statements are equivalent: jLn;r .f ; x/  f .x/j D O..n1=2 :ın1 .x//˛ /;

(5.1.5)

!'r  .f ; t/ D O.t˛ /;

(5.1.6)

' r: .x/jLn.r/ .f ; x/j D O..n1=2 ın1 .x//˛r /;

(5.1.7)

where ın .x/ D '.x/ C

p1 n



p  max '.x/I p1n ; '.x/ D x; x  0:

70

5 Pointwise Estimates for Linear Combinations

It was pointed out in [37] that the generalization of Zhou’s result could not cover the range between r and 2r. On the other hand, for  D 1; 2r can replace r in Theorem 5.8 and to obtain corresponding equivalent relation (5.1.5), (5.1.7). It is similar to [50, (9.3.3)].

5.1.3 Bernstein Polynomials In 1994 Z. Ditzian proved in [47] the following pointwise estimate for approximation by Bernstein polynomials: Theorem 5.9 For f 2 CŒ0; 1/; x 2 Œ0; 1; 0    1; we have jBn .f ; x/  f .x/j 

C:!'2

  1 1 f ; p ' .x/ ; n

(5.1.8)

p where 0    1; '.x/ D x.1  x/: However Ditzian did not consider the inverse result of (5.1.8). Guo et al. (also see the references therein) in [86] obtained the following equivalence: Theorem 5.10 For 0 < ˛ < 2; 0    1; the following holds true:  Bn .f ; x/  f .x/ D O

1 p ' 1 .x/ n

˛ 

, !'2 .f ; t/ D O.t˛ /:

(5.1.9)

In 1996 the last was generalized by Guo and coauthors for linear combinations Bn;r , defined in Section 1.1 as follows (see [89]): Theorem 5.11 For f 2 CŒ0; 1/; 0 < ˛ < r; 0    1; we have  jBn;r .f ; x/  f .x/j D O

1 p ' 1 .x/ n

˛ 

, !'r  .f ; t/ D O.t˛ /;

(5.1.10)

where ın .x/ D '.x/ C p1n : For r D 1;  D 0 as a corollary we get the result of Berens and Lorentz [25]. The proof of Theorem 5.11 is based on the following two estimates: A. Direct Theorem (see [89, Theorem 1]):   jBn;r .f ; x/  f .x/j  A!'r  f ; n1=2 :ın1 .x/ :

(5.1.11)

5.1 Approximation by Linear Combinations

71

B. Inverse Theorem For f 2 CŒ0; 1; 0 < ˛ < r; 0    1; if  ˛ 1 jBn;r .f ; x/  f .x/j  B: p :ın1 .x/ ; n then !'r  .f ; t/ D O.t˛ /;

(5.1.12)

(see [89, p. 400]). We recall that the related K-functional, equivalent to !'r  .f ; t/ was given by Ditzian and Totik [50, p. 24]: K '  .f ; tr / WD

inf

gr1 2ACloc

fjjf  gjj C tr jj' r g.r/ jj C t2r=.2/ jjg.r/ jjg: (5.1.13)

Proof of A. Direct Theorem According to (5.1.13) we may choose gn  gn;x; , for a fixed x and  such that   1 jjf  gn jj  A1 !'r  f ; p :ın1 .x/ ; (5.1.14) n 



1 p :ın1 .x/ n

1 p :ın1 .x/ n

r

jj' r g.r/ n jj

2r=.2/



A2 !'r 

  1 1 f ; p :ın .x/ ; n

  1 1 r f ; jjg.r/ jj  A ! .x/ ; :ı p 3 ' n n n

(5.1.15)

(5.1.16)

From the definition of Bn;r , it follows: jBn;r .f ; x/  f .x/j  jBn;r .f  gn ; x/j C jf .x/  gn .x/j C jBn;r .gn ; x/  gn .x/j  .C C 1/jjf  gn jj C jBn;r .gn ; x/  gn .x/j

(5.1.17)

From the fact that Bn;r reproduces all polynomials of degree  r  1 and the integral representation of the remainder in Taylor formula, we get ˇ  ˇ Z t ˇ ˇ 1 ˇ jBn;r .gn ; x/  gn .x/j  ˇˇBn;r .t  u/r1 g.r/ .u/du; x n ˇ .r  1/Š x



r1 X iD0

j˛i .n/j:jjınr g.r/ n jj1 :

