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Table of contents :
CONTENTS
CHAPTER I: BASIC PROPERTIES OF TRIGONOMETRIC INTEGRALS
§1. Trigonometric Integrals Over Finite Intervals
§2. Trigonometric Integrals Over Infinite Intervals
§3. Order of Magnitude of Trigonometric, Integrals
§4. Uniform Convergence of Trigonometric Integrals
§5. The Cauchy Principal Value of Integrals
CHAPTER II: REPRESENTATION — AND SUM FORMULAS
§6. A General Representation Formula
§7. The Dirichlet Integral and Related Integrals
§8. The Fourier Integral Formula
§9. The Wiener Formula
§10. The Poisson Summation Formula
CHPATER III: THE FOURIER INTEGRAL THEOREM
§11. The Fourier Integral Theorem and the Inversion Formulas
§12. Trigonometric Integrals with e^x
§13. The Absolutely Integrable Functions. Their Faltung and Their Summation
§14. Trigonometric Integrals with Rational Functions
§15. Trigonometric Integrals with e^x
§16. Bessel Functions
§17. Evaluation of Certain Repeated Integrals
CHAPTER IV: STIELTJES INTEGRALS
§18. The Function Class B
§19. Sequences of Functions of B
§20. PositiveDefinite Functions
§21. Spectral Decomposition of PositiveDefinite Functions An Application to Almost Periodic Functions
CHAPTER V: OPERATIONS WITH FUNCTIONS OF THE CLASS JO
§22. The Question
§23. Multipliers
§24. Differentiation and Integration
§25. The DifferenceDifferential Equation
§26. The Integral Equation
§27. Systems of Equations
CHAPTER VI: GENERALIZED TRIGONOMETRIC INTEGRALS
§28. Definition of the Generalized Trigonometric Integrals
§29. Further Particulars About the Functions of Jk
§30. Further Particulars About the Functions of Ik
§31. Convergence Theorems
§32. Multipliers
§33. Operator Equations
§34. Functional Equations
CHAPTER VII: ANALYTIC AND HARMONIC FUNCTIONS
§35. Laplace Integrals
§36. Union of Laplace Integrals
§37. Representation of Given Functions by Laplace Integrals
§38. Continuation. Harmonic Functions
§39. Boundary Value Problems for Harmonic Functions
CHAPTER VIII: QUADRATIC INTEGRABILITY
§40. The Parseval Equation
§41. The Theorem of Plancherel
§42. Hankel Transform
CHAPTER IX: FUNCTIONS OF SEVERAL VARIABLES
§43. Trigonometric Integrals in Several Variables
§44. The Fourier Integral Theorem
§45. The Dirichlet Integral
§46. The Poisson Summation Formula
APPENDIX
Concerning Functions of Real Variables
Measurability
Summability
Differentiability
Approximation in the Mean
Complex Valued Functions
Extension of Functions
Summation of Repeated Integrals
REMARKS  QUOTATIONS
MONOTONIC FUNCTIONS, STIELTJES INTEGRALS AND HARMONIC ANALYSIS
Introduction
I: MONOTONIC FUNCTIONS
§1. Definition of the Monotonic Functions
§2. Continuity Intervals
§3. Sequences of Monotonic Functions
II: STIELTJES INTEGRALS
§4. Definition and Important Properties
§5. Uniqueness and Limit Theorems
III: HARMONIC ANALYSIS
§6. FourierStieltjes Integrals
§7. Uniqueness and Limit Theorems
§8. PositiveDefinite Functions
§9. Spectral Decomposition of Square Integrable Functions
SYMBOLS  INDEX
Annals o f Mathematics Studies N um ber 42
ANNALS OF M ATH EM ATICS STUDIES Edited by Robert C. Gunning, John C. Moore, and Marston Morse 1. Algebraic Theory of Numbers, by H e r m a n n W
eyl
3. Consistency of the Continuum Hypothesis, by K u r t G o d e l 6. The Calculi of Lam bdaConversion, by A l o n z o C h u r c h 10. Topics in Topology, by S o l o m o n L e f s c h e t z 11 .
Introduction to Nonlinear Mechanics, by N.
15.
Topological Methods in the Theory of Functions of a Complex Variable, b y
16.
Transcendental Numbers, by
C
L
arl
u d w ig
K
ryloff
and N.
Fourier Transforms, by S.
B ochner
and
K. C
L
Contributions to the Theory of Nonlinear Oscillations, Vol. I, edited by S. Functional Operators, Vol. I, by J o h n
22.
Functional Operators, Vol. II, by
24.
Contributions to
25.
Contributions to Fourier Analysis, edited by A. d e r o n , and S. B o c h n e r
von
o r se
ia p o u n o f f
2 1.
J ohn
M
arst o n
h an d r ase kh ar an
20.
von
M
S ie g e l
17. Probleme G eneral de la Stabilite du Mouvement, by M. A. 19.
Bogohuboff
L
e fsc h e t z
N eum ann N eum an n
the Theory of Games, Vol. I, edited by H. W . K u h n and A. W . T u c k e r Zygm un d,
W.
T r an su e ,
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M
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Contributions to the Theory of Nonlinear Oscillations, Vol. IV, edited by S.
42.
Lectures on Fourier Integrals, by S. B o c h n e r
43.
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A. W .
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and
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a
LECTURES ON FOURIER INTEGRALS BY
Salomon Bochner W ITH AN AU TH O R’S SUPPLEM ENT ON
M onotonic Functions, Stieltj es Integrals, and Harmonic Analysis
TRANSLATED FROM T H E ORIGINAL BY
Morris Tenenbaum and Harry Pollard
PRINCETO N, NEW JER SEY PRINCETON UNIVERSITY PRESS
1959
Copyright © 1959, by P r in c e t o n U n i v e r s i t y P r e s s All Rights Reserved L. C Card 595589 Printed in the United States of America by W e s t v i e w P r e s s , Boulder, Colorado P r in c e t o n U
n iv e r sit y
P r e ss O n D
em an d
E d it io n ,
1985
TRANSLATORS1 PREFACE
In undertaking this translation of Bochner1s classical hook and its supplement (Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse, Mathematische Annalen, Volume 108 ( 193 3 )> pp. 378—iv10), our main purpose was to make generally available to the present generation of grouptheorists and practitioners in distributions the historical and concrete problems which gave rise to these disciplines. Here can be found the theory of positive definite functions, of the generalized Fourier integral, and even forms of the important theorems concerning the reciprocal of Fourier transforms. The translators are grateful to Professor Bochner for his encouragement in this work and for his many valuable suggestions. Morris Tenenbaum Harry Pollard Cornell University
v
CONTENTS Page CHAPTER I: §1. §2. §3 * §4 . §5 
BASIC PROPERTIES OF TRIGONOMETRIC INTEGRALS.......... Trigonometric Integrals Over Finite Intervals........ Trigonometric Integrals Over Infinite Intervals...... Order of Magnitude of Trigonometric,Integrals....... Uniform Convergence of Trigonometric Integrals....... The Cauchy Principal Value of Integrals..............
1 1 5 10 13 18
CHAPTER II: §6. §7 * §8. §9. §10.
REPRESENTATION — AND SUMFORMULAS............... A General Representation Formula................. The Dirichlet Integral and Related Integrals The Fourier Integral Formula......................... The Wiener Formula.. ........................... The Poisson Summation Formula.............
23 23 27 31 35 39
CHPATER III: THE FOURIER INTEGRAL THEOREM........................ §11. The Fourier Integral Theorem and the InversionFormulas
46 46
§12. §1 3 *
51
§1 4 .
Trigonometric Integrals with e”x .................... The Absolutely Integrable Functions. Their Faltung and Their Summation................ Trigonometric Integrals with Rational Functions......
54 63
§1 5 * §16. §17.
Trigonometric Integrals with e“x ................... Bessel Functions.......................... Evaluation of Certain Repeated Integrals.............
67 70 74
CHAPTER IV:
STIELTJES INTEGRALS.................................
78
§18. §1 9 * §20. §21.
The Function Class ............................... Sequences of Functions of $ ........................ PositiveDefinite Functions.......................... Spectral Decomposition of PositiveDefinite Functions. An Application to Almost Periodic Functions........
78 85 92
CHAPTER V: §22. §23. §2 4 . §25. §26. §27. CHAPTER VI: §28. §29* §3 0.
...........
104
The Question......................................... Multipliers.......................................... Differentiation and Integration...................... The DifferenceDifferential Equation .......... The Integral Equation................................ Systems of Equations.................................
104 1 08 114 120 130 134
GENERALIZED TRIGONOMETRIC INTEGRALS.................. Definition of the Generalized Trigonometric Integrals.. Further Particulars About the Functions of ....... Further Particulars About the Functions of .......
138 138 145 153
OPERATIONS WITH FUNCTIONS OF THE CLASS
§3 1  Convergence Theorems................................. §32 . Multipliers........................................... §33  Operator Equations................................... §3 4 . Functional Equations............... CHAPTER VII: §3 5 . §3 6. §3 7 * §38.
97
ANALYTIC AND HARMONIC FUNCTIONS Laplace Integrals.............................. ..... Union of Laplace Integrals........................... Representation of Given Functions by Laplace Integrals. Continuation. Harmonic Functions....................
160 166 173 178 182 182 189 194 202
CONTENTS Page §3 9 » Boundary Value Problems for Harmonic Functions.
... .
CHAPTER VIII: QUADRATIC INTEGRABILITY. ........... . §bo. The Parseval Equation.................... ....... . §bl . The Theorem of Plancherel ............. ......... §b2 . Hankel Transform .... ......... .
208 21b 21b 219 22b 231
CHAPTER IX:
FUNCTIONS OF SEVERAL VARIABLES. ....................
§b3 . §bb. §b5 §b6.
Trigonometric Integrals in Several Variables. .... . The Fourier Integral Theorem............................ The DIrichlet Integral................ ......... . The Poisson Summation Formula. ...... .
231 239 2b9 255
....... . Concerning Functions of RealVariables.................. Measurability ..... . Summability. ............ ........ . Differentiability.............. ......... .......... . Approximation in the Mean................ . Complex Valued Functions Extension of Functions........ ...... ..... ..... . Summation of Repeated Integrals .......... .
26b 26b 26b 2 66 270 271 276 277 279
APPENDIX.
REMARKS  QUOTATIONS. ............................... MONOTONIC FUNCTIONS, STIELTJES INTEGRALS AND HARMONIC ANALYSIS...... I: §1. §2 . §3 • II: §b. §5 . III: §6. §7 • §8. §9 
292
Introduction. ........... ...................... . 292 295 MONOTONIC FUNCTIONS. .............. ...... . Definition of the Monotonic Functions.................. 295 Continuity Intervals............... . 299 Sequences of Monotonic Functions ..... 3 03 STIELTJES INTEGRALS. ..... Definition and Important Properties Uniqueness and Limit Theorems
..... ......... .
307 307 312
HARMONIC ANALYSIS........ ......... .... .............
316
FourierStieltjes Integrals ..... Uniqueness and Limit Theorems ........... . PositiveDefinite Functions.......................... Spectral Decomposition of Square Integrable Functions..
316 320 325 328
SYMBOLS  INDEX. .... .... .......... ........... .............. .
332
LECTURES ON FOURIER INTEGRALS
CHAPTER I BASIC PROPERTIES OF TRIGONOMETRIC INTEGRALS §1 .
Trigonometric Integrals Over Finite Intervals
i. We denote as trigonometric integrals expressions of the form b (l )
$ (a ) = J
(2)
Mr { a ) = J
f (x ) cos ax dx
b f( x ) sin ax dx
It is frequently more convenient to use the exponential factor eiax in place of the trigonometric factors cos ax and sin ax. The trigonometric integral will then read b J(a) = J
f(x)
e lax dx 1
For typographical simplification we shall always denote the function by e(‘)« Hence we shall write J(a) as b (3)
e1 ^
J(a) = jf f (x ) e ( a x ) dx .
It Is also customary to denote trigonometric integrals as Fourier integrals C1j because J. J. Fourier provided the first incentive to the study of these'Integrals [2].
We shall also frequently use the symbols r, H, J to denote respectively the gamma function, the Hankel function and the Bessel function. These special uses at times will be evident from the context. 1
2
CHAPTER lo
TRIGONOMETRIC INTEGRALS
Whenever the contrary Is not evident from the context, a "number" will be a complex number, and a function a complex function of a real vari able* A function f(x) will therefore be an expression of the form f 1 (x) + if2 (x) where f 1 (x) and f2 (x ) are real valued functions as usually defined* For dealing with such functions, cf» Appendix 12 « We shall assume once and for all that each function which occurs under an in tegral sign, will first of all be integrable on each finite interval, and we shall take as a basis the integral concept of Lebesque. Thus we assume that, automatically, any given function Is measurable (Lebesgue) in its entire extent and "summable" (Lebesgue) in every finite interval« > 2 * If the limits of integration integrals (1) to (3), then
a
and
b
ere the same for the
J (a ) = 0 (a ) + ItKa ) , and 2< £>(a) = J (a ) + J (a );
2 l\U(a
) = J(a)  J(~a)
Because of the similarity in construction of 0>(a) and tA(a), we shall frequently prove a statement for only one of the three integrals, and when the transfer is an obvious one, assume its correctness for the other two* Since, in addition, 0>(a) = ®(a);
# (a ) = ~ ^ ( a )
and
1
j(a) = j y s i
where b J1 (a) = J fTxTe (ax ) dx a with the function f^(x) = TTxJ, it will suffice for the study of the functions ®(a), \p(a) and J(a) to limit ourselves to one of the half lines or As a rule, we shall favor the right half line. 3 * At least one of the two limits of the definite integral with which we shall be concerned, will in general, be infinite. To simplify writing, we shall omit the upper integration limit when its value is + 00, and the lower limit when its value is  0 f(x)
is differentiable in
as
(a, b)
a> + 00 and if we denote by
M,
a bound of
and also of b J
f1(x)dx
,
a then it follows from
1 (x, n) will mean the interval X < x < \i, [X, \±]the interval X < x < n. Mixed brackets will also be employed so that (X, n] will mean X < x < n. 2
We shall write for the limit, with no difference in meaning, either litn f()  h or f(l)  > h.
CHAPTER I.
J(a) =
TRIGONOMETRIC INTEGRALS
J
[f (b )e (ab) f(a)e(aa)] ~
f f(x )e (aX ) clx
a that (6 )
J(a ) < yj—y
and from this (5 ) follows. If we write c J (a) = f + a and if (5) valid for involved. particular
b f = J, (a) + Jg (a) c
is valid for J 1 (a) and J2 (a), then it is evidently also J(a).A similar reasoning would apply if more intervals were Hence (5) is valid for a piecewise differentiable function, in for a piecewise constant function ("step function").
By a limiting process it is now possible to prove that (5) is valid for any (integrable) function. Let f(x) and f ^ (x) be two func tions such that u
(7 )
I
f (x)
f 1 (x) dx < £
Then for the corresponding integrals J(a) and J^a), one has b J(a)  J 1 (a ) = (f(x)  f 1 (x))e(ax ) dx
J
a (e)
f^x).
f (x)  f 1 (x) dx
0 a
as
a
An analogous relation also 'holds for the functions
> + «>
s>(a )
and tfr(a) [3 ].
5 • We observe that J(a) is a continuous function, and this fact can be proved as follows: b b J(a + p)  J(a) < f f (x) e(px)  1 dx < M(p ) f f(x) dx , a a
where M(p) is the maximum of e(px)  1 in the interval (a, b). But if p  > o, then M(p) — > o. — (a) and &(a) are also continuous. functions, cf. 2^. §2. 1. the integral
Trigonometric Integrals Over Infinite Intervals. We say that the function
g(x)
is integrable in
A
f g()
dx
a approaches a finite limit as
(i )
A
> ».
We denote this limit by
) dx •
We shall also say that the integral (1) "exists" or that it "converges". Since the function f(x) occurs under the integral sign, it will be tacitly assumed, as agreed to in our previous statement, that it is in tegrable . 2 Paragraph 2 of the present section is meant. Each section is divided into several paragraphs. A simple number denotes a paragraph, and a round bracketed number a formula. Therefore (5 ) denotes the formula (5). If the paragraph or formula is quoted from other sections, then the number of the section is stated in advance. Thus §51, 3 denotes paragraph 3 of §51 , and §51, (9), the formula (9) of §51
[a, »],
if
CHAPTER Xo
6
TRIGONOMETRIC INTEGRALS
Whenever a function g(x) has a certain property in a sub interval [A, oo] or [ oof b] of its interval of definition, then we shall also say that it has this property as x ——y oo or as x — >  ooB Since for each
A > a,
(2)
the integral (1) along with Jg(x) dx A
either converges or does not converge, it follows that the function is integrable in [a, oo] if it is integrable as x ~  > »•
g(x)
It Is a basic property of the Lebesgue integral, that in a finite interval, each Integrable function is also absolutely Integrable. Hence each of the functions considered heretofore is, in each finite interval, absolutely integrable. The same assertion, however, cannot be made if the Interval of Integration is Infinite. If g(x) Is integrable as x — > «> in the sense of our definition, then lg(x) need not be alsointegrable as x — > oo, although the converse does hold. Next, if f(x) is absolutely Integrable in (a, »] then because f(x) sin axj< f(x), the integral tf(a) = J f (x) sin ax dx a
(3 )
converges for all values of is deducible from
a.
Again # (a) —
;> o as
a — > + o and J(a), the simplest of the three. From
0
J(a). a = o,
and let us
§2. one obtains, by letting
7
INFINITE INTERVALS
A  > oo,
and by separating the real and imaginary
parts
(4 ) Both expressions actually approach zero as
a > +■ »
[4 ].
As regards behavior at infinity, an important class of functions which need not be absolutely integrable are monotonic functions.
Let the
(real valued) function f(x) converge monotonically to zero as x — > », i.e., let it be monotonic in a certain interval [ A , o o ] , and convergent to zero as x  > oo. Since we already have at our command integrals over finite intervals, we can assume, therefore, that the point x = A co incides with the initial point x = a. A function monotonic in [a, oo] which converges to zero as
x  > »
is, in its entire range, either
positive and decreasing, or negative and increasing. Since an increasing function becomes, by a change of sign, a decreasing one, we need consider only the decreasing one.
2. We shall need the following theorem of analysis; the socalled second mean value theorem of integration. If, in the interval (a, b), the function q>(x) is continuous, and the function p(x) is positive and monotonically decreasing, then in the interval (a, b) there is a value c between a and b for which b
c
a In particular, let
a
cp(x) = sin ax, a > o.
From
c sin ax dx
p
< —  a
it follows that
(5) Now in' (a, oo],
let the function
p(x)
decrease monotonically to zero.
From A' (6)
p(x) sin ax dx
< ~ p(A)
a > 0,
8
CHAPTER I.
TRIGONOMETRIC INTEGRALS
in conjunction with the fact that that the integral ip (a) = J
p(A) — > 0 as
A —— > °o,
it follows
p(x) sin ax dx
cl
is convergent for and we have
a > o*
We can now allow
Hence it follows that \p(a) — late the following theorem.
> o as
A*
in (6) to become infinite,
a — —> oo. Summarizing, we formu
THEOREM 1. If in [a, »], the function f (x) under consideration, as x — A> oo;, either 1. is absolutely integrable, or 2. converges monotonically to zero, then the integrals 0 (a ), xp(a ), J (a ) exist for 1. all a or 2* all a 4 Q> and converge to zero as a —— > + oo [5 J. The restriction a 4 °, made under 2 applies only to (a) and J(a). For a function decreasing monotonically to zero, it is not nec essary that the integral
which should represent the value f(x) = 1 /x). Now let
f(x)
$(0)
or
J(o),
be representable In the form f(x) = g(x) sin px
where p Is a constant, and means of the relation 2&(a)
converge (for example
g(x)
,
approaches zero monotonically.
= / g(x) cos (a  p)x dx  / g(x) cos (a + p )x dx a a
one recognizes again that, with the possible exceptions of
By
,
a = p
and
§2.
a =  p, the Integral ip(a) The same assertion holds for
INFINITE INTERVALS
exists and converges to zero as
f(x) = g(x) cos px
a  > +
,
and more generally for f(x ) = g(x ) sin (px + q)
where
p
and
q
,
are constants.
THEOREM 1a. The assumption 2 in Theorem 1 can be generalized by setting f(x) = g(x) sin (px + q)
,
where p and q are constants, and g(x‘) approaches zero monotonically as x  > oo. However, the In tegrals need not converge for the values a = + p [5]. THEOREM 1b. A results if the and P ( a ) , is sin a(x  t), [5 3 .
further generalization of the theorem factor cos ax or sin ax in 0>(a) replaced by cos a(x  t) or where t is an additional constant
This generalization can be justified by the transformation y = x  t. [ 00, b],
3. Analogous'statements are valid for a left half line and for the entire interval [ «>, «]. We call the integral
f
g(x) ax
convergent If both integrals 0
J' g(x ) dx
and
J
g(x) dx
0
converge. In this sense, we shall later on attach to the function the special integral E(a) =
f(x)
f f(x)e( ax) dx
and denote it as the (Fourier) transform [6] of
f(x).
The integral
E(
CHAPTER I.
TRIGONOMETRIC INTEGRALS
is therefore normalized somewhat differently from the integral namely
J(a),
J(a) = 2 jt E(~ a ) From the above, we see that
E(a) exists for all
a / o,
and converges
to zero as a — > + °°, provided f (x) is either absolutely convergent or approaches zero monotonically not only as x —— >  A.