1 Bn .r  1/Š i



jt  xjr ;x ınr .x/



72

5 Pointwise Estimates for Linear Combinations



r1 X

r r=2 r j˛i .n/j:jjınr g.r/ ın .x/ n jj1 ın .x/:M1 n

iD0

 C:M1 nr=2 ınr.1/ .x/:jjınr g.r/ n jj1

(5.1.18)

In the last inequality we have used (b) property of linear coefficient ˛i .n/, upper bound (1.2.9), for the moments Bni .jt  xjr ; x/ and the estimate (see [51, Lemma 5.3]): ˇZ t ˇ ˇ ˇ r1 Z t ˇ ˇ ˇ ˇ r .r/ ˇ .t  u/r1 g.r/ .u/duˇ  ˇ .t  x/ ˇ: ı .u/g .u/du n n n ˇ ˇ ˇ ı r .x/ ˇ x

n

(5.1.19)

x

From [51, Lemma 5.3], it follows that the last estimate holds true with ın .x/ replaced by '.x/. Therefore jBn;r .gn ; x/  gn .x/j  C:M1 nr=2 ınr .x/' r .x/jj' r :g.r/ n jj: If x 2 En D

1 n

;1 

1 n



(5.1.20)

, then ın .x/  '.x/ and (5.1.20) and (5.1.15) imply

jBn;r .gn ; x/  gn .x/j  M2 Œn1=2 ın1 .x/r :jj' r :g.r/ n jj  M2 !'r  .f ; n1=2 ın1 .x//: S   1  1n ; 1 , then ın .x/  If x 2 0; 1n have

p1 n

(5.1.21)

and by (5.1.15), (5.1.16), (5.1.18), we

r=2 .r/ gn jj jBn;r .gn ; x/  gn .x/j  M3 Œn1=2 ın1 .x/r :jj' r :g.r/ n jj C M3 jjn

 M4 :Œn1=2 ın1 .x/r :jj' r :g.r/ n jj CM4 :Œn1=2 ın1 .x/2r=.2/ :jjg.r/ n jj  M5 !'r  .f ; n1=2 ın1 .x//:

(5.1.22)

From (5.1.22), (5.1.21), (5.1.14) and (5.1.17), we complete the proof of the theorem.  Proof of B. Inverse Theorem To prove the inverse theorem the author in [89] introduced the following notations: C0 D ff 2 CŒ0; 1; f .0/ D f .1/ D 0g; jjf jj0 D sup jın˛.1/ .x/f .x/j; x2.0;1/

C0 D ff 2 C0 W jjf jj0 < 1g;

5.1 Approximation by Linear Combinations

73

jjf jjr D sup jınrC˛.1/ .x/:f .r/ .x/j; x2.0;1/

Cr D ff 2 C0 ; jjf jjr < 1; f .r1/ 2 ACloc g: For f 2 C0 the following K-functional is defined K˛ .f ; tr / D infr fjjf  gjj0 C tr jjgjjr g: g2C

(5.1.23)

The proof of (5.1.12) is based on the following lemmas (for details see [89, Lemmas 3.1, 3.2]). Lemma 5.1 If n 2 N; 0 < ˛ < r, then

Lemma 5.2 For 0 < t < Z

t=2

Z

Z

t=2

jjBn f jjr  B1 nr=2 jjf jj0 ; f 2 C0 :

(5.1.24)

jjBn f jjr  B2 jjf jjr ; f 2 Cr :

(5.1.25)

1 rt ; 8r 2

t=2

 t=2

t=2

t=2

x1

ınˇ .x C

r X

rt 2

and 0  ˇ  r, we have

uk /du1 du2 :::dur  C.ˇ/tr ınˇ .x/: (5.1.26)

kD1

Since Bn .f ; x/ preserves linear functions only f 2 C0 for r > 1 will be considered. For r D 1 f .x/ D ax C b we have !'1 .ax C b; t/ D a'  .x/:t  at˛ .0 < ˛ < 1/. So let f 2 C0 : From (5.1.23), we have K˛ .f ; tr /  jjBn;r f  f jj0 C tr jjBn;r f jjr ;

(5.1.27)

and we choose g 2 Cr such that jjf  gjj0  2K˛ .f ; nr=2 /; nr=2 jjgjjr  2K˛ .f ; nr=2 /:

(5.1.28)

By the assumption, we have jjBn;r .f ; x/  g.x/jj0  Bn˛=2 : From Lemma 5.1 and (5.1.28), we get jjBn;r .f /jjr  jjBn;r .f  g/jjr C jjBn;r .g/jjr  MŒnr=2 jjf  gjj0 C jjgjjr   4Mnr=2 K˛ .f ; nr=2 /:

(5.1.29)

74

5 Pointwise Estimates for Linear Combinations

From (5.1.27)–(5.1.29), it follows K˛ .f ; tr /  M1 Œn˛=2 C tr nr=2 K˛ .f ; nr=2 / and this implies by Berens–Lorentz lemma that if ˛ < r, then K˛ .f ; tr /  M2 t˛ :

(5.1.30)

It remains to show j rt'  .x/ f .x/j



Cın˛.1/ K˛

 f;

tr ı r.1/ .x/



D Mt˛ ;

by (5.1.30), which implies (5.1.12). Using Lemma 5.2 for g 2 Cr ; 0 < t'  .x/ < 1=8r; rt:'  .x/ rt:'  .x/ x1 2 2 we have j rt'  .x/ f .x/j ˇ ˇZ  Z t'  .x/=2 ˇ ˇ t' .x/=2 ˇ ˇ .r/  g .x C u1 C u2 C    ur /du1 du2    dur ˇ ˇ ˇ ˇ t'  .x/=2 t'  .x/=2 Z  jjgjjr

Z

t'  .x/=2

t'  .x/=2

 t'  .x/=2

t'  .x/=2

ınrC˛.1/ .x C u1 C    ur /du1 du2    dur

 Mtr ' r .x/ınrC˛.1/ .x/jjgjjr  M 0 tr ın.˛r/.1/ .x/jjgjjr :

(5.1.31)

Now by choosing an appropriate function g, we get j rt'  .x/ f .x/j  j rt'  .x/ .f  g/.x/j C j rt'  .x/ g.x/j    Mın˛.1/ .x/ jjf  gjj0 C tr ınr.1/ .x/jjgjjr ! r t  4Mın˛.1/ .x/K˛ f ; r.1/ ; ın .x/ which implies (5.1.30). Thus the proof of Theorem 5.11 is completed.  For the result in Theorem 5.11—Ditzian showed in [50, Chapter 9] that for  D 1 instead of (5.1.11) it is possible to compare Bn;r .f ; x/  f .x/ with !'2r .f ; t/. On the other hand, in [46] Ditzian showed that for  D 0 it is not possible to replace

5.2 Simultaneous Pointwise Approximation by Bn;r

75

!'r  .f ; t/ with !'2r .f ; t/. So the question arises which case of , !'r  .f ; t/ can be replaced by !'2r .f ; t/; in Theorem 5.11? The answer to this question was given in [88] by S. Guo et al. in 2000. We formulate their result as: Theorem 5.12 For f 2 CŒ0; 1; r 2 N; 0 < ˛ < 2r; 1 

1 r

   1, we have

Bn;r .f ; x/  f .x/ D O..n1=2 ' 1 .x//˛ / , !'2r .f ; t/ D O.t˛ /: For 0   < 1 

1 r

(5.1.32)

the equivalence (5.1.32) is not true.

5.2 Simultaneous Pointwise Approximation by Bn;r The relation between the rate of convergence for the derivatives of combinations of Bernstein operators Bn;r and the smoothness for the derivatives of functions was investigated in 2005 by L. Xie in [198]. His main result states the following: Theorem 5.13 (see [198, Theorem 1]) Let s; r 2 N; 0    1 and s < ˛ < 2r s C 2 : Then for all f .s/ 2 CŒ0; 1 the following equivalence holds true: ns .n  s/Š .s/ Bn;r .f ; x/  f .s/ .x/ D O..n1=2 ın1 .x//˛s /; n ! 1 nŠ , !'2r .f ; t/ D O.t˛s /; t ! 0

(5.2.1)

p where ın .x/ D x.1  x/ C p1n : The implication ( (direct estimate) follows from the following statement (see [198, Theorem 2]): Theorem 5.14 Let s; r 2 N; 0    1 and J D maxfj W r2rCj  0; j  2r1g: Then for all f .s/ 2 CŒ0; 1; n 2 N with n  M, we have ˇ s ˇ ˇ n .n  s/Š .s/ ˇ .s/ ˇ ˇ B .f ; x/  f .x/ n;r ˇ ˇ nŠ C

X J

  ! i f .s/ ; .nr ' 2.ir/ .x//1=2 C !'2r .f .i/ ; n1=2 ' 1 .x//

iDr r 2r.1/

Cn '

.x/jjf jj ; .i/

(5.2.2)

where M is a positive constant and ! i .f ; t/ is the i-th order classical modulus of smoothness.