If
a > o,
then the integral J s l o a x dx a
falls "under Theorem 1, because in the Interval (a, 00, the limit on the right side of (2) exists, and hence also the limit of f (x). We denote this limit by f (°o). If, therefore, with x fixed In (2), we now let b — > », we obtain f(x) = f (00) + h 1 (x)  h2 (x)
,
with h 1 (x) and h2 (x) monotonicallydecreasing functions, andwith the parameter b In these functions having now the value + oo. if therefore f (x) decreases to zero as x — > 00, one can then represent f(x) as the difference of two monotonic functions which also decrease to zero [8 ]. 3 • The inquiry can be extended to include the case in which f(x) has infinite discontinuities at Isolated points. In an interval (a, b ) and for any c, let
f(x) = x~cM where g(x) is of bounded variation and One can then easily show that
\i
is a positive number
< 1.
b / a
f(x) 3in axdx =
•
We shall not, however,, prove this [9]. t. Let the function f(x) be differentiable In [ 00, 00], together with its derivatives be absolutely integrable. Since f ’(x) absolutely integrable, the integral
and Is
x g(x) =
J
f 1( ) d
exists, and since g ’(x) = f 1(x), we have f(x) = g(x) + c. As x — >  °°, g(x)  > 0. if now c were 4 then f (x) could not be absolutely integrable as x — >  «>. Hence f(x) = g(x), and In particular f (x) —— > 0 as x >  oo. If we now put J r 1(1) de = c1 then
,
§*♦.
UNIFORM CONVERGENCE
f (x) = c 1  J
f f U ) d£
13 .
x Again we obtain
c1 = o,
and hence
f(x)  > o
also as
x  >  oo.
Let
us now consider the transform E(a) =
f(x)e( a x ) dx
By partial integration, we have iaE(a) = •— J
f ,(x)e( ax) dx
.
Applying Theorem 1 (cf. also §2, 3) to the integrand f*(x) f(x), we see that iaE(a) converges to zero as a > +
instead of to i.e.,
E(a) = o(a”1) If the function f ’(x) has In its turn an absolutely Integrable derivative, then by the same reasoning, one obtains iaE(a) = o(a1 ) and therefore E(a) = o( a f 2 ) Continuing in this manner, we obtain the following general theorem: If the function f(x) is ktimes differentiable In [ ~ o o , 00], k = 0, i, 2, ...,1 and together with its first k derivatives, is absolutely integrable, then for its transform we have E(a) = o( a T k )
Uniform Convergence of Trigonometric Integrals 1.
Consider the convergent integral f
Six,
x) dx
a which depends on a parameter
1
X.
The integral is called uniformly
By the oth derivative of a function, we mean the function itself.
CHAPTER I.
u
convergent, if to each
e,
TRIGONOMETRIC INTEGRALS
one can find an
A(e),
such that for
A > A(e)
x
and for all considered values
J
g(x,
X)
dx
£ €
A An analogous definition is valid for an integral extending over
[ °°, then the uniform convergence for a ^ a Q (> 0) follows from §2, 2. A similar statement is valid for e>(a). More generally, if f(x) = g(x)sin(px + q) and g(x) — — > 0 monotonically, then 9 ( a ) and tfr(a) are uniformly convergent in each interval (a1, a2 ) which does not contain the points a = + p and a =  p. The same assertion, moreover, holds for the general integral (1)
J
f(x)cos a(x  t) dx;
a
J
f(x)sin a(x  t ) dx
a
Furthermore, each of these integrals, in each interval in which it converges uniformly, is a continuous func tion of
a.
The last statement is easily verified. in which the function
Consider an ainterval
a+n & n ( a) = J *
f(x)sin ax dx
a converges uniformly to
&(a)
as
n — — > «>.
Since iPn ( a)
is continuous
everywhere in a , cf. §1, 5, the limiting function & ( a ) , by a well known theorem, is also continuous in this interval. A similar statement is valid for the other integrals. 2. Uniformly convergent integrals can be differentiated and in tegrated under the integral sign with respect to the parameter, in accordance
§1*.
UNIFORM CONVERGENCE
with the following rules [10]: a) XQ < x < X1,
If a function
g(x, x)
is continuous for
a
< x
0, let f(x) be integrable in the intervals (a, c  €) and (c + €, b), and let the sum U— /
u f(x ) dx + I
f(x) dx
approach a limit as € — > 0. We denote this limiting value as the Cauchy principal value of the integral
§5 .
CAUCHY PRINCIPAL VALUE
J
19
b f(x) dx
.
a If f(x) is integrable in the neighborhood of the point c, and hence integrable in the entire interval (a, b), then the Cauchy principal value exists and equals the usual value of the integral. A similar definition is valid if more than one singular point exists, namely points c ..., c^, and also if the integration limit a or b is not finite. A special case of the Cauchy principal value occurs when the point c by chance coincides with the end point a; that is f(x) is integrable for e > o in the interval
(a + e, b),
and b
exists. —
If
f(x)
is integrable in each finite interval, and N
exists, then it is also customary to speak of the Cauchy principal value of the integral
o do not need to converge.
a are of especial interest.
a Here
t
is a point of the interval
(a, b)
and
f(x) is, first of all, integrable. First let t = o , a =  1, b = 1. The integral ^(a) then exists in the usual sense. The existence of 0, ♦ (a) + *"(a) = J 0
d
x
= f
.
The general solution of this differential equation is (2)
&(a)
= — + A sin a + B cos a
where A and B are constants. Since by letting a > o, (2) has the value a > 0 in t 1(a) = A cos a  B sin a,
a) is continuous everywhere, B =  it/2. < Similarly by letting we obtain
dx
Ao '  x2 Now writing for the last integral 2
2
dx 2 ■
x2
CHAPTER I.
22
TRIGONOMETRIC INTEGRALS
there results A = o + 12 log 3J  i2 log % 2

a > 0, a > 0 :
A simple transformation yields finally [15] for f sin ax / p p 0 x(a xfA x y )
n ,. „N ~ ■'inrirp, r 0 ” cos aa), 2a
= o
1
f< / p '
r J0 a
0
 x
p 2a
u
(3 ) x sin ax /
k*
dx =  4 cos aa
These integrals could have been calculated more quickly by
complex integration. In a similar manner, the following formula [16], can be obtained, also for a > 0 and 0 < R(\) < 2 :
C)
f JQ
a
 y
dx a
+ y
5.
J0
— *■ a
 x
For later needs, we shall now prove that the real part of r
converges, as
.
+ x
dy =  i  ax2e_ao:e(x. «/2 ) + e(X. n/2 ) f x
L_ a
dy =  i f ax'2 e(aa) + e U */2 ) f J
J
e~x
a > 0
a
e
> 0,
to the principal value of
Z^dx
.
a The real part of the difference of (7 ) and (6 ) amounts to a
2 r
e ' x  ex
/
(o, a ) ,
And since x~1(e~x  ex ) is bounded in smaller than a constant times 2 f e
1
,
{iF T ?* * '
Its absolute value is
« e
?
•
CHAPTER II REPRESENTATION — AND SUM FORMULAS §6.
[ oo, oo].
A General Representation Formula
1. Let K() and f() be given functions in the interval Under suitable conditions, the integral
(1)
fn(x)=J f(x+i)k(s)
d
exists for
n > n . Letting
n
> »
under the integral sign, we may
expect to obtain (2)
f(x) T
k
J
() d! =
lim n >oo
•
That is
r(x) f Ui) ds
(2 ')
=
J
lim
f
(x+
i ) K(6 )d
We call this relation a representation of the function the kernel
K().
(*0
f(x)
by means of
We can also write for (i)
fn (x ) = n f
(3 ) or
.
f ^x +
'
fn(x)=nJ f()Ktn(x)]d
The relation (2) is to be expected only if f(i) is continuous at x = . But a representation is still possible in the general case where f(§) has the right and left limits f(x + 0) and f(x  0). In this case, the expression (2) is to be replaced by (5 )
f(x + 0)
J
n
K( 6 ) d 5 + f(x  0 )
J
0
23
K() d6 =
lira fn (x) n —>°°
.
CHAPTER II.
2k
If
REPRESENTATION  AND SUM FORMULAS
K() is even, i.e.,
K( £) = K(I),
and if in addition
(6) then it reads (7 )
•2 [f (x + 0) + f (x  0)] =
lim f f ( x + i j K U ) d £ n —>00 o
.
THEOREM 3 [1 7 ]• For the validity of (5 ) at points x for which f(x + o) and f(x  o) exist, one of the two following assumptions is sufficient. (a) f() is hounded, f(l) £ G, and K() is absolutely integrable. (b) f U ) is absolutely integrable, K(i) is absolutely integrable and bounded, and o( U r 1 ) as U I > oo. PROOF. Setting it is evident that f.,(x)
K(§) =
f 1(I ) = f () for  ^ x, and = 0 for I < x, also satisfies assumption (a) or (b). Hence
(5) reads (8)
f(x + 0) / K(5) dj = lim / f ( x + 1 ) K({ ) ^6 o n —>« J0 ' n /
.
Similarly one obtains
0
0
(9 )
Conversely, (5 ) is a consequence of (8) and (9). It Is, therefore, sufficient to prove formulas (8) and (9), and since they are symmetrically constructed, we need concentrate on only one of them. PROOF OF (a).
Then, for
The Integral
fn (x)
We choose (8).
exists for all
n.
Let
A > 0 A
If / K()d 4 °> a constant.
then It can be normalized by multiplying
K() by
§6.
A GENERAL REPRESENTATION FORMULA
25
Since K() is absolutely integrable, the second terra on the right can be made, by a suitable choice of A, smaller than an arbitrary number € > 0. The first term is A < 6(n ) J 0
K(s)d{
,
where 5 (n) denotes the upper limit of  f ( x + t )  f ( x + o )  in the in terval o < t < An”1 . But with A fixed, the length of this interval be comes arbitrarily small with n~1, and by the definition of the limit f(x + o), 5 (n) also becomes arbitrarily small. Therefore for fixed A, the first term is smaller than € for n ^ n(e). Hence cp(n) g € + € = 2e for
n ^ n(e).
Expressedin another cp(n)  > o
PROOF OF (b). nJ
fix
For
way
as
n
> «
c > 0, we write c
+ %)K(ng)d = n /
0
0
+ n / =gn (x) C
+V
x)
*
Since the limit f(x + o ) exists, f(x + ) is bounded in a certain in terval o <  < c. if for this c, one sets the function f(x + ) equal to zero outside of this interval, and momentarily denotes the resulting function by f(x +  ), then gn (x) agrees with the corresponding ex pression (3 )« Now using the proof of (a), we obtain asn > oo gn (x)  > f { x + 0) J
k
i t ) d£
.
o We must still show that hn (x) exists for n > nQ, and that ^n (x )  > 0 as n > oo. For c < £ < , we have ng >nc. Since nc > oo, and K{ 5 ) = o(’) there exists, for
n > nQ,
an
€R
with
en
K(ng) < €n (ng)"1, Therefore to hn (x)
ln/c l < € /cl"”1 f(x + l)dg, follows. 2.
(10)
, > 0,
such that c 1 ,P
(,2 )
[ f ( i ) \ s i n JlL irlL l ]
f^  c) ■ litn L 2 2
n —>“
p J
L
s  X
j
where °p If
p = i,
i.e., if
(1 3 )
K(S) = ^
,
then (12) is the socalled Fourier integral formula, which must be treated separately since it does not come under Theorem 3 , cf. §8. 3.
We shall now prove an important criterion for the case of the
Fejer kernel. Let finite. If we set
f()
be absolutely integrable, and the value of
5
2 f
[f(x + I)  f(x)] (3^ g e) 0 °o gn (x),
for fixed
we introduce the function £
P(5 ) = f
f(x + 5)  f(x) d5
,
0 and assume for the point
x
under consideration, that
.
§7 . DIRICHLET INTEGRAL j F(i) > 0
(1 5 )
as

27
> o
Making use of the inequality sin x
X
l
>
0
o < x < 1 and
(which can he easily proved by considering the two cases 1 < x),
,
one obtains
(16)
U)de
ns I F(8 ) + 2 f (1 +n5 ) 2 66 J
(i+n ) 2
1 F() ■—
1
(i+ m )3
If, In addition, we take (15) into consideration and make use of the relation
0
f J
n £d
we find from (16) that for each (1 7 )
Since
^
f
(1 + n l) 3 ■ J
xdx
_ 2
(1+x)3
€ > 0,
'
there is a
Tim Ig (x ) < € n —>00 €
2
such that
.
can be made arbitrarily small, it follows from ( l b ) and (17) that 8 n(x) + hn (x)  > 0
Hence at each point (18)
x
f(x) =
.
for which (15) is valid, we have lira
f(s)
at
.
But Lebesgue has proved that for any arbitrary (integrable) function (15) is valid for almost all points x [19]* We have, therefore,
f(x),
THEOREM For an absolutely integrable function f(x), the representation (18) exists for almost all points x of the entire interval [.19]. §7. 1.
The Dirichlet Integral and Related Integrals Let
f(x)
be a given function in
(0, »],
and monotonically decreasing. We denote its limit at the end point as is usual, by f(+ 0). For any p, 0 < p < 1, the kernel
which is positive x = 0,
28
CHAPTER II.
REPRESENTATION  AND SUM FORMULAS
does not fall under Theorem 3.
However, by Theorem 1, the integrals
(2)
J
F(n) =
dx ,
f(2)
0 exist for all
n > 0.
(3)
x
We shall now show that the relation
lim F(n) = f(+ 0)
J
3i^ 
0
is also true.
dx ,
n > »
x
Let Kn) * F(n)  f(+ 0)
J 1sinxpx dx 0
x
Then + (n) =
f
A
f dx +
[f(5)f( + o)l
0 L
J
JU
x
= V,(n)
f()sln x dx 
A x + +2 (n)
J Lf(+o)sln
x dx
A x .
By §2, (5)
M
Hence for suitable
n >l < J
+ f "o n Jn
a > 0
.
§T .
29
DIRICHLET INTEGRAL
This formula is valid in particular for f(x) = i, and therefore also for f(x) = c, where c is any arbitrary constant. Further, if it holds for fjCx) and f2 (x), then it also holds for f ^ x ) + f2 (x). The following theorem now follows easily. THEOREM 5 * The relation (5 ) is valid for each func tion of bounded variation. COROLLARY'.
For
0 < a < 2«
.
a (6 )
c =
lira i f f(x)( — L— n >» J \ sin 
 —
1 sin nx dx
This result follows from Theorem 1, since the factor of sin nx tegrable. With the addition of (5) and (6 ), we have [20]
is in
£L
(7 )
f(+ 0) =
3.
lim i f n >«> *
f (x)
0 < a < 2*
dx,
.
sin 
We return to the function
f(x)
specified in (1).
For the
kernel (6)
K(x) =
,
0 < p < 1,
one obtains (9 )
lim n —>00
J
f(2)K(x) dx = f(+ 0) / K(x) dx
We have, for the present, removed the value K(x)
p = 1,
because for this value,
is not integrable in the neighborhood of the point
x = 0.
But it is
indeed possible to remove this point by introducing, for fixed a > 0, the kernel: K(x) = x“1 cos x for x > a and = 0 for x < a. Then (9 ) again holds. To prove this, we proceed just as we did in 1. However for the component (n), a slight change is necessary. It will now read A + ,(11) > J
l(f)  f(+ o)Jk(x) dx
,
and here again, one finds that it becomes arbitrarily small with n”1. If (9) is valid for the kernels K ^ x ) and K 2 (x), then it will also be valid for K = K 1 + Kg. Bringing in Theorem 3, assumption (a), one finds that (9) is valid for each kernel, which, as x  > 00, can be written as
CHAPTER II.
30
REPRESENTATION  AND SUM FORMULAS
K ( x )
(10)
=
a
^
+
b
£ i n x
where
p > o, q > 0
and
H(x)
+
H (x)
^
X1
XP
Is absolutely integrable as
x > oo,
and instead of requiring that f(x) be positive and monotonically de creasing, it is possible to allow f(x) to be a function of bounded vari ation in
(o,
oo].
For example, one may allow
integrable derivative in
which, as
[o,
f(x)
to have an absolutely
oo].
In particular, each function KQ (x) has the property (io), x > oo, admits an asymptotic expansion
with p > o and q > o (as for example the Besel function Jv (x) for .9l(v)> — i). By this is meant that for each m > 0, the difference
has the order of magnitude Hence
K 0(x)
0(x”m “1 ) as
> 0
x > oo.
as x  > oo,
and the integral
(12)
x exists, and is a function of the same character as recognized without difficulty by the formula
K 0(x).
This fact is
(r > 0, a 4 °)
1 e(ax) d5 = ' 15
x One can thus repeat the process (12), resulting in functions 2
(13) X
all of which are similar in character to THEOREM 6. integral
For
X = 1, 2, 3 ,
F(n) =
J 0
K Q(x). •••>
we consider the
f(i)xXK 0 (x) dx
§8 . FOURIER INTEGRAL FORMULA
31
In order that F(n) exist for n > o and approach a finite limit as n > oo, it is sufficient that 1. f(x) be xtimes differentiable in (o, oo], and f^^(x) be continuous in (o, oo], and that 2. each of the functions g(x) = x*T(x), g l(x), g"(x), ..., g^^(x), have an absolutely in tegrable derivative in [0, oo]. This limit has the value x l \ + ^ (°)f (+ °)
PROOF.
•
By assumption 1, it easily follows that
g( o) = g *(o) = ...  g (x"1 }(o) = 0, By assumption 2, it follows that the functions are bounded as x — > oo. Writing
g (A,)(0) = x!f (+ 0)
.
g(x), g'(x), ..., g^x \ x )
F(n) = nX f g(~)K0(x) ax , '"0 it follows that the integral F(n) exists. Hence one can integrate it partially xtimes one after another, and obtain F(n) =
J
g (X)(j)Kx(x) dx
•
0 By (9 ), we have lim F n>»
§8.
= g (X)(o) f K (x) dx ,
Q.E.D.
The Fourier Integral Formula
i. This formula reads (1 )
i [f(x + 0) + f(x  0)] = lim 1 f f (x + 5 )  ILI ^ n —>oo 0 s
It will'suffice to discuss the part formula f(x
+
0)
=
lim  f f(x n >» * J
+ I)
s ln 
n dj
4
.
CHAPTER II.
32
REPRESENTATION  AND SUM FORMULAS
Let us consider, for fixed *(n)
 I /
f(x
x
and variable
+ I)
n,
ds
the integral
.
0
For arbitrary
*
»
=
!
a > 0,
/
decompose it into the two sums
f(x
+ S) ^ i p t
d S,
*2 (n)
slnns
o
d
.
a
We can make use of §2 to evaluate $2 (n). By Theorem 1, $2 (n) exists for n > o and converges to zero as n — — > oo, provided the function (3 ) is either absolutely integrable or convergesmonotonically to zero as I — ■> oo. By the valuation (A > x + 1 )
/A
f(x+)
d =
/
A+x
f(£) I
5  x d ^ A A xJf
f(s)
dl
+1
one observes that the first condition mentioned above can be modified to one requiring that the function fU)
(*0 be absolutely integrable as
_ >
dependent of the considered point
oo.
x,
In this form, the condition is in and refers only to the infinite be
havior of (t). It is also possible to modify the above second condition by requiring the function (A) to approach zero monotonically as  _ > oo. In this case, the function (3 ) itself need not be monotonic, but one recognizes by the decomposition f (x+ ) = f (x+g ) + x f (x+ ) i
x + 
x + 
that it is the sum of two functions, each of which approaches zero mono tonically as   > oo. By Theorem 1 (a), it is even possible, as is easily verified, to generalize the second condition by requiring that the function (t) should be representable in the form () sin (p + q), where
g()
approaches zero monotonically as
p > 0   > oo.
But in this case
§8. the integral
®2 (n)
33
FOURIER INTEGRAL FORMULA
need not exist for
We must still state
n = + p.
conditions under
which
therelation
lim « 1(n) = f(x + o) n —>°o holds.
In §7 > we became acquainted with the most important of these con
ditions, namely that f(x +  ) be of bounded variation in the interval o <  < a. We shall not, however, enter into a discussion of still other conditions which are deduced in the theory of Fourier series. Summarizing, we have the following. THEOREM 7 [21]. If the function f() is of bounded variation in the neighborhood of trhe point x, (5)
then the formula
i [f(x + o) + f(x  o)] =
^
lim 1 f f(e) — n _>oo K u s " *
dg
is valid,provided the function f() not only as Z — — > 00, but also as £ >  °°, fulfills one of the following conditions: a ) it is absolutely integrable 8) it is monotonically convergent to zero, or more generally, is representable in the form g(l) sin(p£ + q), where g U ) converges mono tonically to zero. REMARK. It is obviously possible to generalize condition 8) by requiring that the function ( k ) or g(£) be representable as a linear combination of functions converging monotonically to zero. This repre sentation is possible, for example, if the function in question converges to zero as  > «>, and has an absolutely integrable derivative. A kind of special case of condition 8) is therefore the following: 7)
it converges to zero and has an absolutely in
tegrable derivative, or the function these properties.
g()
has
Hardy [ 22 ] has shown that the requirement that (U) have an abso lutely Integrable derivative as £ > «> is equivalent to the require ment that l ^ f 1 (£) be absolutely integrable as £  >
1
3^
CHAPTER II. 1
JtJ
2•
f(x) =
REPRESENTATION  ANDSUM FORMULAS
f f{i)
3 . f(x) =
n —
.
slnx X
Then by §4 , A, for
fU ) M p i
e
I f t d )
d
g “ X
cos x. i f
As
slnnl^xi
for
ds = 1 =
x > o
.
f(x)
_
n >1 f(O) .
and = o
for
x < o.
l l p l d = 1 arc tan  ,
Then by §k,
k>0
> oo, the right side actually has the limit ~
.
[f (+ 0) + f ( o)]«
h. By our theorem, for example, the lim 1 f L 3% i l l. 2L> de, n —\ 5  X  > oooo X J t P
o < p < 1
n
exists for
x ^ o and
= x“^
for
x > o,
and
= o for
x < o.