76

5 Pointwise Estimates for Linear Combinations

The proof of the inverse implication ) relies on the following result (see [198, Lemma 2.5]): Theorem 5.15 Let r 2 N; 0    1 and 0 < ˇ
0 W jf .x/j  M.1 C xk /; 8x  0g; k 2 N: Consequently from (6.2.1) and (6.2.4), we obtain Ln .jRr .f I t; x/jI x/   D Ln jt  xjr Rr .f I t; x/I x

92

6 Voronovskaja’s Theorem in Terms of Weighted Modulus of Continuity

p   2 1 r Ln .jt  xj I x/ C p :Ln jt  xjrC1  rŠ x:h m    jt  xj 1C xC I x !' .f .r/ I h/: 2

(6.2.5)

Using Cauchy–Schwarz inequality, we get Ln .jRr .f I t; x/jI x/ p  1 2 p r  Ln .jt  xj I x/ C p : Ln .jt  xj2rC2 I x/ rŠ x:h v u  2 !   u jt  xj m tL I x !' .f .r/ I h/: 1C xC n 2

(6.2.6)

Gupta and Tachev [186] have considered Taylor formula for an arbitrary r  2, although in our main statements and in applications we need only the case r D 2, because similar Voronovskaja-type theorems can be proved for all r  2. First quantitative theorem considered in [186] in terms of weighted modulus !' .f I h/ for r D 2 is the following result: Theorem 6.2 Let Ln W E ! CŒ0; 1/; Ck Œ0; 1/  E; k D maxfm C 3; 6; 2mg be sequence of linear positive operators, preserving the linear functions. If f 2 C2 Œ0; 1/ \ E and f 00 2 W' Œ0; 1/, then we have for x 2 .0; 1/ that ˇ ˇ ˇ ˇ ˇLn .f ; x/  f .x/  1 f 00 .x/L .x/ˇ n;2 ˇ ˇ 2 v 3 2 u m 2 !   u p 1 jt  xj  4Ln;2 .x/ C 2tLn 1 C x C Ix 5 2 2 0 !' @f 00 I

Ln;6 x

!1=2 1 A: 

Proof The proof follows immediately from (6.2.6), if we set h D

Ln;6 x

1=2

Without using the Cauchy–Schwarz inequality, we proceed as follows:    jt  xj m jt  xjrC1 1 C x C 2

:



6.2 General Form of Voronovskaja’s Theorem

rC1

93

! m X m mk jt  xjk x 2k k kD0

D jt  xj

rC1

C jt  xj

D jt  xj

rC1

! m X m mk jt  xjkCrC1 C : x 2k k kD0

Hence      jt  xj m rC1 Ln jt  xj 1C xC Ix 2 ! m L X .x/ m mk Mn;kCrC1 L D Mn;rC1 .x/ C x k 2 k kD0 " # ! m L X .x/ 1 m mk Mn;kCrC1 L D Mn;rC1 .x/ 1 C L x 2k Mn;rC1 .x/ kD0 k L WD Mn;rC1 .x/Cn;r;m .x/;

(6.2.7)

L .x/ D Ln .jt  xjk I x/ are the absolute moments of order k: where Mn;k For r D 2 by Cauchy–Schwarz inequality, we get L Mn;3 .x/ 

r Substituting h WD

Ln;4 .x/ Ln;2 .x/

q L L Mn;2 .x/Mn;4 .x/:

(6.2.8)

in (6.2.5) and using (6.2.7) and (6.2.8) we obtain the

proof of our quantitative variant of Voronovskaja theorem studied in [186] as: Theorem 6.3 Let Ln W E ! CŒ0; 1/; Ck Œ0; 1/  E; k D maxfm C 3; 4g be sequence of linear positive operators, preserving the linear functions. If f 2 C2 Œ0; 1/ \ E and f 00 2 W' Œ0; 1/, then we have for x 2 .0; 1/ that ˇ ˇ ˇ ˇ ˇLn .f ; x/  f .x/  1 f 00 .x/L .x/ˇ n;2 ˇ ˇ 2 s # " ! p L  .x/ 1 2 n;4  Ln;2 .x/ C p Ln;2 .x/:Cn;2;m .x/ !' f 00 I : 2 x Ln;2 .x/ where ! m L X .x/ m mk Mn;kC3 : Cn;2;m .x/ D 1 C L : x k 2 Mn;3 .x/ kD0 k 1

94

6 Voronovskaja’s Theorem in Terms of Weighted Modulus of Continuity

We suppose for the operators Ln that L Mn;k L Mn;3

;4km

is a bounded ratio for fixed x and m, when n ! 1.

6.3 Applications of General Form In the present section we apply Theorems 6.2 and 6.3 of the previous section to some of the classical positive linear operators which reproduce constant as well as linear functions (may call as genuine operators). Here E is the domain of maximum definition of the operators and in all the cases we have Ck Œ0; 1/  E; k  0. In Theorems 6.4, 6.6 and 6.8 we suppose that k D maxfm C 3; 6; 2mg and in Theorems 6.5, 6.7 and 6.9 we suppose that k D maxfm C 3; 4g.

6.3.1 Szász–Mirakjan Operators The Szász–Mirakjan operators Sn defined by (1.2.2) (also see (3.1.6) with c D 0): The central moments of the Szász–Mirakjan operators are well known and satisfy for m  1 the recurrence relation: x mx S Sn;mC1 .x/ D ŒSn;m .x/0 C .x/:  n n n;m1 It follows that

mC1  Sn;m D O nΠ2  ; n ! 1;

In particular Sn;0 .x/ D 1; Sn;1 .x/ D 0; Sn;2 .x/ D Sn;4 .x/ D

x S x ;  .x/ D 2 ; n n;3 n

x 3x2 S x 10x2 S x 25x2 15x3 C ;  .x/ D C ;  .x/ D C C : n;5 n;6 n3 n2 n4 n3 n5 n4 n3

We verify that    2     jt  xj m jt  xj m jt  xj 2m 1C xC D1C2 xC C xC D 2 2 2 !  !    m 2m X m k jt  xj mk X 2m k jt  xj 2mk D 1 C 2: C : x x 2 2 k k kD0 kD0

6.3 Applications of General Form

95

Hence  2 !   jt  xj m Sn 1 C x C Ix 2 ! ! m 2m X X 1 1 m k S 2m k S D 1 C 2: x Mn;mk .x/: mk C x Mn;2mk .x/ 2mk 2 2 k k kD0 kD0 D An;m;x :

(6.3.1)

It is easy to verify that for fixed x and m, the term An;m;x defined in (6.3.1) is bounded when n ! 1. For example, if 2  k  2m using Cauchy–Schwarz inequality we infer q

k S .x/  Sn;2k2 .x/  Sn;2 .x/ D O n 2 ; Mn;k q

1 S Mn;1 .x/  Sn;2 .x/ D O n 2 : Now we apply Theorem 6.2 and obtain the proof of our next result, which states that: Theorem 6.4 If f 2 C2 Œ0; 1/ \ E and f 00 2 W' Œ0; 1/, then we have for x 2 .0; 1/ that ˇ ˇ 1 hx p i x ˇ ˇ C 2An;m;x ˇSn .f I x/  f .x/  f 00 .x/ˇ  2n 2 n 0 s 1 2 1 25x 15x !' @f 00 I C 4 C 3 A: n5 n n To obtain the quantitative variant of Voronovskaja’s theorem for Száasz–Mirakjan operator similar to Corollary in [7], we apply Theorem 6.2. We show that the term Cn;2;m .x/ is bounded for fixed x and m when n ! 1. Using Cauchy–Schwarz inequality we verify that S Mn;4 .x/ S Mn;3 .x/



q Sn;2 .x/  Sn;6 .x/ Sn;3 .x/

D O.1/; n ! 1;

and in a similar way we proceed for 4 < k  m to confirm that all ratios

S Mn;k .x/ S Mn;3 .x/

are bounded when n ! 1. Thus we verified the condition for the operator Sn , formulated at the end of Theorem 6.3. Consequently our next quantitative result states the following:

96

6 Voronovskaja’s Theorem in Terms of Weighted Modulus of Continuity

Theorem 6.5 If f 2 C2 Œ0; 1/ \ E and f 00 2 W' Œ0; 1/, then we have for x 2 .0; 1/ that ˇ h iˇ x ˇ ˇ ˇn Sn .f I x/  f .x/  f 00 .x/ ˇ 2n ! r h i p p 1 3x 1  x C 2 xCn;2;m .x/ !' f 00 I C 2 : 2 n n Remark 6.1 The argument of the modulus !' .f I :/ in Theorem 6.5 is of order O.n1=2 / like in [7, Corollary 1]. The advantage here is that instead of modulus

.:I ı/ we apply the modulus !' .:I ı/ which makes the estimate valid for much larger class of functions.