2 • THEOREM 8. If f(x) has the same infinite be havior as in Theorem 1 , and if, for almost all x in the interval (a, b), the expression i f
(6)
f (n
dt
is convergent as n — ~> oo, then the limit function is, for almost all x in (a, b), identical with f (x). PROOF.
We decompose (6) into the terms b b
If in (6) f(x) is replaced by zero in (a, b), Bg and B^ results. Hence by Theorem 7 * B2 + B^ converges to zero for all x in [a, b]< Therefore, by the hypothesis, the expression
*(n, X ) = 1 /
is convergent for almost all
x
in
f({)
(a, b)
dE
as
n — ~> oo.
it remains to
35
§9 • WIENER FORMULA show that it converges to
f(x).
We form 2n
x) =
cp(v, x) dv
.
o Since we can interchange the order of integration with respect to i (Appendix 1, 1 0 ), we obtain
»(.. »>■ If
«>,
then ^(n) = ~
J
n cp(v ) dv
0
approaches the same limit; as per a general theorem (Appendix 17). this reason lira cp(n, x) = 11m >Kn, x), for almost all
x
In
almost all x in x in (a, b).
(a, b).
(a, .b);
>00 r
n
But by Theorem h,
hence also
For
lim \r(n, x) = f(x)
lim 0.
Assuming that the "mean value" of the func x
3»{f} =
(1 )
exists (andis (2)
lim i f x >00 0
f( ) d£
finite),the Wiener formula then reads
f
lim [ f(£)K(x) dx = mi f) K(x) dx J n —'v 0 J LL
0
0
.
CHAPTER II.
36
REPRESENTATION  AND SUM FORMULAS
THEOREM 9 « For the validity of (2), the following assumptions are sufficient: 1)
that
K(x)
be differentiable in
and that there be a constant x K(x)  < H
(3 ) 2)
H
(0, «>],
such that
for
1 < x < 00
that there be a constant
G
.
such that
x ™ jf 0
(*0
f (I)  d£ < G
for
0 < x < 00
PROOF. Subtracting from the given function f(x) its mean value there results a new function which again satisfies 2, and whose mean value vanishes, i.e., x .im I f lira V > x™ x 0
(5 )
f (1 ) d = 0
Since formula (2) holds for each constant function
f(x),
and is "additive
it is sufficient to prove our theorem for functions satisfying (5 ). ever, two preliminary remarks are required for the actual proof. a)
Introducing the function .A
(6 )
®(x) =
J
f(! ) d
,
0 and taking (A) Into consideration, we obtain for
f
If (x)
dx  f ^
d* , ±(B) . H A ) + 2 f x^
B
A
*(x) dx
A
X
Hence also
(7 )
i
By (3), we find for
A > 1,
J
( 8)
3)
If (x)
with
3G
x = n£
f(I)K(x) dx
Introducing the function
0 < A
^
f
0
J
and(5 ), we obtain
®n(t) £ Gt n
and
lim . tSitil = o n >o L
Also, (5) Implies as follows. To each a > 0 and to determine an nQ > o such that foro < n < nQ, ®n (t) < T)t
.
n > 0, it is possible and for t > a
.
We are now ready for the proof itself.
Because of (5 ) we need
show that
J f(I)K(x)
(10)
dx
0 converges
to zero asn
> o.
Let an
e > o be given.Decomposing
f
f
the
integral (10) into the terms
and
o
A
we determine, by reason of (8), a fixed
A k 1
such that
/ We now set A (11)
A
J o
=
J
A K(x) d*n (x) = 4>n (A)K(A) 
®n (x)Kf(x) dx
.
o
o
By (9 ), there is an
J
n1
such that for
o < n < n1
®n (A)K(A) < e . • There still remains the right side integral in (11 ). a A
/ * /
■
We decompose it into
38
CHAPTER II.
REPRESENTATION  AND SIM FORMULAS
On the one hand, by (9 )
f
£ Ga
which can be made smaller than hand, for this a and for
o
K' (x) dx
,
e
for a suitable fixed
f
xK'(x) dx
a.
On the other
1 T] = €
we determine an
nQ
in accordance with observation H.
rt.
J
Hence for
< nJ
0 < n < minimum
J
3 . Then
xk!(x) dx i
(n , n 1)
f()K(x) dx
o
nx2’
.
From this, one easily finds the following:
2.
If
f(x)
is a given function in
[ oo, oo],
then by its
"mean value", we shall understand the limit A
(1 2 )
an{f) =
insofar as this limit exists.
If
lim
~
f(x)
J
f() ds
has such a mean value, and is
bounded, say, then (13)
lim n —>o
 f f(x) Slnnx dx = nJ nx
_
§10. §10.
POISSON SUMMATION FORMULA
39
The Poisson Summation Formula [2^]
1.
Let the function
f(x)
be defined for all
x.
We form the
transformation1 (1)
„ ( « )  / f(x)e( 2 itax) dx
Then the Poisson formula reads (here
a
+ 00
(а)
and
k
are integers)
+00
= X fCk) '
y
a= oo
k= oo
Before proving (2) under suitable assumptions for
f(x),
we shall formally
deduce from it certain other formulas. Until further notice, will be any two positive numbers for which
X
and
m
Xu = i. Replacing
f(x)
byf(^)>
thereresults
from
+ 00
+00
(3)
(aX) = J f(tji
We consider the function Hence
+ x )e(2 jtaxx) dx = e( 2*at ).
ing over the whole Interval
[
Analogously, we call an integral extend
«,
convergent if its Cauchy principal
oo],
value exists, cf. §5, i. Hence for integers
(li)
T / /s
a
and Integers
jr 1
f (x )e(2 nax) dx =
pi/2
/ 2 /
1/2
J f)
p > o,
\ f(x + k)Je( 2 jtax) dx
\ p
.
'
If the series + 00
(15)
f(X + k) —00
converges uniformly in the interval  ^ < x < 1, call g(x) the limit function, then the expression on the right of (it) is convergent as p  > oo. Hence the expression on the left is also, i.e., the integral q>(a)
exists and cp(a) =
r /2 j g(x)e(2nax) dx
°l/2 From this it follows that
t "'“>• //a n
1/2
“Mr.;1"
•
CHAPTER II.
REPRESENTATION  AND SUM FORMULAS
If furthermore g(x) is of bounded variation, then the right side con verges to g( o ) as n — > ooj cf 8 §7 , 2. Hence the left side also con verges, and If one now inserts the value of g(o), (2) results. Applying a formal transformation stated in 1, we obtain THEOREM 1Oo For the validity of formula (A), it Is sufficient that the series +00
(16)
2^ f (x + tn + k(i ) k= 00 converge uniformly in
(17)
.1
. For k ^ s, and x ^  1,
we then have k
f(x
+ k)
0
p+r f(x + k) < p
J
f(e) d
.
k2
Therefore the series
X converges uniformly in
+
 1 < x < °°. Moreover, since all terms of the sura
are monotonically decreasing, the function representing the sum is also monotonically decreasing. b)
Let the function
f(x)
be differentiable in the Interval
§10. i
 1 g x < oo,
POISSON SUMMATION FORMULA
^3
and let the integrals
f
(1 9 )
f(!) d 5
f
and
f U) d
J01 be finite.
Hence
m
m+1
Y f (x + k) 
f
P
P
If we now set, for
m
1
f
f (x + ) d P o < 
S(a)sin xada
(7b)
o By substitution, there results (8 )
f (x) = 
J
0
da cos xa / « i )cos ad£ 0
or f (x) =  J
(9 )
da sin xa o
J
f ( )sin ad£
o
Let X and n be any two numbers whose product is 2 /it. Multiplication of C (a) by X n /2 , transforms (6 ) into the symmetrical pair of formulas (1 0 )
ty(a) = X Jf()cos
ad&,
f(x) = 41
J
\r(a)cos xada
Similarly one obtains from (7 ) (11)
X(a ) = X J
f()sinad£,
f(x) = n J
x(a)sinxada
In what follows, we shall denote the passage from (6a) to (6b) as an in” version of (6a), and shall call the second the inverse (or also the con verse C323 ) of the first; similarly in the case of (7 )« By Theorem 11, there follows immediately THEOREM 11a.
The inversion of (6a) and (7a) is
admissible provided the function f(), defined in [0, 00], satisfies 1 ) or 2 ) of Theorem 11 as I  > 00, and is of bounded variation in the neighborhood of x .
CHAPTER III.
50
THE FOURIER INTEGRAL THEOREM
By inversion of §2 , (it), we obtain for example T cos xa? da = ^ 7t ekx , / gd r CK 0 k + a~
/,o\ (1 2 )
If
f (x) =
for
a >
then by §A, (6 ),
f— a 3ij:1 x“ da *= 5 e"kx / 2 2 2 JQ k + a
C(a) = 1
0 < a < 1
for
and
= 0
1; hence actually 1 f 0 (a) cos axda = f cos axda = f(x)
3 • If f(x) is given in O oo, oo], the pair of relations (5) can also be interpreted as a pair of inversions, after introducing "complex” notation at any rate.
Consider the transform of
(1 3 )
E(a) =
[ f ( )e(~ a t ) d
f(x) ,
Then, cf. the proof of Theorem 11, n J
a
E(a)e(xa) da + a
f
n
E(a)e(xa) da = f
n
[E(a)e(xa) + E( a )e (~xa)] da
a n = 7 jT da J
f ()cos a(  x) d
a
and the following theorem ensues. THEOREM 11b. If f U ) satisfies 1) or 2) of Theorem 11 not only as  — — > oo, but also as I — ~~>  00, then (13) may be converted to (1A)
f(x) = jf E(a)e(xa) da This converse holds for points x in whose neighborhood f(x) is of bounded variation pro vided one interprets the integral (1A ) in a suitable manner as a Cauchy principalvalue. A.
Of interest is
§12.
TRIGONOMETRIC INTEGRALS WITH
e"x
51
THEOREM 1 2 . If f() behaves at infinity as specified (especially if f() is absolutely in tegrable) and if the integral
J
E(a)e(ax) da
converges for almost all
x
in an interval
(a, b) — perhaps as a Cauchy principalvalue — , then the limit function Is, for almost all x in (a, b) identical with f(x). PROOF.
By the assumed property at infinity, we have n n
Now apply Theorem 8 . §12. The formulas
Trigonometric Integrals with [a > 0 , 0 < ^ < 1]
(1)
which can be consolidated in one as
originated with Euler [3^].
Here
r(n)
denotes the Euler gamma function
(3 ) 0 The formulas of Fresnel are important special cases of (1),
We shall discuss these separately later on. [k > 0,  00 < a 0
and then add, we obtain
ekxe( ax)cos Px dx = g _■ p ’
+ a 'T p ’ Ik
'
0 Integration with respect to
p
between
o
and
p
gives, for
0 < p < a,
§12.
TRIGONOMETRIC INTEGRALS WITH
e"x
53
where the imaginary part of the logarithm is to be taken between  n/2 and «/2 . Letting k  > 0 , and taking the real part we obtain for o < a
0 , b > o]
gives
, . e dx ■ loe a T T E
•
Hence follows [37] r ^ax
J—
(n )
(12)
p / J
“bx f, 2 ,  ^ cos ax dx = log J
 ehx
2
,
sin ax dx = arc tan —  arc tan — CL OL
X
.
0 For positive numbers
a, x
\x,
and
we have by (3)
r(n) (x + a)_w = J
(13)
dz
.
0 _kx Multiply by e e( ax), k > 0, and integrate with respect to x between 0 and oo. One can interchange the order of integration in the repeated right integral to obtain
(U)
r(n)
f £—Sls^l1 dx
*•7
((x+a)^ x+a
f
= r e ^ z 11" 1 az JQ z + k + ia
If a 4 0? one can even consider the case where one obtains [a > 0, n > 0, k ^ 0, a 4
r(“ 1
f e f r * a 
The Fresnel integral
J
> 0 . Altogether,
k
j
f
° “} ag
■
CHAPTER II! e THE FOURIER INTEGRAL THEOREM can be evaluated "directly" by making use of § 1 1 , (1 1 ) with Setting f(x) = x~1 we obtain (1 7 )
X(a) = /  V *J
Hence also
f(a) = x(a )J,
d =
X (i ) = f(a)J
k/T
X = n
[38 3 
.
^
and therefore X(a) (1  J2 ) = 0
(18) By the decomposition
( 2 k + 1
0
.
one recognizes that Hence (1  J2 ) = 0
k=0
) jt
2kn
J > 0 , and therefore because of (17), follows from (18), and because J > 0 ,
x(a) 4 0o it follows that
J = 1 . One can treat the cosine integral in a similar manner, only here the proof that J > 0 is somewhat more complicated. The proof rests on 1 /2 the fact that the function x 1 Is its one cosine — or sinetransform. Many other such "self reciprocal" functions exist [3 9 ] • §1 3 . The Absolutely Integrable Functions. Their Faltung and Their Summation" 1 . We shall now consider all those functions f(x) which are de fined and are absolutely integrable in [•«, oo] . We denote the totality of such functions by 3 Q. Since we require of a function of %Q only integrability In the infinite region, we shall disregard its behavior on a null set.
In particular, we shall regard two functions of
30
as Identical, if
the functions have the same value almost everywhere in [ », °o]. If f (x) Is a function of 3 Q, then the following also belong to 3 Q: f ( x), T ( x ), g(x)f(x) where g(x) is bounded (integrable in the fifiite region), in particular the function e(\x)f(x) where x is real, and In addition the function f(x + X ). Moreover the class 3 Q is linear. 2 . We call a collection of functions linear if cjf^ + c2fg is an element of the collection, where c^ and c2 are any two (complex) constants, and f^ and f2 are elements of the collection. In particular, for each constant c, cf is also an element of the collection, where f is an element of the collection. Moreover, 3 0 is closed in thefollowing sense. If a sequence of functions f (x), n = 1, 2, 3 , of 3 0 is convergent In the
§13
ABSOLUTELY INTEGRABLE FUNCTIONS
55
Integrated mean, I.e., if (1 )
lim
f f (x)  fn (x) dx = o
,
IB —> » d
n ~>oo then there is exactly one function
f(x) of
%Q
towards which it converges
in the sense that (2 )
lim f  f (x )  f (x) dx = o n >oo d
cf. Appendix 1 1 . 3 * Let
f1
and
f2 be two given functions of
$ Q. For
brevity, we set
J
(3 )
If
) dx = Ci,
i = i, 2
We now make full use of Fubini1s theorem, cf. Appendix 1, 10. (1 )
.
The function
g(x, y ) = fj (y)f2 (x  y)
is a measurable function of the variables f 1 (y)
which
f
(x, y).
For each point
y
in
is finite
g(x, y ) dx = f,(y) f
f2 (x  y ) dy  f,(y) • C£
.
Therefore
f
dy
f
g(x, y ) dx = C
ZJ
f, (y) dy = C^Cg .
Hence thefunctiong(x, y) is absolutely integrable over the plane. Therefore the integral
(x, y)
f g(x, y) dy =jT f/yjfgfx  y) dy
(5 )
exists foralmost all x, and is again a function of Is independent of the order off 1 and f2 because J
(6 )
f 1 (y )f2 (x  y) dy = J
If for any two functions in all
whole
x,
then the function
[oo, oo],
f2 (n)f1(x  n ) drj
3 fQ.
Moreover
it
.
the integral (6 )exists
for almost
56
CHAPTER IIIo
THE FOURIER INTEGRAL THEOREM
~h f f i(y)f2(x " y} dy
(7 )
is called the Faltung of f 1 and f2 [to] . If f 1 (x) and f2 (x) are functions of then their Faltung exists, and likewise belongs to g To each function of
we attach its transform
E(a)
(8) and denote the totality of these functions a function
E(a)
belong to
7 0,
E(a) by
3 Q.
In order that
it is necessary that it be bounded
(9 )
that it be continuous (Theorem 2 ), and that it approach zero as (Theorem i). c1E 1 + c2E2 »
a —
> +
The class 7 Q is moreover linear: to c1f 1 + c2f2 belong If E(a) belongs to 7sQ, then the following do also:
E(a), E(aJ, E(a + X ) and e(Xa )E(a), X real. They are the transforms of f (x), f'Cx)', e (“Xx )f (x) and f (x + x ). If functions of 3 Q con verge in the integrated mean, then owing to •2 ttE (a )  En (a)  < f f (x)  fn (x)  • e(ax) dx < [ f(x)  fQ (x)  dx
their transforms are uniformly convergent in
,
 » < a < 00 J
can also write for (18)
[f(x + 0) + f (x  0)] =
P
P
lim 1 cp(~) da f() cos a(x n —>00 d J
 ) d
Letting now n — > °o under the integral sign, formula §11, (1) results; but this extension of the limit is inadmissible because for the validity of the last formula, bounded variation in the neighborhood of x was assumed. However, we may view the situation in the following manner. Formula §11, (1) can still be maintained for absolutely .integrable functions
§13f(x)
for points
ABSOLUTELY INTEGRABLE FUNCTIONS
x
at which the requirement of bounded variation is not
fulfilled, provided that ordinary convergence of the integral be replaced by "summability" with the aid of a convergence producing factor 9(a)* The factor can be of very general nature; in the two special cases cp(a) = e~^a l and Integral [t2 ]. 7.
cp(a) = e~a2,
the integral (19) is called a Sommerfeld
Theorem 13 can be employed for the calculation of definite
integrals. THEOREM 15. Let f 1 (x) and f2 (x) be given functions of 3fQ. In order that the relation (20)
2jt hold
f
E 1 (a)E2 (a)e(ax) da =
for all
x,
J
f 1 (y)f2 (x  y) dy
it is sufficient that one of the
following conditions be fulfilled: 1) The function E 1 (a)E2 (a) be absolutely integrable and the right side of (20) represent a continuous function. 2)
One of the functions
f 1 (x) and
f2 (x)
have an absolutely integrable derivative. PROOF of 1). This half of the theorem has already been proved and used in 6 .We repeat the proof briefly. By Theorem 1 2 , (20) holds for almost all x. Since the functions on both sides ofit arecontinu ous, it holds for all x without exception. PROOF of 2). Because of Theorem 11b, it is sufficient to prove that the right side of (20) is differentiable. We may assume, from the hypothesis, that f2 (x) Is the function which has an absolutely In tegrable derivative. We set 2 itcp(x) = J ' f 1 (y)f^(x  y) dy and form, for an arbitrary x 2* f a
a,
the integral x
cpU ) d! = f a
J
f 1 (y )f^(£  y) dy
.
One can interchange the order of Integration on the right, to obtain
CHAPTER III.
6o
f
THE FOURIER INTEGRAL THEOREM
f, dyJ fg(5 d£=J f, (y)
y)
(y)fg(x 
y)dyJ f,(y)f2(ay)dy
a = 2rtf (x )  2 rtf(a ) Hence the Faltung If
f (x) is the integral of a function
f 1(x) and
also vanishes; for
f2 (x) both vanish for
x > 0,
cp(x),
x < 0,
Q.E.D.
then its Faltung
it has the value x
(si)
/
V y ) fV x  y) ay
.
0
The first part of the above assertion is also valid if more functions are faltet one after the other. More generally, if for v = 1, 2, ..., n, f (x) = 0 for x < x , then the Faltung = o for x < x 1 + x2 + ... + x . A similar remark is valid if the ""
sign.
From
J
3lngl e(cr) d = 0
for
p
a  >
> 0
,
one obtains
(2 2 )
r
e(ot ) dl
= o
for
+ Pr
+ P.
and in particular, cf. §t, k n (23 ) ^.3l. U.,.P. e(a£ ) d = 0
provided one faltet together the functions to the n functions
for
a  > np > 0
f(x)
of
R0
corresponding
sinpa of
s 0. By §12, 6,
we obtain [t3 ],
for
91 ( k ) > 0 , py > o, v = 1,
§13
( 2k )
ABSOLUTELY INTEGRABLE FUNCTIONS
61
' f  e C o x M o  . 2n J u, in
o
,
'
(k1+ia) 1 ... (kn+ia) n
and
/ s 1 (2 5 ) ±
e ( a x ) da e P Jr . —  J (k,+ia) '(kg+i a) 2
for and
“ (kpk )y p 1 n 1 2 1 y 1 (xy) dy
r ( u 1 ) r(n2 ) 0
x > o. By §12, (6 ) and §12, (7), we obtain, for x > 0
^
a£ „ _  H r e2kV r2 ( n ) i
^ (k +a
For
e
11=1,
1(y  x)^1 dy
0, p. > 0,
[ W
we obtain the already known formula of §11, (12),
/07)
1
1
27
^
J
T e(ax) da _ 1 ^
+ &2
'
27
x
.
T 003 a x
A
.
dt(£)>
0,
00 < x
p > 0,
x
r
e r(2 pri)____ (2x ) ______ yV _rj___________
* J0
r (r+ i )r((xr) 1
2 2 ^ " 1r ( n )
by computing the integral on the right of (26), for x > 0. If for x < 0 , the function (27) is faltet ntimes one after another with the function of §12, (6) for various values of k and p [ b ^ ] f we obtain [ 9t(kv ) > 0, ) > 0, ny > 0, x < 0] n
1 f ___________ e(q*) dQ!
( 2 9)
2«J
2
2
{ i +a )(k+ia)
...(kn+ia)
By §12 (6), and §12, (7 ), we obtain for
(30)
’
f .
(k+ia )p (kia ) If p + a > 1,
one can allow
 e
x > 0
22
u K
1.
1 (i+k^)
[t6]
= _ A i _ [ e2kyy pi (y _ X)a1 dy r(p)r(a)^ x  > 0,
and obtain
.
62
CHAPTER III.
THE FOURIER INTEGRAL THEOREM
d a ________ = 2jt
r (p+a1 )
r(P )r( a )(2k)p+0_1
(k+ia)p(kia)°
By the transformation a = k tan x and the substitution and a  p = q, we obtain (p > 0, arbitrary q) jt/2
f
a + p  i = p
*/2 oosP'U
f
cos qx dx
' jt/2
co b^
x
e(qx) dx 1
ir/2
Prom this, we obtain by inversion, for ,
i,f 9.