6.3.2 Baskakov Operators The Baskakov operators Vn defined by (1.2.3) (also see (3.1.6) with c D 1): The central moments of the Baskakov operators are well known and satisfy for m  1 the following recurrence relation: Vn;mC1 .x/ D

x.1 C x/ V mx.1 C x/ V Œn;m .x/0 C n;m1 .x/: n n

In particular Vn;0 .x/ D 1; Vn;1 .x/ D 0; Vn;2 .x/ D

x.1 C x/ V x.1 C x/.1 C 2x/ ; n;3 .x/ D ; n n2

Vn;4 .x/ D

x.1 C x/ 6x2 .1 C x/2 3x2 .1 C x/2 C C ; n3 n3 n2

Vn;5 .x/ D

x C 15x2 C 50x3 C 60x4 C 24x5 10x2 C 76x3 C 86x4 C 20x5 C ; 4 n n3

Vn;6 .x/ D

x C 31x2 C 180x3 C 390x4 C 360x5 C 120x6 n5 C

25x2 C 288x3 C 667x4 C 534x5 C 130x6 n4

C

15x3 C 105x4 C 105x5 C 15x6 : n3

We apply Theorem 6.2 and obtain the proof of our next result for Baskakov operators (1.2.3), which states that:

6.3 Applications of General Form

97

Theorem 6.6 If f 2 C2 Œ0; 1/ \ E and f 00 2 W' Œ0; 1/, then we have for x 2 .0; 1/ that ˇ ˇ ˇ ˇ ˇVn .f I x/  f .x/  x.1 C x/ f 00 .x/ˇ ˇ ˇ 2n 0 !1=2 1  V  1 x.1 C x/ p n;6 A;  C 2An;m;x !' @f 00 I 2 n x where  V 1=2 n;6 .x/=x D



1 C 31x C 180x2 C 390x3 C 360x4 C 120x5 n5

25x C 288x2 C 667x3 C 534x4 C 130x5 n4  15x2 C 105x3 C 105x4 C 15x5 1=2 C : n3 C

and An;m;x is given by .3:2/ with Sn replaced by V.

mC1  Vn;m D O nŒ 2  ; n ! 1; with the same arguments as for the case for Szász–Mirakjan operator we confirm that An;m;x is bounded for fixed x and m, when n ! 1. To obtain the quantitative variant of Voronovskaja’s theorem for Baskakov operator similar to Corollary in [7], we apply Theorem 6.3 and get Theorem 6.7 If f 2 C2 Œ0; 1/ \ E and f 00 2 W' Œ0; 1/, then we have for x 2 .0; 1/ that ˇ  ˇ ˇ ˇ ˇn Vn .f I x/  f .x/  x.1 C x/ f 00 .x/ ˇ ˇ ˇ 2n i p p 1h  x.1 C x/ C 2 x.1 C x/Cn;2;m .x/ 2 0 s 1 2 1 C 6x C 6x 3x.1 C x/ A: C :!' @f 00 I n2 n In the same way as in the case of Szász–Mirakjan operator we infer that the term Cn;2;m .x/ is bounded for fixed x and m when n ! 1.