\
,
x
p(p )e(xa) ‘(JLLLi*)r (£±li£)
2
' ^
2
'
p > i cosP*"1a,
i“ i < I i« i
>
The following remark will be of use later on.
If the functions f U ) and E(a) are interchanged in Theorem and small changes are madein normalization, the following results. If the function E(a) is absolutely Integrable, and if the function
12,
f(x) = f e(xa)E(a) da o'
belongs to
0?0
(i.e., it is absolutely integrable), then the function f e( ax)f(x) dx
agrees with E(a) for almost all continuous,, then for all a E(a ) = and
E(a),
as the transform of
a.
If, in particular,
J e( ax)f(x)
dx
E(a)
is also
,
f (x) is afunction
of
3 Q.
Hence each function y (a) defined In [*«>, »], which is two times differentiable, and which together with Its both derivatives Is ab solutely integrable, is contained in the class 3 0 • In particular, there fore, each function y ( a ) is In this classwhich is two times differentiable and vanishes outside of a finite interval. Since for the function (3 2 )
K7 (x) = jT e(xa)7(a) da
we obtain, by a n ,appropriate application of
§3*
, the valuation
§U.
INTEGRALS WITH RATIONAL FUNCTIONS K r (x) = o(x2 )
it follows that
K^(x)
is a function of 7 («) = ^
If
7(a)
has
r
J
63
.
S Q. We note again that
e( a x ) K y (x) dx
.
absolutely integrable derivatives,
r^
2,
then
by §3 > ^ (3 3 )
K 7 (x) = o(xr ) If for
r k 1,
(3*0
.
the functions
ctp7 (a),
p = 0, i, ..., r,
are absolutely integrable, then by application of §U, 2, (c) to the integral (32), we obtain the result without assuming differentiability that the function K^(x) is rtimes continuously differentiable, and (3 5 )
K^(x) =
J
e(xa)(ia)p7(a) da,
p = 0, 1, ..., r
.
If in addition, the first two derivatives of the function (3*0 are absolute ly integrable (which happens, for example, if 7(a) vanishes outside of a finite interval), then by the above, the functions (35) are likewise ab solutely integrable. §11+.
Trigonometric Integrals with Rational Functions
1. In order that a rational function come under § 11, it is nec essary and sufficient that the degree of the numerator be smaller than the degree of the denominator, and that no real poles exist. — By §12, (6) and §12, (7 )> the transform
m
1
2i j
(1 }
is known for any complex number (2) and for (3 )
0
for
a > 0
fxe ( aIkx ) dx, k. and
It is, for ieko!
9t(k) > 0,
for
a < 0
for
a < 0
9i(k) < 0,  ±e*a
for
a > 0
and
0
,
CHAPTER III.
64
THE FOURIER INTEGRAL THEOREM
By Faltung, or more simply by differentiation with respect to k, one can compute from this the transform of (x  Ik)~n, n = 1, 2, . By a partial fraction decomposition, one can obtain from this, in principle, the transform of every rational function. For example, one thus obtains for a > o, formula §11, (12 ), but this time more generally for 9t(k) > 0. By integration with respect to results
a,
of the first expression therein, there
2. The partial fraction decomposition is burdensome. It is possible in certain special cases to proceed differently® For example, setting k = re(cp),  ~ < cp < ~, in §11, (12), and separating reals and imaginaries, we obtain
jt
JT
i0 x 4 1
cos ax dx " p 2 4 + 2r x cos 2cp + r
2 x cos ax dx T~ 2” 2 T x + 2r x cos 2cp + r
In particular, if
i3. ©^ra cos cp sin(cp— + ra— sin cp) — 2r sin 2  ra sin cp) 2r sin 2cp
cp = j~, [a > 0, r > 0] ra '
By differentiation with respect to
41
a,
COS
it follows that
Numerous other integrals can be evaluated In this manner [4 7 ].
If
f(x)
3. is rational,
The residue theory gives the most practical method to follow. z = x + yi, then
has the value 2nizR+,  2 jtIzR
for
a < 0
for
a > 0.
§U. Here
ZR
INTEGRALS WITH RATIONAL FUNCTIONS
65
is the sum of the residues of the function f (z )e(az)
for those poles which lie in the half plane y > o, and those poles which lie in the lower half plane. If f(x) ~ J
£R~ the sum for is even, then a > 0,
f(x) cos ax dx = izR+ =  izR~1, o
and if
f(x)
is odd, then —
f
f (x) sin ax dx = LR+ = LR“,
a >. 0 .
"b
o.(x)
For example, let us select the simplest case; f(z) = z If o(x) > 0 ,and if x is replaced by a, a by  x,
4 o.
and we
obtain \ J
and if
r e(ax) , a  \
$(*.)< o,
for for
x < 0 x > o,
we obtain similarly
J
(6)
_ f 0 ^nieCxx)
r e(ax) .
f 2 itie(\x) a~C“T da = ( o
for x < for x >
o 0 .
These results are in agreement with the value (2) or (3 ) of 1. From this we shall make an application which will be useful to us later. Consider any function f(x) of S 0 and denote it S'transform by cp(a). Since the functions on the right of (5) and (6 ) likewise belong to S Q , there is, for ts(i) i 0, a function of 3 0 which we denote by g(x), whose transform agrees with 17>
l7)
cp(a ) ilak)
•
It has the value x e(\x )
J
e (
Xx1 )f (x1 ) dx1,
or
 e U x ) ' j ' e( Xx1 )f (x1 ) dx1 , x
according as Is(x) > 0
or
oU) < 0
.
This process can be repeated. For each integer p > 0, tion of g 0 whose transform agrees with
there is a func
CHAPTER III.
66
THE FOURIER INTEGRAL THEOREM 1
J
V i e(~ Xxp )f (xp ) dxp
,
or
(n) (1)vJ dx1J dx2...J dXp_1J" e(\xp)f(xp)dxp , x according as i*.
xi
xp2
3 (x) > o
or
For k > o,
0 1
§(x) < o.
a partial fraction decomposition yields
 kx
(1 3 )
xpl
.
*
„y
r
dx = / y ^ T 0
dy  / y ^  r a dy 0
a result which is also evidently valid for k = € + ai, a > o
and
€ > 0,
^(k) > o.
'
We now set
and write for the first integral on the
right 2a f
J
y
y e“y
,
_
r r
 a + ei dy + J
oy e”y
y  a + ei dy
Separating the real part on both sides of (13 ), and allowing there results for a > o, cf. §5, 5 2 P sin_ax ^ J0 1 + x
= P _ e j _ dy J0 y + a
f _ e _ L dy J y  a
€ — > o,
^
where for the second integral on the right, we have to take the Cauchy principle value. If now we introduce the function
Hi y =
(logarithmic integral of
y),
e6 J V d5 ~ log y
it is possible to write for this [1*8]
then
§ 15 .
f
The function a“vJv (a) is an even function of (11 ), one obtains by Faltung, for the expression
a.
By (2) and
sin p(at) J (a)
/
(9 )
a  t
the value [58]
Div r~ pp
0 v1/2 j cos ty • (1  y )
(1 0)
J (t) dy
or
r(v+i/2 ) J 0 according as 0 < p < i Here 9t(v) >  1/2 and
t
is any real
or number.
[ 31 (p + v) >  1, z any
From the formula
*—
tv
,
1
j t .
Hence in particular [6o] f
(1 3 )
[ 1/2
0] [61]
(HO H (l)(x) = r(l/g.y.>. V rT i n/jf5
0 5 )
Hy1 ^(x) =
H^oo,
).I [, + e( 2 vit)J / (t2  1 )v_1^2e(xt) dt , J 1
. 2W 2 ) : V. ...,eix, t ) ,./2, dfc i n/T r(i /2v ) i (t2i )
The corresponding integrals for and
H
/2 ) (x) are obtained by interchanging
 i. This implies [61]
0 6 )
j(x) = „ ? J x / 2 T l r ... dt v V7 r(i /2v) ^ (t2i )v+1'2
(1 7 )
Y v (x)  f _ c o 3 rt dt v ■ 4/7 r(i /2v) J (t2i)v+1/2
For.the function
Kv(z)=^nie )•f/1^iz) we have
[ 9t (v ) >  1 /2 ] [6 2 ]
i
MRi
k fx)  gVr 0 ,
$
(xyz)
j (a2 + X 2) (b2 + y2 )(c2 + z2 )j
=
Then by § i3 ^ (27) E(a) = T e (a x ) dx P e(ory) dy f e(az) dz _ a0 ,  la I(a+b+c) J a2 + xa J ba + y2 J c2 + z2 “ abc Therefore
* ( _ y) = * L r e i « i ( a +b+C)e ( _ ya) da = abo. abc b y
abc (a+D+c)2 + y2
Hence
fff
p— g— g "^pd'Z P— p =  — { arc tan — — — +x +y +z ) abc I a+b+c
—
\°°
t y ,
(v
= 1, 2 , 3 * •••).>
and that it is uniformly bounded (3 )
(n = 1 , 2 , 3 , ...)•
Vn (a) < M ,
The totality of values of
ry
is of itself also monotonically increasing,
i.e., if < ay, then < Ty. Since the are by assumption every where dense, the xy determine uniquely, as one finds without difficulty, a distribution function V(a) which has the following property. For each a,
V(a + o)
is the lower bound of the numbers
ay is larger than of the numbers xy
t
for such
v
for which
a; and correspondingly V(a  o) is the upper bound for such v for which a y is smaller than a. It is
also evident that ( k)
V(a ) < M
.
We shall show that the relation V( a ) =
(5 )
lim V (a) n —>oo
holds at every point of continuity of a < a x v
V(a).
< Vn K )
Pick an arbitrary
a.
For
;
therefore TIE
V (a)
oo
H H V (a ) = t n >oo
The left side of the inequality is independent of v ; Ty can stand for which a < a y. But since V(a + o) of such t , we have US V (a) g V ( a + 0 ) n >» Similarly, one obtains
.
.
on the right side any is the lower bound
§19.
SEQUENCES OF FUNCTIONS OF
V(a  o) < 1 M  V(a) n —>a>
$
87
.
Hence V(a  o)
o o
n
Tim V (a) < V(a + 0 ) n > o o
a, V(a  0 ) = V(a + o).
But at a continuity point
.
Hence all four terms
of the Inequality are equal to one another, a result which is synonymous with (5 ).
and
V(a),
If we are given distribution functions Vn (a), n = 1, 2 , ..., and if relation (9 ) holds at all points at which V(a) is
continuous, then the sequence Vn (a) is said to converge to V( a ). Since two distribution functions which agree as their common points of con tinuity, also agree at all other points, It follows that the limit func tion of a convergent sequence V ( a ) is unique. THEOREM 2 0 . If the distribution functions Vn (a), n = 1, 2, 3, ..., converge to a distribution func tion
VQ (a),
(6 )
and if lim n —>00
V
(+ 0 0 )
=
V Q (+ o o )
then the functions fn (x) of ^ belonging to them converge at each point x to the function f*0 (x) of
^
PROOF.
which belongs to For fixed
a > 0
a fn (x) = J
VQ (a)
C67 3 
and for
n = 0, 1, 2, ...,
set
a e(xa) dVn (a:) + J
a
+ J
= fn (x, a) + Qn (x, a) + Pn (x, a)
where a is selected in such a way that and a =  a. Therefore (7 )
lim V (a) = Vn (a) , n >00 11 u
Hence as
n — > 00
,
a VQ (a)
Is continuous at
lira Vn (a)  V (a) n —>00
Tim fQ (x)  fn (x) < A(x, a) + B(x, a) + C(x, a)
,
a = + a
88
CHAPTER IV.
STIELTJES INTEGRALS
where A(x,
a) = Tim fQ (x, a) C(x,
 fn (x, a), B(x, a) = H I a) = Tim IQq  Qnl
PQ  PR ,
•
On the one hand, we have a fn (x, a) = e(xa)Vn (a)  e(xa)Vn (a)  ix J a and because of (7 ) and the convergence of
V (a)
e(xa)Vn (a) da
to
V0 (a),
,
it follows
that (cf. Appendix 7 , 8 ) A(x, a) = 0 On the other hand B(x, a)  < and because of
P0 1 + Tim
Pn < VQ (oo)  VQ (a) + Urn [Vn (oo)
for each
,
(6 ) and (7 ), it follows that IB(x, a)  < 2 [VQ (oo)  VQ (a)]
Therefore, for B(x, a)  < e.
 Vn (a)]
.
a given € > o,we canchoose ana so large that A corresponding assertion is valid for C(x, a). Hence
e
TIE  f 0 ( x )  f n (x )l < 2e
•
Prom this it follows that fQ (x) =
lim f (x) n —>00
Q.E.D. 2.
By assumption (6 ), it follows very easily that the
are uniformly bounded (8 )
VR (a) < M ,
(n = 1, 2 , 3, •••)
But one must not replace assumption (6 ) in Theorem 2 0 , by the weaker assumption (8 ), as can be seen from the following counter example: Vn (a) = 0 for a < n and = 1, for a > n. Hence fn (x) = e(nx), VQ (a) = o, and therefore fQ (x) = 0 . Note that in this example the sequence ^n (x ) only fails to converge to fQ (x ) "but it fails to converge at all. Now, this is not an accidental phenomenon, and the following assertion can be made in fact.
V (a)
§19* 3. and if the
SEQUENCES OP FUNCTIONS OF
89
If the Vn (a) are uniformly bounded, and converge to fn (x) converge to a continuous function F(x), then F(x) = fQ (x)
We proceed
$
to prove this statement*
fn (x) §X z 9* L . ~,A. = J
.
By §18, 7, wehave
for fixed
(n = 0 , 1, 2,
e(xa) dWn (a),
p,
...),
where the functions W n (a) =
J
a [Vn (p + p )  Vn (p)] dp
0 are again distribution functions. functions
By ''iteration”, one finds for the
gn (x)  fn (x) ( SS. ^ 1  1 ) 2
,
the relation (9)
gn (x> = / e(xa)En (a) da
,
where the functions En (a) = Wn (a + are absolutely integrable.
" Wn (a)
By (8 ), It follows that, cf. §18, (12), fn (x) < 2M
(10)
.
Hence the functions gn (x) belong to S Q, and since the En (a) are also absolutely integrable, (9) states, cf. §i3 , 9, that En (a) is the trans form ofgn (x)• Because of the convergence of fn (x) to F(x), the con vergence of
gn (x)
to 2
G(x) = P(x) ( — follows. Since by (10), the a > 0 (cf. Appendix i0) a
f
gn (x)
~ ^)
are uniformly bounded, therefore for
lgn (x)  G(x )  dx > 0
.
VQ (a),
90
CHAPTER IV.
STIELTJES INTEGRALS
And because
one also easily finds that
J Hence for the transform
gn (x)  G(x)  dx — > 0 o>( a ) of (a) =
But by the convergence of It follows that
vn (a )
G(x),
we have
lim E (a) n ~>oo to
VQ (a)
and Its uniform boundedness,
lim E (a) n >«> Prom this, we obtain successively: k.
THEOREM 21 [68],
n = i, 2, 3, . (11 )
.
E0 (a ) = ®(a), gQ (x ) = G(x), fQ (x) = P(x).
If the functions
f*n (x)
of
$■,
are uniformly bounded Ifn (x ) < M
and converge, for all x, to a continuous limit function f(x), then the following assertions are valid: 1 ) The function f(x) likewise belongs to . By (11), one can normalize, after §i8, (12), the additive constants in the distribution functions Vn (a) of f*n (x ) such a way that the are also uniformly bounded (12 )
Vn (a) I N
V (a)
.
2 ) There is at least one such normalization in which the Vn (a) a^e convergent, and 3) If in any such normalization the V (a) con verge, then its limit function is equivalent to the distribution function of f(x). PROOF o f .1 ). Let the Vn (a) be given in a normalization (12). We take on the aaxis any countable set of e v e r y w h e r e dense numbers (2 and determine by the well known diagonal argument a subsequence
§19(1 3 )
SEQUENCES OF FUNCTIONS OF
V ^ (a), V ^ a ) ,
w h ich c o n v e r g e s a t a l l t h e s e p o i n t s . tio n
V (a)
By 1 ,
V (a) =
S in c e b y a s s u m p tio n ,
th e s e t
f(x ),
i t f o l l o w s b y th e
V (a)
is
...
th e re i s a d is t r ib u t io n fu n c
f
lim k > «
(x )
k p ro o f in
V
(a )
.
k
as a su b set o f
3 th a t f ( x )
f
(x),
co n verg es to
i s a fu n c tio n o f
SV ,
and
i t s d is t r ib u t io n fu n c tio n . PROOF o f 2 ) .
c o n tin u o u s .
We t a k e a f i x e d p o in t
We c a n add
to th e
v n (a)
V (a and th e new f u n c t i o n s
V (a )
) =
a t w h ich
V (a)
is
V (p ).
aQ a t w h ich
c o n sta n ts
lir a V (a n —>oo
V (a).
c o n v e rg e to
th e
p r o o f o f i ), s p e c i f y a
I f n o t,
Vm M ,
1 ), s i n c e
U (a )
a co n sta n t
c
w ould
...
w h ich w ould c o n v e rg e t o a d i s t r i b u t i o n f u n c t i o n a t th e
v n (£)
su b s e q u e n ce V^ia),
V (p )
We c la im
t h e r e w ould be a p o in t
We c o u ld th e n , u s in g th e argu m en ts em ployed i n
(16 )
v e r g i n g to
is
)
c o n tin u o u s and a t w h ich th e s e t
not
V (a)
a n so t h a t
c o n tin u e t o be u n if o r m ly boun d ed .
t h a t th e y a r e a l s o c o n v e r g e n t to
p o in t
U ( a ) , b u t w it h o u t co n
a = p . A g a in r e f e r r i n g
w ere a d i s t r i b u t i o n f u n c t i o n o f
t o th e p r o o f o f
f ( x ) , th e r e
w ould be
su ch t h a t
(17 )
U (a ) = V (a ) + c
th e r e fo re ,
U (a )
w o u ld be c o n tin u o u s a t
(18 )
a = a ,
U(a ) = lim V (a) ° k > rak 0
By ( 1 5 ) and ( 1 8 ) ,
U (a Q ) = V ( a Q );
T h e r e fo r e tin u o u s a s
c = 0,
I.e.,
a = 0, and
But b e c a u s e o f
a ss u m p tio n t h a t
.
.
U ( a ) = V ( a ) . F u r th e r m o r e , th u s th e se q u e n c e
( 16) i s
U( p) = V( p )
is
th is
th e su b s e q u e n ce ( 1 6 ) d o e s
PROOF o f 3 ) .
and h en ce
on th e o t h e r hand b y ( 1 7 )
U (a 0 ) = V ( a Q ) + o
a = 3.
91
su ch t h a t
( ^b)
a  3
V ^ a ),
‘
%
U( a )
i s a l s o co n
c o n v e r g e n t t o U( 3)
at
a c o n t r a d i c t i o n t o th e
n ot con verg e to
V( 3)
T h is p r o o f f o l l o w s im m e d ia te ly from 3.
a t a = 3.
92
CHAPTER IV. 5 . THEOREM 22.
The product of two functions of
is a function of
cp(x)
STIELTJES INTEGRALS
.
PROOF. Let f(x) and g(x) be any two functions of % . If is some function of $ which has a continuous distribution function,
then by Theorem 19, q>(x)f(x) is also a function of $ with continuous distribution function. Repeated application of Theorem i9 results in cp(x)f (x)g(x) also belonging to
Now by §1 5 , (*0
J ' e(xa ) dV(a)
where V(a) = 2 = / ^ o and therefore the function “
2 x__ 2
e
f(x)g(x)
is contained in . But now consider this function for variable values of n. As n > oo, this set converges to f(x)g(x). It is moreover uni formly bounded; hence by Theorem 21, f(x)g(x) is also a function of §20.
PositiveDefinite Functions
i. We call a functionf(x) positivedefinite [69] if continuous in the finite region, and is bounded in [ », 00], 2 ) "hermitIan”, i.e.,
1 )it is it is
f (~x j = f (x)
(1 )
and 3) it satisfies thefollowing conditions: For any points x 1,xg , ..., xm , (m = 1, 2, 3, ...), and any numbers Pl, p2, ..., Pm m
m
Y Z f(t p=1 v=1
(2 )
2.
Each function of
$
" Sv )pl
* 0
•
is positivedefinite.
Property 1) is
§20.
POSITIVEDEFINITE FUNCTIONS
already known and 2) is easily verified.
93
In regard to 3 ), the left side
of (2 ) has the value J ' $(a) dV(a)
where m
m
' ¥(a) = Z
Z
Ui
Z
e(tXlJ " X vla)pn"^ “
e 0
)pM
(U= 1 v=1
and is therefore actually
^ o.
3 . Conversely, we shall now show thateach positivedefinite function f (x) belongs to $ . We shall first prove this for all positivedefinite functions f(x) which happen to belong to 3 , f(x) ~
in
In addition to f(), [ °o, co]. For each A > o, A
J' e(xa)e(a)
da
let g U ) be a given continuous function the expression
A
J J fix 
y)g( x)g( y) dx dy
A A is nonnegative.
In fact, by thedefinition of theRiemannintegral,
is the limit*
n — > «,
as n 1
it
of
n 1
h2
f(nh  vh)g( iah)g( vh),
(h =~ ) ,
n=n v=n and because of the assumption of the positivedefiniteness of double sum is
^
o.
If further
(3 )
f(),
this
is also absolutely integrable,
gU ~J (xa (a da , byallowing obtain J J fix y)g(x)g(—y)dxdy^o )
then
g U )
A — —> °°,
e
)r
)
we
Denoting by F() the Faltung of the functions f(), all three of which belong to 3 , we have
g()
and
g( 7 ,
CHAPTER IV.