98

6 Voronovskaja’s Theorem in Terms of Weighted Modulus of Continuity

6.3.3 Phillips Operators The Phillips operators e Sn defined by (1.3.1): The central moments of the Phillips operators satisfy the following recurrence relation: x e m e e S S 0 S S .x/; m  1 e n;mC1 .x/ D Œ.n;m .x// C 2mn;m1 .x/ C n n n;m In particular, we have 2x e e S S S ; e n;0 .x/ D 1; n;1 .x/ D 0; n;2 .x/ D n 6x e 12x2 24x S S e C 3 n;3 .x/ D 2 ; n;4 .x/ D n n2 n 120x 72x 48x2 S e C C ; n;5 .x/ D n4 n3 n3 720x 576x2 432x 120x3 S e .x/ D C C C : n;6 n5 n4 n4 n3 From [118, Lemma 2.1] it follows that

mC1  S Œ 2  ; n ! 1: e n;m D O n By the same arguments as in the case of Szász–Mirakjan operator we verify that for the Phillips operator again the terms Cn;2;m .x/; An;m;x are bounded for fixed x and m, when n ! 1. We apply Theorem 6.2 and obtain the proof of our next result for Phillips operators, which states that: Theorem 6.8 If f 2 C2 Œ0; 1/ \ E and f 00 2 W' Œ0; 1/, then we have for x 2 .0; 1/ that ˇ ˇ 1  2x p x ˇe ˇ C 2An;m;x ˇSn .f I x/  f .x/  f 00 .x/ˇ  n 2 n 0 s 1 2 720 576x 432 120x A :!' @f 00 I ; C 4 C 4 C n5 n n n3 S. where An;m;x is given by (6.3.1) with S replaced by e To obtain the quantitative variant of Voronovskaja’s theorem for Philips operator similar to Corollary in [7], we apply Theorem 6.3 and obtain

6.4 Voronovskaja’s Estimate for Functions with Exponential Growth

99

Theorem 6.9 If f 2 C2 Œ0; 1/ \ E and f 00 2 W' Œ0; 1/, then we have for x 2 .0; 1/ that ˇ h iˇ x ˇ ˇ e ˇn Sn .f I x/  f .x/  f 00 .x/ ˇ n ! r h i p p 1 12 6x  C 2 : 2x C 2 2 x.1 C x/Cn;2;m .x/ !' f I 2 n n At the end we point out that recently Bustamante, Quesada and Cruz obtained in [33] general theorem for weighted approximation for a broad class of linear positive operators, including the Szász–Mirakjan operators for example, but their results are in terms of weighted K-functional.

6.4 Voronovskaja’s Estimate for Functions with Exponential Growth Here we point out that in case of functions satisfying polynomial growth on the interval Œ0; 1/ other weighted moduli, different from the moduli !' .f I h/ may be applied to obtain Voronovskaja-type theorems. For example, in [132] the following modulus of continuity was introduced

˛ .f ; ı/ D

sup x2Œ0;1/;0 0, i.e. kf kA WD sup jf .x/eAx j < 1; x2Œ0;1/

the second order modulus of continuity, considered by Ditzian in [45] is defined by !2 .f ; ı; A/ D

sup hı;0x 0; n 2 N: It is easy to observe that ˇn < eˇ  1: Corollary 6.2 For f 2 C2 Œ0; 1/ \ E and f 00 2 Lip.˛; A/; 0 < ˛  1 we have for all x 2 Œ0; 1/ ˇ ˇ ˇ ˇ ˇn ŒSn .f ; x/  f .x/  1 f 00 .x/xˇ ˇ ˇ 2 ! r ˛ 1 3x  x  M.A; x/ C n2 n and M.A; x/ D eAx C

C.A;x/ 2

p

C

C.2A;x/ : 2

6.5.3 Phillips Operators For Phillips operators, we have e Sn ..t  x/2 eAt ; x/ D



2n2 x A4 C 4n2 A2  4nA3 /x2 nAx=.nA/ e C : .n  A/3 .n  A/4

Let us suppose that n > 2A. Then n  A >

n 2

and it is easy to verify that

nAx

e nA < e2Ax ; 2n2 x 2x  8;  .n  A/3 n .A4 C 4n2 A2  4nA3 /x2 x2 4 < 16 .A C 4n2 A2 / < .n  A/4 n4  2x   8xA4 C xA2 : < n So we conclude that the condition (i) in Theorem 6.10 is true with C.A; x/ described as   C.A; x/ D e2Ax  8 C 8xA4 C 32xA2 : For the well-known Phillips operators Theorem 6.10 takes the following form: Theorem 6.13 Let e Sn W E ! CŒ0; 1/ be sequence of Phillips operators, preserving the linear functions. If in addition f 2 C2 Œ0; 1/ \ E and f 00 2 Lip.˛; A/; 0 < ˛  1 then we have for x 2 Œ0; 1/