(2 jt)2F U ) =
=
J J f( JJ
STIELTJES INTEGRALS
 x  y )g(x)g( y ) dx dy
f (6 + x  y)g( x)g( y ) dx dy
.
The transform of F() has the value r (a) 2E(a). If r(a) is in addition also absolutely integrable, then since F(§) is a continuous function, we have by Theorem 15 H i)
= / e ( £a ) r(a ) 12E(a) da
,
and in particular F(o) =
J
r(a)2E(a) da
But since, by (3), F(o) ^ 0 , we conclude as follows. The transform E( of f(x) Is such that for each absolutely integrable function r(a) of *0
(«0
J
r(a)2E(a)da^0 .
The transform of f ( x ) is E (a). Since f( x) = f(x), therefore E(a) = E(a), and hence E(a) is real. Moreover, it follows by (4 ) that E(a) > 0 for all x. If It were not so, there would bean Interval aQ > a > a 1 in which E(a) were < 0 . A function 7(a) which is two times differentiable and which vanishes outside of (aQ, a 1) is a func tion of .x .
But for such a function
J
7 (a )12E(a) da < 0
in contradiction to (4 ).  We shall now show that Integrable. Consider the function 2
E(a)
Is absolutely
2n
(‘as;70') a>/ •(■ 2n
Since f() Is bounded, there is a constant A such that Hence, because E(a) ^ 0, it follows for 2 n > a, that
fn (0) < A.
§20.
POSITIVEDEFINITE FUNCTIONS
/ (12nME(a)da^A
,
a
and 'by allowing
n —
> °o,
that a E(a) da < A a
One now replaces
a
by
Q.E.D.
But if the transform
is positive and absolutely integrable, then
E(a)
of f(x)
f(x) is a function of ? ,
cf. §18, k . *4. Let
f (x) be any positivedefinite function, and let
be positive and absolutely integrable. F(x) = f(x)
J
j (a)
Consider the product
e (xa )y (a ) da
F(x) is likewise positivedefinite. In fact, the boundedness and con tinuity of F(x) and the relation F( x) = F(x) are at once verified. Moreover 11
F(x M  X v )p ~p~ = 'p'v
H,iv=1
I I I f(x„ xv)e(X(ia)p^
and since the cornered bracket is fore also
L °*
y o,
e T T a jr ^
7(a) da
the whole expression is there
In particular, the functions x2
fn (x) = f(x)e belong to the functions definite, functions to f (x) tion of
n ,
F(x).
(n > o)
,
These functions are therefore positive
and since they belong to g Q, they are, as already shown, of ^ . Furthermore, they are uniformly bounded, and converge as n > oo. By Theorem 21, therefore, f(x) is also a func . We have thus proved THEOREM 23 [69]. In order that a function belong to class 33 > it is necessary and sufficient that it be positivedefinite. 5.
One can state exactly for which
p
the functions
CHAPTER IV.
(5 )
= e” ^
STIELTJES INTEGRALS
>
(° < P < 00)
t
belong to and for which they do not. For p = 1 and p = 2, the transforms of (5 ) are positive and absolutely integrable. Hence the func tions themselves are in $. We shall show below that they will also be long to $ if o < p < i [70], but will not if 2 < p < oo [71], They are likewise positivedefinite if 1 < p < 2, but the proof of this assertion is somewhat troublesome and we shall, therefore, not give it [7 2 ]. Let us denote the transform of (5 ) by tt Ep (a) = ~
/
e( xa)fp (x) dx =
We obtain by partial integration for
J
Ep (a).
cos (xa) • e”x
sin xa
dx
0 < p < 1,
n a E p (a) = / s i n (xa) • £ pxp“1e~x
The factor of
Therefore
j
dx
is positive and monotonically decreasing.
Since
/ sin (xa) • g(x) dx
Tf/a °
X
/
n=o 0
sin (xa) ■
s
+ x ) ■ g
+ x )
dx
it follows that E (a) is nonnegative for a > 0. As in 3*/ the absolute integrability of Ep (a) now follows because of the boundedness and con tinuity of f(x). Hence fp (x) is a function of y for 0 < p < 1,
f(x)
Let E(a) be the transform of a function f(x) of 3 Q . If is ptimes differentiable, and if the derivatives up to the pth order
also belong to S Q, then the transform of f ^ ( x ) is (ia One recognizes from this the following. If a function f(x) of has 2n absolutely integrable derivatives and if the function f^2n ^(x*) is con tinuous and bounded, then thefunction f(x) is positivedefinite if and only if the function ( 1 )nf ^2n ^(x) so is. For p > 2 (5) has two derivatives which are absolutely integrable, continuous and bounded. Be cause fp (o) = 0,  f ”(x) is not a function of sp, since such a function has a value > 0 for x = 0, unless it vanishes identically. Hence for p > 2, (5 ) is not a function of .
§2 1 . §2 1 .
1®
SPECTRAL DECOMPOSITION
97
Spectral Decomposition of PositiveDefinite Functions. An Application to Almost Periodic Functions A distribution function
V(a) has at most a countable number
of discontinuous points, but these can pile up in any manner on the aaxis. We denote them in some sequence by Xq,
(1 )
X.J , Xg > •••
and the corresponding function jumps by
ay.
Thus
av = V(\v + 0 )  V U v  0 )
(2 )
,
and (3)
y \
oo
0) h(x) dx = 0
,
03
or in the notation of §9 , 2 2ft{h(x)
(9 )
But if S(a) does not vanish outside of a finite interval, then we consider the functions n h^x) =
f
e(xa) dS(a)
n By what has just been proved
aftCh^x)}=o . If a set of functions, each of which has a "mean value”, converges uni formly in [ oo, oo], then the limit function also has a "mean value” which can be calculated by taking the limit. most general function (7 )• bution function S(a2 + x),
Hence (9 ) is valid for the
Since 'h(x)e( ax) has the continuous distri the following is also valid m {h(x)e( Xx)} = 0
(1 0 )
.
On the other hand 2ft{e (Ox )} = 2ft{i } = i
2ft{( )}=o,
p4o .
e px
The following assertion is therefore valid for the function n (11 )
3f t ) e (~
gn (x) = £ a ve(xvx)
)) = o
for
.
X i xy, v = o, 1, ..., n
and
= ay
for
100
CHAPTER IV.
X = Xy. (8)
Since as n —
in
> co,
STIELTJES INTEGRALS
the functions (11 )converge uniformly
to
[« , « ] , we have o
for
W ^ v = o ,
1,2,
2ft{g(x)e( xx)3 = \
(12)
ay for
x = Xy
.
We therefore obtain by (1o ) and (12) THEOREM 2b .
For each function f(x) =
J
e(xa) dV(a)
,
the following relation holds for all real
x
9ft{f (x)e( xx)) = V(X + 0)  V(X  0) . 4.
If
f(x)
Theorem19, the product function Is again
is positivedefinite, then h(x )h (x)
is too.
By
whose distribution
continuous. By(9 ), it therefore follows that
(13)
"(t  x) dt = 2ft {« t
du
Since the inner limit in each interval  o> < u < cd is uniform, it is permissible to interchange the limit with respect to t and the inte gration with respect to
u.
Therefore
(1 7 )
in
f cp(t )
1
a(x) =
lim 03 —^>00 T
lim > c »
r
cu
f [
d a)
We now make a further assumption in regard to x,
cp(t  u)e( Xu) du
dt
^
cp(t ),
namely that for each
the limit u jru 03+t c (x ) =
f
lim 03 —>00
/
0
9(1 )e( X) d s
m {9(1 )e( X))
 03+1
exists uniformly with respect to all
t
in
[ 00, °c 203
I
9 (t  u)e( Xu) du = e ( xt)c(x)
Hence OJ
J* 9(t  u)e( Xu) du = e( Xt )c(x) + e(t, 03) ,
where the error term for all t in [
e(t, 03) converges uniformly to zero, as 003. Substituting this in (17), we have
03 — >
CHAPTER IV.
102
STIELTJES INTEGRALS T
a(x) = c (x ) •
Those values of
X
lim 4 * / T >« %
for which
a(x)
0 ,
the Bessel inequality
v
therefore holds for the function
(a) = E(a)
The question as to whether at least one solution exists can be separated from the question as to how many solutions exist. Since the operation Ay
is additive ACc^y, + c2y2 ) = c1Ay1 + c2 Ay2
,
the general solution of (1) can therefore be obtained out of a particular solution by adding to the latter any solution of the homogeneous equation (9)
Ay = 0
3. The homogeneous equation is settled immediately since it corresponds to the relation (10)
G(a)cp(a)
If now G(a) isdifferent (
from zero in
11)
cp(a) = 0
=0 [ 00, 00],
then itfollows
that
.
This means that (9) has only the trivial solution y(x) = 0 If
G(a) doesvanish
atpoints
of
.
t », «],
then thetrivial solution
remains the only one, If the zero points of G(a),  that is the values of a for which G(a) = 0 , are nowhere dense, as for instance if they are Isolated. Because in this case, (11) is valid for an everywhere dense set of values of a, and since cp(a) as a function'of >Q is continuous, therefore (11) must be valid throughout.  The function (5) is an analytic function of the variable a. Hence its zero points are isolated, and therefore the general equation (A), in its homogeneous case, has no solution (in our sense).  If we admit "solutions" other than those stipulated here then the homogeneous equation may well have some. In the case of equation (B) say, if the (complex) zeros of the polynomial cr (iT)r + cr 0,
f(x)
can be "multiplied"
then the function
must be rtimes differentiable in
y(x)
of
by G(a)“1
belonging to
gQ.
These conditions are not only necessary, but also sufficient. For if they are satisfied, then the function y(x) ~
J G(a)~1E(a)e(xa)
da
is a solution of (A)as can be verified by substitution.
We have the
problem, therefore, of stating criteria for the fulfillment of these con ditions .
§23. 1• (1) be a given continuous tuted that denote the (2)
Multipliers
Let f(x) ~
J
E(a)e(xa) da
function. By a multiplier of E(a) or of f(x), we mean a function r (a) defined in  « < a < then this piece need not again be a function of z Q (i.e., it need not again form the oscillation group of a function of $ 0 )* But for many problems, it is important to be able to " I s o l a t e a finite piece of a function E(a). This "isolation" can be accomplished, provided the cut is not carried out too sharply on both interval ends a 1 and a 2 . For as a Is laid close to on Its left and a£ close to «2 on Its right, there Is a function r(a) which has the value 1 in (c^, a2 ), vanishes outside of (a*, a^), and in [oo, oo] has derivatives up to a previously assigned order
r ^ 2.
The product
r(a)E(a)
Is a function of
ZQ
which agrees with E(a) in (a1, a2 ) and vanishes outside of the interval (a], ajp. For the function of 5 Q belonging to it, we have (1 1 )
where
rtf) = a.
J
Kr (j)f(x  I) cU
Kp (l) has the order of magnitude
o(Jx~r ).
, One can even construct
the function r(a) in such a way that it has derivatives of arbitrary higher order (Appendix 1 5 ). Then (12) 
K r (x) =
0(*rr ),
(r = 1, 2, 3, ...)
and by reasonof (n ), many "smoothness properties" And for each function of z Q,
of
,
f(x) will carry
continuity is a necessary condition.
m
CHAPTER V.
OPERATIONS WITH FUNCTIONS OF THE CLASS
over to the function 10.
obtained from it, see for instance §2 4 , i.
r[f]
Let
(1 3 )
t,, v
V
...
be a finite set of real numbers which differ from one another. index
x,
we can determine a multiplier
For each
rx (a ) which is differentiable
arbitrarily often, which has the value 1 in some (sufficiently small) neighborhood of tx and which vanishes outside of a (somewhat larger) closed interval, the latter interval not including any of the other points (13). The multiplier r 0 ( a) = 1  ^ r x (a) x is differentiable arbitrarily often, has the property that it vanishes in a certain neighborhood of each point (13) and has the value
1
outside of
a (sufficiently large) finite interval. 11.
Let a function (1) be given.
Two continuous functions
r 1 (a) and r2 (a) which differ from one another only at such points a for which E(a) = 0, are either both multipliers of (l ) or neither of them is. We call them "equivalent" in regard to (1 ). 12. terval
If the transform of (i) vanishes outside of a finite in
(a, b),
then by Theorem 12 b f(x) = ^
(14 )
E(a)e(xa) da
a If
r(a)
(a, b),
is a function defined and rtimes
(r ^ 2)differentiable
in
then b J
(15)
r(a )E (a )e (xa ) da
a is again a function of S Q * For if one extends r(a) to a function which is defined and rtimes differentiable in [ °o, 00], and which
r*(a)
vanishes outside of a finite interval, then r(a) and r*(a) are equiva lent as regards (1 4 ), and (15) is then the function r*[f]. r*[f] can also be written in the form (11) where Kr*(x) = o(x~r ).
1.
If
§2 4 .
Differentiation and Integration
K(x)
is rtimes differentiable in %Q
and
f(x)
belongs
§2^. to 5 Q ,
DIFFERENTIATION AND INTEGRATION
115
then the function g(x) =
K()f(x  ) d
is also rtimes differentiable in
2fQ,
and indeed
g ( p ) (x) = ■ij y ' K ( p ) (  ) f ( x  e) d,
0 < p < r
.
As is easily seen, it is sufficient to prove this assertion for the case r = 1.
The function h(x) =
I* K ‘ ()f(x  I ) d
is by §1 3 > 3 a function of Theorem 15,2
SQ.
=
f
f ^ ) K ! (x  n) dt]
As has already been shown in the proof of
x I
h(x) dx = g (x)  g (xQ ) ,
xo and therefore
2.
h(x) = g ’(x),
Q.E.D.
To each function f(x)
there is,
for eachinteger
r >
o,
an rthintegral, i.e., a function Fr (x) such thatFpr ^(x) =f(x). The function Fp (x) is unique excepting for a random additive polynomial of (r1 )th degree in x. We call a function f(x) rtimes integrable in %Q if f(x) belongs to 8 0, and the additive polynomial in Fp (x) can be selected in such a way that Fp (x) together with derivatives Fpp ^(x}, o < p < r, belong to 3 0, i»e., Fp (x) is rtimes differentiable in S0 . In §3> ^ we showed that if f(x) is 1time integrable in 5 0, then the integral F ^ x ) is unique and has the value x J
(1)
f(x) dx =  / f ( x ) dx x
More generally the following is valid. 3 q, then
Fp (x) is unique and x /
x
f(x)
has the value
X1
xr2
toi /
= (i)r / d x ,
If
•••/
f
.
is rtimes integrable in ^
Xr1 dxri /
f(xr ) dxr
dx2
f(xr ) dxr xr2
xrl
.
116
CHAPTER V.
OPERATIONS WITH FUNCTIONS OF THE CLASS
AQ
The proof is very simple. We prove it say for r = 2. By hypothesis F^(x) belongs to 3 0 and is the integral of the function F ”(x ) = f(x), hence F^(x) is unique and has the value (i ). Now Fg (x) as the in tegral of F^(x) is again unique and results from F^(x) in the same way as F^(x) results from f(x). Therefore x F2 (x) =
J
.
xi
dx1 ^
THEOREM 2 5 
f (x2 ) dx2 = (1
f(x) ~ J
J
f (x2 ) dx2
,
E(a)e(xa) da
is rtimes differentiable in f (p)(x) ~
dx1 J
If the function
(2)
(3)
fJ
(ia )pE(a )e (xa) da,
8?Q,
then p = 0 , 1, ..., r.
Conversely if a function (t)
cp(x) ~
J
(ia )rE(a )e (xa ) da
exists (i.e., if (ia)r is a multiplier of then f(x) is rtimes differentiable in hence
E(a)), (and
f ^ ( x ) = cp(x)).
PROOF. The first part of the theorem has already been proved in §3, A, and used many times previously. The second part is more difficult. We shall prerequire the following two theorems which are also useful of themselves. THEOREM 26.
If
E(a)
vanishes outside of a finite
interval (A, B), then f(x) arbitrarily often in 5 . PROOF. That f(x) from §k , 2, c ). Indeed
is differentiable
is differentiable arbitrarily often follows
B f (p)(x) = j ' (la )pE(a )e (xa) da, A And that these derivatives belong to
p = 0, 1, 2, ...
follows from
§23, 12.
§21*.
DIFFERENTIATION AND INTEGRATION
THEOREM 27. If E(a) vanishes in an interval [A, B] which contains the point a = 0 (A < 0 < B), then f(x)
%Q,
is integrable arbitrarily often in
and
the pth integral has the form Fn (x ) ~ F e(xa) da p J (ia)p
(5)
PROOF. in in
Since together with E(a), also
(ia)”1E(a)
vanishes
[A, B], it is sufficient to prove that f(x) is 1times integrable 8 0 and that this integral has the value
(6)
in
.
g(x)

The function (ia)”1 [ °o, co] a function r(a)
e(xcu) da
.
isregular oustide of [A, B].Therefore can be found (cf. Appendix 16) which Is
two times differentiable, and outside of [A, B] agrees with (ia)” 1. This function is a multiplier by §23, 7 , and since in regard to E(a), it is "equivalent" with (ia)”1, cf. §23, 11, the function (6) exists in any case, and indeed g(x) = r(f). We still have to prove that g(x) is differentiable and that g*(x) = f(x), and It suffices to prove only that it Is differentiable In $ 0 « For if this is so, then the derivative, by the first part of Theorem 25, has the transform hence g ’(x) = f (x). Set
rl^ “2i(ai)’
(ia)r(a)E(a) = E(a)
and
r2(]. PROOF.
We write r (a) = ~ t L
A A
a:q~p A + a“ 'q(a)
By hypotheses,
k(a)
=
aq'p
k(a)
is a bounded, continuous function.
.
Because
q  p > 2, r(a) is therefore absolutely integrable in [ct , co]. For m = o, therefore, the proof is finished. For m > 1, one can diff erentiate r(a), and set for the derivative P1 , a h1 (a) (11 )
r1 (a) = A 1a
QLi
Ui” 1 + oc g1 (a)
^ = p +q, q 1 = 2q, A 1 = A2 and t^Ca) and g ^ a ) are certain detailed expressions from which one can easily see that these functions have m  1 continuous and bounded derivatives, and that the denominator of (11) differs from zero in (aQ, °o] . Since p 1 > o and q 1 ^ P 1 + 2, the quantities p 1# q1, m 1 = m h ^ ( a ) , A 1 again satisfy the hypothesis of the theorem. It is evident therefore, that the proof of the conclusion can be arrived at by induction from m  1 to m, Q.E.D. We will now treat equations (E), (B), (C) and (A) in this order. The first three are special cases of (A), to besure, but we will also obtain special statements in return. k.
Equation (E).
Since the exponential function e(5a) is bounded for real the order of magnitude of the characteristic function
6,
§2 5 .
THE DIFFTIRENCEDIFFERENTIAL EQUATION
123
r1 apa(ia )r e ( b a +
0=0
by the highest termar . There
is, there
a ^ ccQ
such that for
G(a). > Mar
.
In particular, G(a) ^ 0 for a ^ a , and thereforeG(a) ” 1 as an analytic function, can have only a finite number of zeros. We assume for the time being that no real zeros at all exist. We shall prove in this case, that the functions (12)
H (a) = (ia )pG(a)”1 P
are general multipliers for
0 < p < r.
cases:
2.
1.
0 < p < r  2.
p =
For
its proof we distinguish three
r  1and 3.
can be applied directly in case 1. In itwe set h(a) = ip, rl s g (a ) = _ 1 _
a
Y
Y
p=0 CT==0
The derivative of each order of
g(a)
p
p= r. = p, q
a p a ( l a ) P e ( 8 1. §2 7 ■
Systems of Equations
1o We consider the system of differential equations
m (l)
^
M = 1, 2 ,
anvy U x ^ + ff x)’
m
,
v—1 where f (x) are functions of . By a "solution”, we mean a system of functions y 1(x), •.., ym (x) which are1time differentiable in a , and satisfy the system of equations in [»].' A characteristic function G(a) again plays a role in the question of its solvability, and it is the polynomial of the mth degree la  a,
a i2
* * 9
~
am2
• • •
~a im
G(a) amra
THEOREM 3 5 ® In order that the system (1 ) have a solution, one of the following two conditions Is sufficient. 1. The characteristic function zeros. 2. At each zero ix, the functions e(~ Txx)f^(x),
tx
of
G(a)
G(a)
have no
with multiplicity
pi = 1 , 2 , ..., m
be ^x~times integrable in 3 0 * There is always one solution at most. It Is sufficient to assume that the y^(x) belong to g Q, are differ entiable, and satisfy the system of equations. The derivatives y^ are then of themselves functions of %Q by reason of equation (1).
§2 7 .
135
SYSTEMS OP EQUATIONS
REMARK. In case 2., as contrasted with equation (A), cf. Theorem 29, the given condition is not necessary. This can be seen by an example. Consider the equations y} (x) = i[y1 (x) + y 2 (x)] + f 1 (x) y2 (x) =  i[y1 (x) + y2 (x)] + f2 (x) Here  i
la  i G (a) =
ia 4 I
i Hence
G(a)
has a twofold zero at the point
twice and q(x) is once differentiable in is solvable for the functions f 1 (x) = p n(x) + q l(x),
a = 0.
3f ,
But if
p(x)
Is
then the equation system
fg(x) = p"(x)  q !(x)
.
Its solution is yj (x ) = 2ip(x) + p 1(x) + q(x) y2 (x) =  2ip(x) + p*(x)  q(x)
.
But the functions f 1 (x) and f2 (x) are two times integrable in only if q(x) is one time integrable in S Q; however this does not hold if for example » ■
(
^
f
since
J
q(x ) dx / 0
,
cf. §3, U. PROOF.
Set, for
n = 1, 2, ..., m
(3)
~
E^(a)e(xa) da
O)
~f
cp^(a)e(xa) da
Substituting in (1 ), we obtain the equations
CHAPTER Vc
OPERATIONS WITH FUNCTIONS OP THE CLASS m
(5 )
1} 2
iaq^(a)  )  a^vcpv(a)  E^Ca), v1
Denoting the algebraic complement of the determinant (2 ) by follows by (5 ) that
G
, (a),
it
UJ
(6)
G(a )q?v (a) = ^
Hence for all
a,
G^v (a )E^ (a),
v * 1, 2,
apart from the (Isolated) zeros of
G ST)
(«) IToT
G(a),
v = 1, 2,
Therefore there can he only one solution at most. Assume now that the hypotheses of the theorem are fulfilled. Then by §25 * J 46, the functions
t P  f E («) G(a) belong to x Q for Q < p < rm But since the polynomials G^v (a) are at most of degree (m  1 ), it follows that the right sum in (7) is a func tion of %qP and that the function y y (x) of p Q belonging to it, is differentiable in d Q . It is verifiable at once by substitution that the yv(x) thus obtained, actually form a solution, Q,EoD. 2.
The proceedings just described can be employed in the solu
tion of verygeneral systems of functional equations •> We shall treat one more in extenso, namely the system of differenceequations m
(8 )
V
x + 8d
=
) _ , V y v(x) + V v"™1
with arbitrary real differences
x)'
The characteristic function reads
e(8 a )  a, 1 a?1
(9)
e(52a)  a22 .«
G(cO = ml
,,
1 2
m2
1m a2m e (8 a )  a vm mm
§27
SYSTEMS OP EQUATIONS
137
It is an expression of the form m (1 0 )
aae(maa) a=0
with real exponents a . We now make the assumption that the expression does not vanish identically. This always occurs for example if all 5 ^ have one and the same 3ign, say
5 ^ > 0.
THEOREM 36. If the characteristic function is essentially different from zero (11)
 G( a )  > S > c,
(
00
0 (a)
< a
i,
(6)
E (i \ a ,
E(a, k) The class
is additive: k E(a, k) x
2Jtiff(a),
t)
.
f = c1f 1 + c2f2
ciE1 (a, k) + c2E2 (a, k)
If the first term on the right of (4 ) second by
k +
is denoted
implies . by
2*$(a)
and the
then the function
1 $ ^ ( a ) = 27 J"' f(x)e( ax) dx 1 is a Otransform, and the function #(a) is likewise contained in x q . Hence both are continuous and converge to 0 as a  > +oo. By the well known formula (for an arbitrary ktimes continuously differentiable func tion (a)) k 4 (a)  x
^
■(k' _Y‘jT f
(“  P)k_1i>(k)(P) dp
,
o one concludes without difficulty that (a) =o(ak ) as a > + oo. therefore follows that each function E(a, k) is continuous, and (?)
E(a, k) = o(ak ) as
a
it
>+ oo
5. A first justification for the introduction of the ktrans forms is offered by the following theorem. THEOREM 3 7 *
If the ktransforms of two functions of
3 ^ agree (i.e., are kequivalent), then the functions are identical [79]. PROOF.
Consider the difference function
CHAPTER VI.
GENERALIZED TRIGONOMETRIC INTEGRALS
Akcp(a) = cp(a + k )  (k )cp(a + k  1 ) + ... + ( i )k (^)cp(a)
(k  1 );
It vanishes for a polynomial of degree

hence
AkL^.(a, x) = o where
L^fa, x ) is defined by (t1 ).
On the other hand
Ake ( a x ) = e ( a x )[e ( x )  i3k It therefore follows from (5 ) that
AkE(a, k) =
J
g (x )e (~ ax) dx
,
where k (8 )
g(x) = f(x) (
The function
g(x)
belongs to
and
AkE(a, k)
is Its 0 transform.
If now two functions f(x) have equivalent k~transforms, then the corre sponding functions g(x) have the same 0 transform and are identical by Theorem 1A . But then the functions f(x) are also identical, Q.E.D. If we are given a function
f(x)
and its ktransform
E(a)
then the fact, of their belonging together will be denoted by the symbolic relation1
f(x) ~ J
e(xa )dkE(a)
.
We also call this relation or its right side a "representation" of
f(x).
Cf. §31 in regard to the possible "convergence" of this representation to f(x).
If It were not for typographical considerations, we should have written
J
da
1
instead of J ' e(xa )dkE(a)
§2 8 . GENERALIZED TRIGONOMETRIC INTEGRALS 6.
One easily finds (
,e(ax)  L2 (a,x)
(Hf
n
n \ e (_a x ) + e (ax)  2 ,
{{
+J I
0 P
FET
dx
jJ ( . ^ , 2
sinf x \ 2
= 4 /   1 _ j
dx  2 = 2 a  •   2
.
"0 Hence for
f(x) = 1 E(a, 2 ) X
^
a 
From (6) there follows more generally for E(a,
k)
X
k b 2
1
ak _ 1
,
where by i~ ik 17 I
we mean, both here and In what follows, that function which has the value k k 7 for 7 > o and the value  " for < 0.
y
For
y
f (x) = x^, (i = o, i, 2, k
2 itE(a , k )
i
and k >
ne(ax)  L,,(ax ) /  — — ~ y ~  dx J (ix)K"M
k x
i
's '
jj, +
2,
we have
r e(ax)  K. i^ / — i>—
dx
.
J
Therefore
E(“ > k > Let
f (x) be a function of 5 ^,
f(x) • e(rx).
t
jt X
tv
:
real and
$ (a)
the ktransform of
Then
n e ( (aT )x )  e ( T X )L,,(a,x ) 2n$(a)x / f(x) £dx J (~ix)k
X
r e ( (aT )x )  L^(ax,x ) r LiLar,x)  e(xx )K (a,x) / f (x ) ^  dx + / f(x) g □£— _ dx
J
(Lx)k
J
The second integral is a polynomial of degree
(ix)k (k  i) in
k (t o )
oo J
f
fn (x)  f*(x)
dx = 0
xo
is valid for finite numbers
xQ ,
and
x^.
.
1A8
CHAPTER VI. GENERALIZED TRIGONOMETRIC INTEGRALS
PROOF. We shall make use (without proof) of the following theo rem ofLebesgue [80]. If f(x) is integrable in (A, B ), then b lim j i 5 ™>o Ja for
A < a < b < B.
.f(x) ~ f(x +  ) dx = 0
,
It expresses the fact that each integrable function
possesses a certain "continuity in the mean". Let
f(x) be a function of
3 ^.
By thetheorem
just stated.,
b lim [ {>°i
f (x )  f(x + £ )p, (x ) dx  0
for every two finite numbers
a, b .
On the other hand a
f
lim b >00 ^
f(x +  )pv (x ) dx = lim K a >™oo J
f
f(x + z )pv (x) dx  0
,
K
and it is found without difficulty that this limit relation holds uni formly for all  in each interval  < Q . From this, we obtain the result that the function (1 7 )
5(0
converges to zero as Let

= %—
J
f(x)  f (x + I )]pk (x) dx > o.
be a fixed number f n K( nO K^i) = j
(18)
> 00
For variable
for
Ul < l0
for
III > i0
n,
set
,
and (19)
H ^ l )  nK(n)  Kn ()
.
We set correspondingly1 gn (x) = f
+ 0^(0
g*(x) = f(x) f ^ U )
The functions
g*(x)
and
d,
d,
h*(x)
=f
? (x + 1 ) ^ ( 0 d£
h*(x) = f(x)
J
also depend on
1^ ( 1 ) d
n.
,
§2 9 . FUNCTIONS OF 3?k
U
9
so that fn (x)  f*(x) = gn (x)  g*(x) + hn (x)  h*(x)
.
Since ./
lfn  f*Pk ^
«J
and f
(21 )
lgn  g*IPk dx < t,(eQ )
where *iU0 ) is independent of (10), it follows that
n
,
and approaches zero as
_ >
0.
By
I V }I i T y  p  p a
(22) where (23)
lim n “>00
= 0
.
By 3.
J
hn (x)pk (x) dx < AnCk
f
h*(x) pk (x) dx
f
6 (§ )
f(x)pk (x) dx
f 
 tQ < i < Q,
i
Q.E.D.
by
we have
\&n ~ 6*Pk dx < f f
Jr
,
Further, denoting the upper limit of
defined by (17) in the interval
lf(x + t)  f (x) pk (x)
dx d
s0
+ oo,
—
>
+
oo)
then there would be
A > o, and a set of points 1 < x ] < x2 < x 3 < ... —
(25)
> oo
,
such that (26)
f(xv ) >
a!
.
We can assume that (2 7 )
xv+1 
xv > 1 ,
V
[otherwise one takes a suitable subset of (2 5 )3 .
J
(28)
= 1 , 2 , 3 ,...
By the finiteness of
dx
1
x
it follows easily that x +1 f
Now If
x v < x < x v t 1,
f'(x) dx = o(xk )
then x +1
f (x)  f (xv )I
vQ,
;
,
§29.
FUNCTIONS OF 3 ^
in contradiction to the assumption that
f(x)
151
belongs to
3 ^«
6 . We call f(x) rtimes ,, if the / \ differentiable in 3 K derivatives f*(x), f n(x), ..., f (x) exist and together with f(x) belong to 3 We call f(x) rtimes integrable in if the kth in tegral of f(x) can be so normalized that it is rtimes differentiable in 3 ^. 7 . Let K( ) be rtimes differentiable, and together with the first rderivatives satisfy (1 0 ). The function g(x)
2~
=
f
K(
formed with a function
e )f (x
. i)
f(x)
of
di
J
= ~
3 ^,
f(l)K(x  e)
d
,
is rtimes differentiable in
3 ^,
and what is more
r = 1; 5 ^)
The proof can be limited to the case show that the function (belonging to
h ( x ) = 1 ; / K'(5)f(x
is the derivative of
g(x).

i)
de
= jT 0
J
f(e)K'(x

e) de
We form the function
x (a) In the normalization
a
/
••• J
d“ l ' 7
darl /
® (“r ) d“r
q:
In particular therefore a J'
®(a) da = f
(1 ) For
k = 1,
the formula of partial integration
x(a )\(f1 (a ) da = X(a
/
$(a) da
an
(a ) 
( 1)
/ X ’(a )\r(a ) da  x(aQ )t(a ) (O
is valid.
\ We can also write for this a /
1 x(a)ijr,(a) da x x ( a )  f a ) 
(1 ) For general
(1)
J
k,
(1 ) the analagous formula reads
a
(k)
a / x f(a)^(a) da
^ x(a )\r^ ^(a ) da
k x(a)t(a) + ^ ( 1 r= i
f (£ ) J
a x^r ^(a)\jr(a) da
.
(r)
We leave its verification, by differentiating both sides ktimes, to the reader. Now let \r(a) be a continuous and *X(a) a ktimes continuously differentiable function. We then consider the function
(2 )
k cp(a )K X (a )ir(a ) + ^ (1 )P (p) jT X ^ (a H(a ) da r =i (r)
determined to within a polynomial of degree (k  l ). ktimes continuously differentiable, then by (1)
If
y(a)
is also
§30.
FUNCTIONS OF £ k
155
00.
Then the relation (9 )
dk9(a )  x(a )dkiif(a)
also holds. For by (1)
k (1 0 )
a
(J>n (a)s=:
f Xn (a)dkin (a) ,
(k) and by (2), the left side or the right side is convergent to
a 9(a)
j x(a )dk\Ka)
or
(k) Hence by (1 ) (11)
k q>(a)x
p [
X(a)dk\jf(a)
,
(k) Q.E.D. If the functions each subinterval (A1, B® ) of defined by the relation
tn (a) [A, B],
converge uniformly to and if the functions
dk9n (a ) = X(a )dktn (o:)
,
y (a) in (a) da
,
(/)• x( a )
is(k+£ )times continuously
differentiable,
dk+A(a) = x(a)dk+V ( a ) . This relation also can be proved most quickly if ir(a)is approximated by ktimes continuously differentiable functions ^(a), if 9n (a) is defined by (13) and then
n
is allowed to become infinite.
9. If the functions fn (x) of* kconverge to f(x), then their ktransforms En (a) are convergent in suitable normalization to the ktransform E(a) of f(x), and indeed uniformly in each finite interval. On the other hand, each function f(x) of can be represented as a klimit of functions f (x), each of which belongs to 5 , cf. §29, 1. We shall make two applications from 10. E(a,
k) and
Let us denote the ktransformsof (a, k).
(16)
the f(x)and
above. f(x
+ \)
by
Then dk(a:, k) = e(\a)dkE(a, k)
In fact, if
f(x)
belongs to
A Q,
then
3>(a, 0) = e(Xa)E(a, 0 )
,
and by 8 ., (16) follows. In the general case, we represent klimit of functions of A , and make n > « in dk $n (a,
= e(Xa)dkEn (a, k)
1 1 . THEOREM ^0 . With a function a function
K()
f(x)
for which
f(x)
of
and
as the
CHAPTER VI. (1 7 )
GENERALIZED TRIGONOMETRIC INTEGRALS
K(6 ) < A( 1 + U k + 2 ) _1
,
we form the function (18)
g(x) = ~
J
f ( )K(x  I) d
,
which also belongs to
Between the ktrans
forms E(a) and (a) of there is the relation
f(x)
(19)
and
dk0>(a )  7 (a )dkE(a ) where (That
g(x),
,
7(a) denotes the Otransform of K( £ ). 7 (a) is ktimes continuously differentiable
follows from §13* 9 ) PROOF.
If
f(x)
even belongs to
S Q,
(19) is the Faltung rule
of §13 • For general f (x), we consider functions fn (x) of are kconvergent to f(x). The corresponding functions gR (x), likewise belong to 3 0, (19)
are by Theorem 38, kconvergent to
which which
g(x).
Hence
results from dkn (a) = 7(a)dkEn (a)
by letting
n  > «. §31. 1. in
(1 )
Convergence Theorems
THEOREM A1 . Let [00,
00].
$(a)
be absolutely integrable
Then the function g(x) = J
e(xa)4>(a) da
,
since it is bounded, is contained in S 2; its 2 transform E(a) is two times differentiable, and satisfies the relation E" (a ) = *(a) PROOF. We shall need only the special case, and therefore shall only prove this part, in which o>(a) vanishes outside of a finite in terval. However the general case can be deduced from the special one without difficulty, if one 'makes use of the fact that the functions
§31•
CONVERGENCE THEOREMS
Sn (x) = /
are 2convergent to
e(xa )4>(a ) da
g(x).
The function *(0, a) = ^
for fixed
0,
P e(xa)  L? (a,x ) / e (x0)  p dx J  x
is the 2transform of
e(x0).
,
Therefore by §28, 6
tfr(0, a) = ~ a  0 1 + A(0) + aB(0 ) The coefficients continuous in 0
A(0) and B(0 ) are continuous since #(0, a) is for a = 0 and say for a = 1. By introducing the
function
A(a,
0)
we obtain # ( 0 , a) = A(a,
0)
+ A(0 )+  + a(jB(0 )  0
If one now substitutes (1) in r e(ax)  L 0 E(a) ^ __ / g(x ) J  x
dx
and (as is permissible) interchanges the order of integration, one obtains a E (a) x j Hence
E(a)
$(0)^(0, a) && x f A(a, 0)$(0) d0 = is actually the second integral of
J
(a  0 M 0 ) d0
$(a).
2. THEOREM b 2 . Let the function f(x) of 3 ^. be such that its ktransform E(a) is ktimes con tinuously differentiable not only on a left half line [ 00, a0 ^ also on a right half line [bQ, 00]. For a < aQ , bQ < b, we define the generalized integral b J(x, a, b) = J
e(xa)d^E(a)
.
CHAPTER VI.
GENERALIZED TRIGONOMETRIC INTEGRALS
by the expression k1 e(xb 1
J
(
£ ( ix f E ( k  r " 1 }(a) r =o
 e(xa)
rlo
(2)
b + (» Ix )k j ' e(xa)E(a) da
Then in regard to the limit value (3 )
lim J(x,  A, A) A —>oo the following assertion holds. If this limit exists for almost all x in an interval (x , x ] ), thenits value agrees with f(x) for almost all x in (xQ, x 1 ) [81]. REMARK.
Our definition of the generalized integral
J(x, a, b)
is such that it has the following properties. 1 ) If E(a) is ktimes differentiable throughout (not nec essarily continuously), then b J(x, a, b) = J
e ( x a ) E ^ ( a ) da a
2 ) For (k)
zero in
a 1 < aQ, bQ < b 1
J(x, a 1, b 1 )  J(x, a, b) + J(x, a 1, a) + J(x, b, b* ) 3 ) If, In particular, [ °o, aQ ] and [b , °°3,
E(a)has Its kth derivative equal to (and thus is ^ o on each of thehalf
lines), then J(x, a ’, a) = J(x, b, b ’) = o Hence the function (5 )
F(x) = J(x, a, b )
Is Independent of the special values of have also (6 )
F(x) =
lim
J(x,  A, A) A —>oo
.
a
and
b.
In particular, we
§31.
CONVERGENCE THEOREMS
1
PROOF. For the actual proof, we first introduce the following theorem * 3. THEOREM t3 . If the ktransform E(a) of f(x) has its kth derivative equal to zero in [ 00, aQ ] and [bo . 00] then f(x) =
(7 )
lim A
J(x,  A, A)
,
> 0 0
(and therefore, among other things,
f(x)
is
differentiable arbitrarily often). PROOF.
Set J(x, a, b ) = t (x ) + ( ix )kg(x )
where
t(x)
means the "trivial" part of the function (2), and b g(x) = J ' a
e(xa)E(a) da
Since g(x) is bounded, the function F(x) defined by (5 ) or (6 ) is contained in now examine the (k+2 )transform (a)
where
f
#(a)
Hence
e(ax)  L]f+a ( ix)kg(x)
dx
means the 2transform of
g(x).
g(x)
But by Theorem ii
= E(a) Consequently a < a < b
Now since (a, b) can be an arbitrarily large interval, and since E(a) is the ktransform of f(x), it follows by Theorem 37 that f(x) and F(x),
considered as functions of
agree.
But this says that the
CHAPTER VI. functions
f(x)
and
GENERALIZED TRIGONOMETRIC INTEGRALS
F(x)
are identical.
k.
function
We can now prove Theorem k 2 . Consider in [00,00], a 7(a), (k+2)times continuously differentiable and vanishing
outside of a finite interval ( aQ, aQ ), which is even [/( a) = 7(a)], monotonically decreasing in 0 < a < 00 and has the value 1 when a = 0. The function K(S ) = jT e (a )7(a ) da is such that K(5 )
00, in each finite ainterval [because y { a/n)  > 7(0)].The derivatives of E (a) are likewise convergent to thecorre sponding derivatives of E(a) at each point a = a < aQ and a= b > b . 3)
Outside of the intervals E^k ) (a) =
( aQn, aQn)
0
.
Because of 3), we have by Theorem k2 fn ^x ) = lim J n (x >  A, A) A —>00
(8) where
Jn (x J a> h)
,
denotes the expression (2) formed with
En (a),
I.e.,
§31.
CONVERGENCE THEOREMS
fn (x) = Jn (x, A, A) + ( j
165
if A >  aQ and A > bQ . Now let x be a fixed point in which the limit (3) exists, i.e., at which the .integral J
(10)
da
+ y ) e (xa O ^ E ^ C a )
(x
q ,
x
1
) at
[e(xa)E^(a) + E( xa)E^( a)] da
A is convergent. Denote (for fixed (10) has the value
x)
J
H =
the integrand by
cp(a).
Therefore
cp(a) da
Next consider the quantity a Qn
\ =f
c p(Q!)^(h)da=f
^7(h)da
We assert that lim H n >00
d O
= H
.
For its proof, we introduce the function H(a ) = J cp(a ) da a and denote the
this
a
maximum value of H(a)
Hh .
From
,
by M.
Then
H(ah(2 ) da  ,(£)h(A) +l / H (a)r'(f) da
we
obtain (11) because > 7 ( 0 ) H( A) = 1
7 ( h ) H( A)
. H( A) = H
and
J H(a )y'
( 2 ) da
< MJ a
Recalling that
y( a ) = 7 ( a ) ,
‘ r' (h)
da < M • 7(0)
L
(11) states that the second term in
166
CHAPTER VI.
GENERALIZED TRIGONOMETRIC INTEGRALS
J(x,  A, A) + ^
+ J je (xa )E^k ^(a ) da
J
is the limit of the corresponding right term in 2 ), the first term is the limit shown therefore that for almost lim fn (x) n ~>oo n
=
of
Wehave
lim J(x,  A, A) A >oo
But since by Theorem 3 9 , the sequence f(x),
(9 )•Moreoverbecause
of thecorresponding firstterm. all x in (xQ, x 1 )
fn (x)
converges in the mean to
the limit lim f (x) n —>oo
so far as it exists for almost all f(x), [Appendix 1 1 ], Q.E.D. §32.
x
in
(xQ, x 1),
is identical with
Multipliers
The additional observations of this chapter will be similar in many respects to those of the previous chapter. Hence we shall often content ourselves with short hints and suggestions, and shall entirely suppress trivial generalizations from
k = o
to arbitrary
k.