6.5 Applications of Voronovskaja’s Estimate for Functions with Exponential. . .

105

ˇ ˇ x ˇ ˇe ˇSn .f ; x/  f .x/  f 00 .x/ˇ n " ! # r p C.2A; x/ 12 6x C.A; x/ 2x Ax 00  e C C  !1 f ; C 2 ;A :  2 2 n n n Corollary 6.3 For f 2 C2 Œ0; 1/ \ E and f 00 2 Lip.˛; A/; 0 < ˛  1 we have for all x 2 Œ0; 1/ ˇ ˇ   ˇn je Sn .f ; x/  f .x/  f 00 .x/xˇ !˛ r 12 6x  x  M.A; x/ C 2 n n and M.A; x/ D eAx C

C.A;x/ 2

p

C

C.2A;x/ : 2

6.5.4 Lupa¸s–Szász Operators Govil et al. [82] recently proposed a modification of the Lupa¸s operators with weights of Szász basis function as Dn .f ; x/ D n

1 X

Z1 ln;k .x/

kD1

sn;k1 .t/f .t/dt C ln;0 .x/f .0/; x  0 0

where ln;k .x/ D 2nx

.nx/k ; kŠ:2k

sn;k1 .t/ D ent

.nt/k1 : .k  1/Š

Lemma 6.3 It is observed that Dn .1; x/ D 1; Dn .t; x/ D x; Dn .t2 ; x/ D x2 C Dn .t3 ; x/ D x3 C

1x 83x2 9x2 18x3 90x C 2 ; Dn .t4 ; x/ D x4 C C 2 C 3 : n n n n n

Proof Obviously, we have Z1

Z

m

1

ent

sn;k1 .t/t D n

n

3x n

0

0

D

.nt/k1 dt .k  1/Š

.k C m  1/Š : nm :.k  1/Š

106

6 Voronovskaja’s Theorem in Terms of Weighted Modulus of Continuity

Thus using the above identity and the fact that 1 F0 .aI I z/ D .1  z/a ; jzj < 1; we get Dn .1; x/ D

1 X

ln;k .x/ C ln;0 .x/ D

kD1

1 X

P1

zk kD0 .a/k : kŠ

D

ln;k .x/

kD0

nx

D2

1 X .nx/k

nx

  1 :1 F0 nxI I 2

D2 kŠ:2k   1 nx nx 1 D2 D 1: 2 kD0

Next Dn .t; x/ D

1 X

ln;k .x/

kD1

1 2nx X .nx/k k D n n kD1 .k  1/Š:2k

1 1 2nx X .nx/kC1 2nx1 X nx:.nx C 1/k D n kD0 kŠ:2kC1 n kD0 kŠ:2k   1 D x:2nx1 :1 F0 nx C 1I I 2 nx1  1 D x:2nx1 1  D x: 2

D

Proceeding along the similar lines, we can obtain the other moments. Remark 6.2 By simple computation, we have D n;2 .x/ D

3x D 27x2 90x ; n;4 .x/ D 2 C 3 ; n n n

and D n;4 .x/ D n;2 .x/

D

30 9x C 2: n n

Remark 6.3 Following the methods as given in the previous lemma, we have Dn .eAt ; x/ D



n  2A nA

nx

; Dn .teAt ; x/ D

n2 x.n  2A/nx1 .n  A/nxC1

and Dn .t2 eAt ; x/ D n3 x.nx C 1/

nx1 .n  2A/nx2 2 .n  2A/ C 2n x : .n  A/nxC2 .n  A/nxC2



6.5 Applications of Voronovskaja’s Estimate for Functions with Exponential. . .

107

Thus, we have Dn ..t  x/2 eAt ; x/ D

1 2 .n  A/ .n  2A/2



n  2A nA

nx

ŒA2 x2 .4A2  12nA C 9n2 / C n2 x.3n  4A/: Remark 6.4 We suppose that n > 3A, then 

n  2A nA

nx

 D 1C  D 1C

A n  2A A n  2A

nx .n2A/x  : 1C

A n  2A

2Ax

< eAx :22Ax D .4e/Ax ; n > 3A

(6.5.1)

Further A2 x2 .4A2  12nA C 9n2 / A2 x2 .3n  2A/2 A2 x2 .3n/2 D < .n  A/2 .n  2A/2 .n  A/2 .n  2A/2 .n  A/2 .n  2A/2 D

A2 3x 3xn3 : 2 n .n  2A/ .n  A/2