For the following paragraphs, cf. §2 3 * 1 . Let (1 )
f(x) ~ f
e(xa)dkE(a)
be a given function of 3 ^* By a multiplier of E(a) or f(x), we mean a ktimes continuously differentiable function r(a) which is such that the function a (2 )
®(a) X
[
r(a)dkE(a)
(k) again belongs to X longing to * (a) by
We shall again denote the function of 3 ^.
be
r [f ] If r(a) is a general multiplier of the class 3 ^ (or then for brevity we speak of it also as a kmultiplier. If r1 and r2 are multipliers, then r =c1ry + c2 r2 is also a multiplier, and
§3 2 . MULTIPLIERS
167
r[f3 = c1r1[f] + c2r2[f] By the associative law (§30, 5 ), the product and r2 is again a kmultiplier, and
r
of two kmultipliers
rtf3 * r1 Cr2 [f]3 * r2tr1[f]3 2 . By §30, 10, e (xa) is a kmultiplier: Hence a finite series of the form (3 )
. rtf (x)]
f (x + x ).
+ cne (nna )+
c 1e (n1a ) + c2e(ki2a)
is a kmultiplier. We shall showthat an Infinite series of this form is thencertainly a kmultiplier, If both series CO eo converge.
00
X c v '’ V^l
I K i k ic vi V=1
It Is easy to verify the inequality 1 < ok 1 + u i k 1 + !xnK  “ 1+ xK
and if we also take into account the finiteness of the series (4 ) then, when putting P k (t)
= (1 +
,
tk )
we obtain n+p
£
lc J P k (x '
* cnPk(x)
v=n+i
and where
Cn
is a number dependent only on ■lim C = 0 n —>00 n
In the notation of §23, 3
n,
r1
for which
168
CHAPTER V I .
GENERALIZED TRIGONOMETRIC INTEGRALS
n+p
/
i
l ^i +p[ f I " V ^ I P k dx
A v=n+i
0 v>
I
f(X + % } IPk(x)
n+p < J
cv pk (x  nv ) dx < cn J
f(x)
f(x)pk (x) dx
n+i Hence the sequence of functions Hn [f] ~ J Is kconvergent.
e ( x a)Hn (a )dkE(a)
We denote Its klimit by F(x) ~ J
e(xa )dk 4>(a)
By §30, 9 k *(a)X
p I H (a )dkE(a )
lim
n >“ (k) and therefore by §30, k
d k 4 ( a ) = H ( a )d kE ( a ) Hence
H(a)
is a multiplier of the arbitrary function
f(x)
of
3 k«
If the function s (5 )
G(a) =
Y
aae (5aa )
a=0 is essentially different from zero, (6)
G(a)
 > S >
0,
(
«
< a < 00)
,
then it can be expanded in an absolutely convergent series of the form (3 )« The function (7)
dk (G(a) M da5
can be written as the quotient of two functions of the type (5). denominator has the value
G(a)k ,
The
and Is also essentially different from
§3 2 . MULTIPLIERS zero. Therefore the function (7 ) can also be expanded in an absolutely convergent exponential series. As is known from the theory of almost periodic functions (and as can also be shown directly without difficulty), the development of (?) results from the development of G(a)~1 by ktimes formal differentiation [82], and it therefore reads 00 £ V=
(inv f o ve(uva)
•
1
This series is absolutely convergent, but that means that the second series ( k ) also converges. We have therefore shown that for arbitrarily large k, the function G(a )” is a kmultiplier. 3.
The Otransform of a function
K()
which satisfies the
relation (8)
K() < AC1 + ik+2)’1
is by Theorem ko, a kmultiplier. the functions
( 9 )
Such Otransforms are on the one hand,
S ~ i~ T
5
;
on the other hand all such functions which with the first k + 2 deriva tives are absolutely integrable, and generally (k+2 )times continuously differentiable functions r(a) which outside of an interval  A < a < A, can be written as r(a) = § + H(a) where
c
is a constant and
,
H(a) along with the first
k+2
deriva
tives is absolutely integrable. The observations of §23, 10 carry over verbatim if "multiplier" is replaced by "kmultiplier".
On the other hand the observations of
§23, 11 are to be modified as follows. Let (1) be a given function. Two ktimes continuously differentiable functions , rQ which differ from each other only for such points a in whose neighborhood dkE(a) = 0
,
are either both multipliers of (1) or neither of them are. "equivalent" in regard to (1).
We call them
For k ^ 1, the result of §23, 12 can be maintained in the following way. Let a function E(a) of I ^ have the property that outside
CHAPTER VI. of a finite interval
GENERALIZED TRIGONOMETRIC INTEGRALS
(aQ, bQ ) the relation dkE(a) = 0
holds, i.e., both in
[ oo, aQ ],
and In
[b , oo]. Let
r*(a)
be a given
(k+2)times differentiable function in a closed interval (a, b), a < aO . b^ O < . b. Then there  is exactly one function $(a) of £,K which satisfies the relation d $(a) = o outside of (a , L>0 ), and the relation d^®(a) = r*(a )d^E(a) in
[a, b]. t. THEOREM Let f(x) be a function of 3 ^ and E(a) its ktransform3 ^,
(10)
1 ) If then
f(x)
f^(x) ~ J 2)
is rtimes differentiable in
e(xa) (ia )pdicE(o:),
p = 0, 1, 2,
r.
Conversely, if a function
(11)
q>(x)
~J
e (xa )(ia )'rdicE(a )
exists (i.e., if (Ia)r is a multiplier of E(a)), then f(x) Is rtimes differentiable in 3 ^. (and hence f ^ ( x ) = cp(x)). 3 ) If dkE(a) = 0 outside of a finite Interval (a , bQ ), then f(x) is differentiable arbitrarily often in 3 ^
(12)
and hence by 1.,
f(p)(x)=J e(xo)(ia)pdkE(a),
p=1,2,3,••• •
4 ) If dkE(a) = o inside an interval taQ, b ] which contains the point a = o, then f(x) is in tegrable arbitrarily often in 3 ^, and the rth in tegral (1 3 )
F p(x) has the representation Fp (x) ~ J
e(xa )(ia )~pdkE(a )
.
REMARK. Formula (10) or (13) is to be understood in such a manner that the ktransform (a) of the function standing on the left satisfies the relation
§3 2 • MULTIPLIERS fu )
dk«(a) = (la )pdkE(a )
or (la)pdh(a) = dkE(a )
(1 5 ) PROOF OF 1 ). r = 1,
.
It is sufficient to limit onself to the case
and indeed to prove for the ktransform dkE , ( a ) = iadkE(a)
( 16)
E 1(a) of
f f(x)
that
.
Hence
Considering that
if (x ) = o( x^),
cf. §29, 5, we obtain by a judicious
partial integration and a slight calculation following it
2 * E 1( a ) X 
f
e(ax)  L^Caqx) fCx)^
Therefore a
o which is merely another way of writing (16). PROOF OF 3). Let r(a) be (k+2 )times differentiable, equal to one In (aQ  e, bQ + e ), and equal to zero outside of a finite in terval. We denote by K(§) the function of QQ belonging to it. By Theorem to, the function
CHAPTER VI.
(1 7 )
GENERALIZED TRIGONOMETRIC INTEGRALS
rtf] = 2 ^ / K()f(x  5) d
±3 on the one hand identical with
f(x);
on the other hand, by §29, 7,
differentiable arbitrarily often in PROOF OF k ) . r(a)
Let
aQ < a < 0 < b < bQ .
One can find a function
which is (k+2)times differentiable, and which agrees with
outside of
[a, b].
(ia)”1
The conclusion now follows as in the proof of Theorem
27, making use of 3* and §29, 7« PROOF OF 2).
The conclusion is reached in a manner analogous
to Theorem 25. Indeed we introduce successively the functions PK(a )> r (a), fK, (x),and it must be proved that f(x) is also an rth integral of
cp(x).
For
this it must be shown that
h(x) = g(x)  f(x) is a polynomial in x of degree amounts to H(a) = F(a)  E(a), (18) By §30, 6,
(r  1 ). The ktransform ofh(x) and there is also the relation
(ia)rdkH(a) = 0 h(x)
.
is a trivial function of 3 k ,
and therefore differ
entiable arbitrarily often in 3 ,. Because of the already proved iY*^ assertion 1 ), it follows by (18) that h v '(x) vanishes, Q.E.D. 5. at the following.
In a manner analogous to Theorem 28 and §2U, 5, we arrive In order that a function F(x) ~
J
e(xa) (i(a  x))~ dkE(a)
exist, it is necessary and sufficient that e ( Xx)f(x) be rtimes differentiable or integrable in ^ (+ is valid for differentiation, 
for integration).
What is more F(x) = e (Xx)(e ( Xx)f(x)] (r)
or x
F(x) = e(xx)
f
e( Xx)f(x) dx
.
(r) If the ktransforms E 1 (a) and Eg (a) are kequivalent in the neighborhood of a point a = x., and if a function
§33•
J
OPERATOR EQUATIONS
1
e(xa) {I(a  X )}“rdkE 1(a)
exists, then there also exists a function J
e(xa){i X}~rdkE2(a (a 
§33 *
[ oo, oo].
)
Operator Equations
1. Let G(a) be ktimes continuously differentiable in By a solution of the equation
(i )
G(a)dkq>(a) = 0
we mean a function 5^
)
9(a)
of
which satisfies it, or the function of
belonging to it, that is the function
(2)
y(x)
~ J
e(xa)dk9(a)
.
2. If G(a) has no (real) zeros, then by §30, 6, there is no solution (i.e., there is only the solution y(x) = 0). If it has only a finite number of zeros (3 )
T , ,
V
Tn
,
then only functions of the form XI y(x) = £
(O
y v (x)
with k2 (5 )
y v(x) = e(xvx)
^
cv^ ,
v = 1, 2, ..., n
can occur. The following is a more precise statement. have multiplicity (6)
I.,
..., £ , ..., £
then the general solution of (1) has the form
,
If the zeros (3)
17^
CHAPTER V I . GENERALIZED TRIGONOMETRIC INTEGRALS
(7 )
y(x) = £ / e ( T vx) £ cv[ixtJ v=1 \ i=0
where m y = rain (iy  1, k  2), and the
cy^
v = 1, 2, ..., n
are arbitrary constants.
For example, if
arbitrary kmultiplier, then together with
ry
,
is an
cp(a),
a
f
d (a)
ry(a)dkcp(a)
( k)
is also a solution of (1).
Now if
r (a)
"isolates" the zero
t
from
the other zeros, and if cp(a) is the ktransform of (t), then according to the structure of the trivial functions discussed In §28, 7, cp^(a) is the ktransform of y y(x). Hence each component y y(x) Is by Itself a solution. Conversely, together with the y (x), y(x) is also a solution. There remains, therefore, only to discuss for which Is a solution. By hypothesis
cy^
the function (5)
I
(9 )
G(a) = {i(a  xy )} VG*(a)
G*(a) 4 0
,
in the neighborhood of relation G(a )dkcpy (a) =
a = From this it follows easily that the 0 is synonomous with the relation
(10)
(l(a 
Tv ))^vd k oo
with well determined multiplicities
^
***
*
v
§33
175
OPERATOR EQUATIONS
then the functions x (1 3 )
yx (x) =
Y:
mv /
. V
( e(tvx)
v==1 \ v~
1
( Y 'i
. ,, \
X = 1, 2 , 3 ,
X Cv^)xU) ’ u =0 /
and each klimit of such functions belong to the solution.
Conversely,
each solution y(x) can then be kapproximated by functions (13). To prove the converse, let 7 (a) and K() be functions as defined in §31f and consider the functions
yp (x) =
f
By Theorem 3 9 , they kconverge to
y(OK(p(x  )) d5 y(x)
as
p — — > 00,
dkcpp (a) = 7    dkcp(a )
. Since
,
we have G(a)d cpp(a) = 0
(lU
.
But now (1
5
)
d k cpp (a) = 0
outside of an interval (a , b p ). Moreover for each solution of (1*0 which satisfies (15), it follows just as In the case of finitely many Ty, that it Is a trivial function of the form (13) — with exponents tv in the interval (ap, bp) — . Q.E.D. If only such solutions are desired whose transforms outside of a finite interval to zero, and If no tv has the value a or the functions (13) result, formed with those terval
(a, b) b,
are equivalent
then the totality of which lie in the in
(a, b).
h. If the zeros of G(a) are of a more complicated character, then the totality of solutions of (1) cannot be so easily described.
If, for instance,
G(a)
vanishes in an interval
[aQ, bQ ],
then
the following belong to the solutions: each function (13) with exponents in the interval [aQ , bQ 3 and each klimit of such functions, but for ex ample also each Punetion b (16) ( ix )k 7(a)e(xa) da a with aQ < a < b < bQ, provided 7(a) is two times differentiable, and together with the first derivative vanishes for a = a and a = b
CHAPTER VI.
176
GENERALIZED TRIGONOMETRIC INTEGRALS
[because (16) is then a function of whose ktransform agrees with 7(a), and is therefore equivalent to zero for a < a and b < a ] . 5.
For a given function
(1 7 ) we now consider more generally the nonhomogeneous equation (18)
G(a )dkcp(a ) = dkE(a)
If qp0 (a) is a particular solution of (18), then the general solution of (18) is obtained by adding to 1,
ap(?(ia )pe(8 ffa ) P
or more generally let r
s
p=o 0=0
for
r ^ 0,
where the "principal part" s 7(a) =
£ cr=0
a rae (8 a“ )
,
§33»
OPERATOR EQUATIONS
17?
is essentially different from zero, (22)
r (ct) y S > 0 ,
3y
(»• 00 < a < 00)
the same lemma and analogous observations as in §25, one shows
the following,
if
has no zeroswhatsoever,
G(a)
(la )pG(a )“1 ,
p  0, 1, ...,
are kmultlpliers, and there exists a solution
3
differentiable rtimes in again valid as in §25,
y
( p ) ( x )
More generally if
=
h
then the functions r,
y(x) which is actually
In the case of (20), the following is
5,
f
K Cp)(5 )f(x 
I )
d s ,
P
=
0 ,
i ,
r.
G(a) has a finite number of zeros, then the above nec
essary condition is also sufficient for the existence of a solution y(x) which is rtimes differentiable in 5 But It is determined only to within an arbitrary additive function of the form (7) • For each function (7) Is differentiable arbitrarily often (therefore especially rtimes) in 5k‘ If assumption (22 ) is not made In (21 ), then analogous to Theo rem 3 3 > the following Is valid. If E(a) is equivalent to zero outside of a finite interval (a, b ), then for the existence of a solution rtimes differentiable in 3 It Is sufficient that the above necessary condition be satisfied in regard to the zeros of G(a) falling In the in terval (a, b ). But by (3), the arbitrarysolution of (1) joined to a particular solution of (l8) need not now be a trivial function. But if only such solutions of (18) are desired whose only trivial ktransform is equivalent to zero outside of (a, b ), exponents in (a, b) are added. 7.
then only trivial functionswith
In accordance with the integral equation examined in §26, we
now consider the case G(a) = X  7(a) where
x
is a parameter and
7 (a)
,
is the Otransform of a function
K()
which satisfies the valuation
 K( e ) I < AO + U l k+2)~1
•
If k l 2, there are always "eigenvalues", i.e., numbers x, for which (1) is solvable. What is more, each value of 7(a) Is an eigenvalue. For example, If a = t I s a zero of G(a), then y(x) = e (
t x
)
CHAPTER VI. is a solution of (i).
GENERALIZED TRIGONOMETRIC INTEGRALS
This can be recognized by the relation
J
\e(xx) 
e(r6 )K(x   ) d£ =
G (
t
)
* e(rx)
.
If 7(a) isanalytic,asforexampleif K() = 0 (eal5 1 ) ,
a > 0
,
x,
then for each eigenvalue G(a) has finitely many zeros (3) with multi plicity (6 ), and moreover the eigen functions are given by (7 )* Concerning the nonhomogeneous equation, the following is valid, similarly as in §26, k. For the solvability of
(x  7 (a ))d^cp(a ) = d^E(a) for a given X 4 it is sufficient that 7(a) ‘be (k+2 )times differ entiable and be absolutely integrable together with the K + 2 deriva tives, and that one of the following conditions be satisfied. 1) G(a) is nowhere zero. 2) G(a) has (what is always true of analytic G(a)) only finitely many zeros tx with multiplicity £x , and for each x, the function e ( Txx)f(x) is
3
&x times integrable in
3) Finitely many intervals the following character. Each zero of
< a < ^ can be specified of G(a) is contained in the interior
of one of the intervals, and in each of these intervals
E(a)
0.
In the case of 1 ), we have again as in §26, the representation by means of the solving kernel. §3^.
Functional Equations
1. If the given function f(x) vanishes in one of the equa tions (A)  (F) of §22 (homogeneous case), or more generally belongs to 3 ^, and if the given function K(g) satisfies a valuation (1 )
KU) < A O
+ U  k+a)_1
in case (F), then one can Inquire about those solutions which are rtimes differentiable in 3 ^. Here it can be assumed that the index k is fixed, or it can be permitted to be arbitrarily large. If k is fixed, the equation
§3^
FUNCTIONAL EQUATIONS
179
G(a)dkcp(a) = dkE(a)
(2)
hold3 for the ktransform, and the results of the previous paragraph are, basically, results concerning the functional equation (
Ay  f(x)
3)
belonging to (2)* the index
classes
k
The results become considerably more significant if
is not held fixed.
By the function class 3 we shall mean the union of all function 3 so that each function of 3 belongs to a certain class 3
All functions therefore belong to 3 which increase more weakly than a power ]xm for a sufficiently large m. Let
3,
f(x) now be a function of
and in the case of equation
(F), let (O
]K(  ) < V '
+ I*!)
n = i, 2, 3
By a solution of (3), we mean a function y(x) which satisfies it and which is rtimes differentiable in 3 , i.e., it, together with the first rderivatives, belongs to 3 . 2.
THEOREM r~i
•
The equation
s
(5 ) p = 0 cr= 0
always has a solution.
This solution is unique to
within an arbitrary additive function of the form t  1
(6)
where the numbers
t1, ..., tn
are the zeros of
G(a) and whose multiplicities are the
1 1, ..., 1
[83]. The same statement is valid for the general equation r s
r k 0
(7) p=0 a=o
CHAPTER VI.
GENERALIZED TRIGONOMETRIC INTEGRALS
whenever s (8 )
X
arae(Baa)
PROOF.
Let
( oo < a < oo)
^ S > 0,
f(x)
be contained in
k ^ k 1.
Set
+ ••• + * n + 2
k ,  ko + ii + and consider a fixed
*5 v . o
’
By §29, 8, the function e( Txx)f (x)
is fx times integrable In
for each
X.
Hence by §3 3 >
our equation
has a solution in 3 ^,. Since k ^ k 1, min. (iy  1, k  2) = &v 1. Therefore the solution is unique to within an arbitrary additive function of the form (6). And since the special value of k does not enter the structure of (6), all solutions have thus been obtained, Q.E.D. — If no zeros at all of G(a) exist in ( 5 ) then one can again write y ( p ) (x)
=
Uf
K ( p ) ( 6 )f(x 
1)
di
,
P
=
0, 1 , 2 ,
One can gather from this representation, that not only the membership of f (x) to a class 3 , Is transmitted to y(x), but also that other more ( ) intimate properties of f(x) are transmitted to y p (x), 0 < p < r  1, and due to r1 y (r)(x) = f(x)  £ P=0
s ^ } V ay (p)(x + 8a ) 0=0
also to y ^ (x). For example, if y(x) is bounded, uniformly continu ous, of bounded total variation, etc., then y^p ^(x), 0 < p < r, is also bounded, uniformly continuous, of bounded total variation, etc. If f (x) = 0 ( x n ), then the same valuation holds also for the y ^ ( x ) . If f (x) Is almost periodic, so also is the y ^ ( x ) . — ■ The same ob servations can also be made for the more general equation (7), but we shall not pursue this subject further. If restriction (8) is not satisfied by (7), but the transform of the given function f(x) is equivalent to zero outside of a finite interval (a, b), then again at least a solution exists. But the arbitrary solution of the homogeneous equation cannot be described as simply as in Theorem b 5 ,_ unless one limits oneself to such solutions whose transforms are likewise equivalent to zero outside of (a, b).
§3^.
181
FUNCTIONAL EQUATIONS
3 o The statement of Theorem ^5 is valid, for fixed for the integral equation vy(x) 
±f
y({)K(x  I) d = f(x)
l / 0, also
,
provided K() satisfies (k ) and G(a) has only finitely many zeros with well determined multiplicity conditions which are fulfilled, for example, if K() < Ae~a 11 1
,
A > 0, a > 0
.
A well known example is [8if] K(t) = 2 *6 ^ 1
,
in which case (9 )
Xy(x) 
J
e~lx ”^y() d£ = f (x) .
In this case \(a )
= — — ?
,
G (a)
=
The numbers
x < 2
X
— =•
1 + a
1 + a are eigenvalues.
The equation
yields f 2 "x “ = t j — r 
and these zeros are both simple. the totality of functions of
°>e (
If
f(x)
belongs to
The totality of eigenfunctions (out of
8f ) consists therefore of the functions
* ) +
5 ,
x )
the nonhomogeneous equation (9) always has a
solution (in $ ). The solution y(x) must belong to the lowest class 8 ^ in which f(x) is already contained, except when X is an eigen value. If f(x) is bounded, y(x) need not likewise be bounded. b . We shall not, in the present chapter, go into the system of functional equations. But in the next chapter, we shall have the oppor tunity to examine certain special systems in a concrete connection.
CHAPTER VII ANALYTIC AND HARMONIC FUNCTIONS §35.
Laplace Integrals [85]
1. We shall examine analytic functions of a complex variable. As frequently done, we denote the complex variable by a =
s = cr + it,
9i ( s ) ,
By a strip
[X, i], we mean all the
x < o < ia.
It is also admissible to
t =
3 (s )
points of the complex plane for which have
X =
 »
or
\x= «>,
in which
case a left half or a right half plane or the whole planeis involved. (X,1, m! ) we shall always mean a real closed substrip of [X, nJ, ( 'X < ) X ] < a i ^ (< n ). 2. In 0 < a < 00, let E(a) every finite interval. If the integral
be a given function integrable in
f s a a E(a) da
(1)
o converges for a certain integral (2)
crQ,
then since for
f(s) =
J
a a a < aQ : e aa < e 0 ,
the
e 3 a E(a) da
o converges absolutely and uniformly for that the function n fn (s) = / esaE(a) da , o is everywhere differentiable.
9t(s) < a .
It is easy to show
n = 1, 2, 3, ••• ,
Therefore it is analytic, and indeed
1 82
By
§35.
183
LAPLACE INTEGRALS n
f^(s) = J
aesaE(a) da o
Since the sequence of analytic functions fn (s) converges uniformly as n > 00 to the function f(s) in the open region 9i(s) < a , it follows hy a general theorem that the latter function Is likewise analytic there. Because of lim a r e a a ~a° a = 0 , a >o
a < u ,
r = I, 2; 3? •••,
the integral J
aesaE(a) da
o is absolutely and uniformly convergent in each partial half plane w (s ) < 01( i;
q > o.
f(.) ■/ (. .*>
.
o Hence for
c > o
1 0 rTq7
c+i«
(12)
271f
Cioc
If
q = i,
 X)q“'
for
o < x < 1
for
1< x
o]
1 88
CHAPTER VII.
ANALYTIC AND HARMONIC FUNCTIONS
1
C + loc
2 k
/
V
d s
■
j
°“ico If
q
’ / 2
I o
is a positive integer
> i,
for
o < x < 1
f o r
x
for
“
1
1 < x
(a) and ip(at), with the aid of a suitable function E(a), stand to one another in the relation 2&( 9* If the function u(0, t) joins itself kcontinuously to a function u(x, t) as 0 — > x, then the transform E(a, oc) goes over to the transform E(x, a) continuously. By §30, 9, (8) is therefore also valid
for0  X.
An analogous remark Is
If the analytic function function
f\(t)
as
0 — > X, fx (t)
f(s)
valid as
0 *> ji.
joins Itself kcontinuously to a
then e (x+it)adkE(a)
.
§3 9 * Boundary Value Problems for Harmonic Functions l. In a strip [X, 11], let a real harmonic function u(0, t) be given, where x andn, until further notice, are finite numbers. We say that the function u(0, t) belongs to the function class 0 in [X, ^], If it belongs to A k for a certain k. If for sufficiently large k, u(o, t) joins itself kcontinuously to the boundary value u(x, t) as 0 — > x, then we say that u(0, t) belongs to A in (x, jj.]• An analogous remark Is valid for 0 = u . The assumption that u(0, t) belongs to ft in (x, n ) is supposed to signify that for sufficiently large k, a kcontinuous joining takes place at the boundary value not only as 0 — x but also as 0 > fi. In this sense, for example, the statement that the function
§39*
2 09
BOUNDARY VALUE PROBLEMS
(1 )
a2 (q,t)
likewise belongs to
A
in
(x, n)
is to be understood as follows.
According to.§38, 7 , not only is the function (i) contained in A ^ in [x, n ] , but also a suitable k = k f exists which if necessary can be larger than the k needed until now, such that (1) by approximation of a to x and \x, is kconvergent to two limit functions. We shall denote them purely symbollically by d2u( X , t ) dcrBt ’
d2u(n,t) ScrSt
For what follows, there will be no restriction of generalness if we set
X  0, (i = i.
THEOREM ^9. If the harmonic function u(a, t) of A joins itself kcontinuously to the value zero at both boundary lines, then it is identically zero. PROOF. (2 )
Since dkE(a, a) = eaadkE+ (a) + eaadkE (a)
is also valid for the voundary value (3) (it)
a = 0,
therefore
dkE +(a) + dkE"(a) = o e+adkE + (a) + e"adkE"(a) = c
.
From this it follows that (5 )
(1  e2 a )dkE+ (a) = c
.
Hence (6)
adkE+ (a) = c* . ,
Because (7 )
2E (a)  E(a),
2E (a) = ( 1 )KE( a)
we obtain
(8 )
adkE(a) = 0
,
210
But
CHAPTER VII.
E (a )
ANALYTIC AND HARMONIC FUNCTIONS
w as th e k tra n s fo rm o f th e fu n c tio n f ( s ) = u (cr, t ) + i v ( a , t )
(9 )
,
an d ( 8 ) s a y s t h a t t h e d e r i v a t i v e o f f ( s ) v a n i s h e s . H en ce f ( s ) I s a c o n s t a n t . T h e r e f o r e u (cr, t ) I s a c o n s t a n t , an d b e c a u s e o f t h e k c o n tin u o u s a p p r o x im a t io n o f u ( a , t ) t o t h e v a l u e z e r o , t h i s c o n s t a n t v a n i s h e s , Q .E .D . A g e n e r a l i z a t i o n o f T heorem THEOREM
5 0.
k9
i s th e fo llo w in g .
Let
m
m
n , v= o
n, v= o
(10)
b e tw o g i v e n f u n c t i o n a l s w i t h r e a l c o n s t a n t c o e f f i c i e n t s a ^ v , b ^ y . L e t i t b e known o f a f u n c t i o n o f ft , h a rm o n ic i n [o , l j , th a t i t d is p la y s th e fo llo w in g b e h a v io r a t th e b o u n d a ry. 1 . The h a rm o n ic f u n c t i o n u0 (cr, t)
i s c o n t a in e d i n m ore
». In [0, ») It Is monotonic. For where its derivative vanished, we would have
But for
a > 0,
4a < e2a  e~2a.
this equality cannot be true since
Further the function on the left of (17) is even. We have therefore the result that for q < 1, G(a) has no additional zeros; for q > 1, It has two symmetrical zeros t2
both of which are single.
=
t
> 0
T
3
T
E(a, 1 )
and because of (6 ) it follows from the first that a
j
(15)
Since
fn (x)
belongs to
a
>J 0
Fn (P) dp
F(p) dp
d ] Q2
tx
En (a , 1 ) = 1 ^ ( 0 ,
+J
1)
F n (P ) dp
,
0
an d fro m ( 1 4 ) an d ( i 5 ) f o l l o w s a
E ( a , 1 ) = E ( 0, 1 )
+J
F ( p ) dp
.
0 S in c e i f
f(x )
b e lo n g s t o
r*^2 ,
th e i n t e g r a ls
(7
TS
converge, therefore 2jtE(a, 1 ) x
Jn
ee(ax) \ a x )  n a,xj L.(a,x)
f (x)
:tTx
dx —
n
,
Prom this, the following theorem results. THEOREM 5 4 . For the Plancherel transform of a function
f(x)
2
of < 1 Q,
,
J f (x) i — ?.jLr, 1 dx
we have the relation [103 3
22k
CHAPTER VIII.
(1 5 ')
QUADRATIC INTEGRABXLITY'
1 Z
F(a) If
f(x)
_  l z « 2L )
d x
is even or odd, then one can also write
for (15')
0 or
m A/»*> ’T 0
” a* •
7  A partial generalization of the theorems of this paragraph reads as follows [1ot]. Consider a function
f(x) J
is finite.
If
1 < p < 2,
(,6>
for which
f(x)p dx
then the derivative
F a, A
J
A
®a (y)2 dy » 
J
A Fa (y)2d 1 =  ^ Pa (A)2 + 2 J '
®a ( y ) g ( y) dy
r A 2 jT
1/2
®a ( y) g ( y ) dy < 2 ^
dy
r /
a
1/2 dy
g ( y ) 2
From this it follows, by squaring the outermost terms of the inequality, and then removing a common factor, that
a jf a Let
b
and
B
.H. * a ( y ) 2 dy < i+ jT g ( y ) 2 dy < i+ jT g(t ) 2 dt .
a
.
o
be two fixed numbers with
B > b > o.
Then for
o < a < b
B
f
(3)
b
°a^
2 dy ^ k /'s(t)2 dt o
But in each fixed interval (b, B), the function to cp (y) as a > 0 . Hence by (3) O D
f By letting
b 2.
dy 
kI
s ^t)2 dt
> 0, B — > », (2) results, Q.E.D. Now consider the function
.
$a (y)
*
convergent
226
CHAPTER VIII.
QUADRATIC INTEGRABILITY 1/a
*g (a) = J
g ( t ) dt
o Hence
j
* 8 ©
= f
f
o
g(t) dt
= V
y)
From this, with the aid of the variable transformation
y = ~,
one finds
/© *g© 2dy=I Va)2da o, and the sequence Fn (a) is convergent in the quadratic mean as n — — > ». The limit function F(a) = I .m. J
(6)
Sy(ax)f(x) dx o
is naturally again a function of 2 ).
32 •
The relation / F(a)2 da = J f { x )2 dx 0 0
(7) holds. 3 ).
The reciprocal relation f(x) = i.m. J
(8)
Sy(xa)F(a) da o
holds. REMARK. with (12 ) of §1+1 .
If
v = 
Sy(t) = J \
then
and ( b ) agrees
cos t,
PROOF of 1). Together with the proof, we shall show the ex istence of a constant c, independent of f(x), such that J
(9)
Because
v) > 
F ( a )2 da < c J
the function
f(x) 2 dx
Sy (t )
And because of the asymptotic behavior of
is bounded in finite intervals. J y(t),
wecan set
sv(t) ■ / ¥  003 (t  i  ¥ ■ ) + RCt)
*
where R(t ) i A
for
t< 1
R (t )
1
and
We now introduce the three functions
(10)4 ,(a ) = t .m. J F J
$2
cos(^ a x  £ (x ) dx , 0 1J a = f R(ax)f(x) dx, 0)^(a)=y^ R(ax)f(x) dx 0 1/a
228
CHAPTER VIII.
QUADRATIC IMTEGRABILITY
Because of = cos ax cos^J +
sin ax sin
,
and by the results of the previous paragraphs, the right integral in (1o ) actually exists, and therefore (11)
J
®1(a )2 da < C] J
0
f (x )2 dx 0
Further 1 /a I$2 (a)  < A
f (x ) dx,
J
2 (a) and ®^(a) are functions of 3 2 . They satisfy certain estimates J'
(12)
(a) 2 da < Cp J
'0
f(x)2 dx ,
(p = 1, 2)
0
But now 1 (a) + $2 (a ) + ®^(a) = Mb.
J
Sv (ax )f (x ) dx
,
0 and we have thus proved that the integral (6 ) exists. a valuation (9) holds for the function F(a) = The function the function
tions of
F(a)
 oo and bx — > + » or as cx — > + ». We shall denote this limit as the value of the integral in question. We call the integral (4 ) absolutely convergent, if the integral in question converges for the function
F(x1, ..., *k ) I• We shall say that the integral J
F(x1, ..., xk ) dx
exists as a Cauchy principal value, if the kfold limit of the integral A
F ( x 1, . . . , x k ) dx
f
A exists as
A —
> + «>*
If the function
F
can be written in the form k
F(x,,
xk ) =
n
F x(xx }
’
X= 1
then the integral (*0 is convergent, absolutely convergent, or convergent as a Cauchy principal value, if for each individual X, the (simple) in tegral f
Fx (x) dx
or
J
Fx (x) dx
o is convergent, absolutely convergent, or convergent as a Cauchy principal value. 2. In the main we will consider a function f(x^, — which is defined and absolutely integrable in the whole space. Then the trigonometric Integral E (a 1, ..., «k ) exists for all
a.
"(2jt
f
f ^x 1' *’’7 xk^e  “ Yj
j
dx
We shall denote it as the (Fourier) transform of
f.
, xk )
§1^3 • It
is
INTEGRALS IN SEVERAL VARIABLES
233
very easy to see that the transform Is hounded and continuous.
Further it is convergent to zero as Ic^  + ... + ak   > oo However we shall rot usethis fac* 3
If f u , , x k) =
and each individual factor
n
fx (xx ) x is absolutely integrable in
fx (x)
[~ «>, «],
then E(a,,
ak ) = J] W X
where Ex (a) = For example, in the interval
J
fx (x)e( ax) dx
0. < ax  < n^ , 2
sin
(6 )
and for other values
(a1, .
)
the integral is
o.
Further if
i A il
a
ax > o X
2
(7 )
r e ^ axxx
dx =
S'
„k /g _
^
X
h.
More generally the integral r
J(a, a) =
je
ly sl,.x Y x ,  i Z Ya Yx Y x,x xx x x x xd x
k
can be computed from the above provided the quadratic form in the a ^ symmetric and positive definite. Indeed we shall show that it has the value [i08] W(a, where
a)
h/2 ,, = 3L_— ep^ /D
,
Is
CHAPTER IX.
a il
••'
FUNCTIONS OF SEVERAL VARIABLES
0
a !k
D(a) =
3
*k1
•••
“1
“l
D(a)p(a, a) =
“Mt
“k
•• •
ak
• “ ik aki " •
akk
Thus we claim the identity J(a, a ) = W(a, a )
(8) Because (9 )
X
a* x V x i A £ xx
x, x
A > 0
'
,
x
the integral J(a, a) Is absolutely Integrable. Hence one can apply to It arbitrary coordinate transformations with continuous partial derivatives. The affine transformation
(1 0 )
XX ‘ X
W
x
with positive determinant A = I ?xxl gives on the one hand
J x x xxx x + XX
■ ^ x ^ x X
+ {ix^x
XX
X
where ■
&
■I".
and on the other hand (11)
J (a, a )
=
J(b, 0 )
• A
But now by formulas in the theory of determinants, we have D(b) = D(a) 8 A2,
D(b)p(b, 0) = D(a)p(a, ct) e A2
and therefore (1 2 )
W(a, a) = W(b, (3 ) • A
.
,
H3‘
235
integrals in several variables
But the transformation (io) can now be determined in such a way that
x . In this, special case, however, the relation J(b, p) « W(b, P) holds because of (7).
By (11) and (12), (8) now also follows.
5 . THEOREM 56 C109]. function f(x1, ..., r = */x? + ... + x,
If the absolutely integrable ) depends only on the quantity i.e.
then the function
in the same way depends only on the quantity a = Jct^ + ... + a^. The kfold Integral (i4 ) can be expressed by a single integral
J(Q!) =ffk'2T72'/0 0. In polar coordinates
is to extend overthe
x 1 = r cos e, r
lies between the limits
0
p = r sin
and
and
0
upper half plane
,
between
0 and
«. Hence
J cp(r)rk~1K(ar) dr
J(a) =
0 where n
K(t) =
e( t cos 0)(sin 0 )k ~2 d0
J
0 But now, cf. §16, (1 )
J
v
(t) =  — —  r r(v+i/2)r(i/2)
n e( t cos 0 )(sin 0 )2v de
From this and (20), (15) results, which is also valid for
Jl/2(t)  / j S 003 t
k = 1.
Then
'
One gains (17) and (18) from (15), if in (15 ) the expression
I =_2 \ m d®_
Jp (“p)
aP
is inserted for
\c
p = — ~—
\ 2
I
Jpm(/3p) \
P I dsm ( vTSPm /
. This expression results from the substitution
238 of
CHAPTER IX. x = /sp
FUNCTIONS OF SEVERAL VARIABLES
in the formula
EXAMPLE.
For odd
k, u > o,
is obtained very easily from (18). (17) for even
k
The same relation is obtained from
by means of the formula
which results from differentiation of [111]
6 . Again the Faltung rule holds. f ^(x 1
Let
.. ., xk ),
be two given absolutely integrable functions with an equal number variables. Then there exists the kfold integral
of
for almost all points of the kdimensional xspace, and the resulting function (the ’’faltung”) f(x1, ..., xk ) is again absolutely integrable.
The proof proceeds just as It did in the
case k. = 1, cf. § 13 > 3> because the theorem of Fublni used in the case k = 1, in regard to the interchange of the order of integration In a two fold integral is still correct if each of the variables x and y runs through not a linear Interval but a kfold interval (cf. Appendix 7> 10). In a similar way the proof carries over that the Faltung of absolutely Integrable functions corresponds to the multiplication of the transforms
§AA. E (ex.j,
FOURIER INTEGRAL
239
ak ) = E 1(a1, ..., ak )E2 ^a i, ..., ak )
.
Thus for example, the transform of the function
■“ >...... a
* / 
.....
has in the interval
0 < ax  < n^,
(22')
E(“ 1< .... ak ) J
and the value
0
for other
the value ( i 
(a1, ..., c*k );
J
while the transform of the
function r J f(jv
/ 0 5 v
(2 3 )
\ ”Exnx^xx"yx^ •••> 7k )e
,
dy
has the value
“ ,/4Z*x
k/2
“x n
E(a1, ..., ak )e
X
^ n i ••• nk The Fourier Integral Theorem 1.
Let
cp( , .•., lk ) be a given bounded function, and
K( 11, ..., lk ) a given absolutely integrable function In the '’octantM
(i )
o < X
xk + o )
For the relation on
Eta,, ..., ak ) = —
 p fI
xv )e (I  J)T a*x* f(x,, ..., xk v S c j dx
the inverse formula (13)
holds.
f(x,,
xk ) = f s i a , ,
■■■,
a k )e 
j
We shall establish criteria for its validity.
integrable
(it)
...,
f,
da
If for absolutely
the integral (12) is substituted in
f
n
E(a,, ..., ak ) ( J" axxx \ \ x /
da
and the order of integration interchanged, we obtain sin n ^ x x ex ) d! The question arises under what conditions this Integral converges to f(Xj, ..., xk ) as nx  > 00. However we shall leave the examination of this question for the next paragraph, and shall now make another statement. Let us form with the absolutely integrable function f, the expression (22) of §1+3, and denote it by f . According to (22' ), the integral (13.), formed with the transform of f , amounts to
CHAPTER IX.
2k2
(1 5 )
f
FUNCTIONS OF SEVERAL VARIABLES
a k ) J]
E(a,,
(
e
l 

j
da
Substituting the integral (12) herein, and interchanging the order of in tegration, the function f results after a slight adjustment. Hence the inverse formula (13) is valid for
f .
By Theorem 5 7 , we now obtain the
following theorem. THEOREM 58.
If
f(x^
..., xk )
is absolutely in
tegrable and bounded, then the inverse formula (13) holds at each point at which f is continuous (or more generally has the limit (n )), provided the in tegral (13) is interpreted as the limit of the in tegral (15) as n^ > 00, i.e., provided the in tegral (13) is summed by the method of the kdimensional arithmetic mean. If the limit of (15) exists, and at the same time the integral (13) exists as a Cauchy principal value, then by a general theorem of summation related to the method of arithmetic mean, the two values are equal to one another [Appendix 18]. Therefore from Theorem 58, we obtain the following. THEORM
59.
if
f(x^
..., xk ) is absolutely in
tegrable and bounded, then the inverse formula (13) holds at each point at which the integral (13) ex ists as a Cauchy principal value and f(x.j, ..., xk ) is continuous (or more generally has the limit (11)). 3.
The following theorem is of a totally different kind.
THEOREM 60. If f(x1, ..., xk ) and E(a1, ..., ak ) are both absolutely integrable, then the inverse formula (13) holds for almost all x. Since for absolutely integrable E, the integral (13) represents a continuous function, therefore f, after correction if need be on a point set of measure zero, is a continuous function, and for continuous f, (13) is valid everywhere. If f Is absolutely integrable and bounded, then, at any rate, E Is absolutely integrable if it is of uniform sign, E > 0.
§kk.
PROOF.
Let
f
and
2k3
FOURIER INTEGRAL E
be both absolutely Integrable.
The func
tion fn considered in 2 . has the value (1 5 ). Since E Is absolutely in tegrable, for fixed x, the integral (15) is convergent to the integral (13). On the other hand, by Theorem 5 7 , fn is convergent to f, if f is known to be continuous and bounded. Thus, for absolutely integrable E, the inverse formula certainly holds if f Is also continuous and bounded. In the general case where only absolute integrability is known of we introduce the function, for fixed
f(x),
h^ > 0
h (16)
Fh ( x } ,
.
It is known from the theory of integration that the function (16) is con tinuous (Appendix 9) . Moreover it is bounded h
If a constant factor is disregarded, (16) is the Faltung of the function f
with that function which has the value
1
in the interval
“ \ \ < 0 and vanishes otherwise. The function, F^, therefore, is also absolutely integrable. Its transform computes easily to Eh (a1, ..., ak ) =
...,
where
Because (1 7 )
l 8 h (cV
“ k
} l
£
1
the transform Is likewise absolutely integrable. already established we have
( 13)
Hence by the special case
da
for all^ x. Relation (13) in the general case will be deduced from this by a passage to a limit. We let h^ — — > 0, for which 5^ converges to 1. Moreover since (17) is valid and because of the absolute integrability of E, the right side of (18) is convergent to the rightside of (13) as h^ > 0. On the other hand, we know from the theory of Integration If in (16) the numbers hx are say, equal to one another,
2kk
CHAPTER IX.
FUNCTIONS OF SEVERAL VARIABLES
h 1 = h2 = ... = hk = h and the common value function
Fh
Is allowed to decrease to zero, that then the
Is convergent to
for almost all
El
f
for almost all
x.
Hence (13) holds
x.
Now let and let
h
,
0.
f
be absolutely Integrable and bounded;
f < G,
We gather immediately from (22) of §43, that likewise
\?n\ ^ G for all
By the observations in 2 , the inverse formula Is now
> 0.
valid for obtain
fR
at all points.
f
E(a,, ..., ak ) [[ ^ 1 
If fixed numbers
ax 1 0
f a is valid for
If it is applied to the origin
da < G
are now taken, then because
Efa,, ..., ak ) JJ (1 x '
n^ > a^.
J
ini
By allowing
xx = 0,
we
.
El
0,
) da < G x
nx > »,
'
we obtain
a J
E(a1, .•
ak ) du < G
.
a And since the numbers
ax
can be arbitrarily large, this says that
E
is
absolutely integrable, Q.E.D. This presents the question of how to recognize for a given function
f
whether its transform is absolutely integrable.
Delicate
criteria do not seem to be known. But a rather obvious criterion [113] is the following. The function f has the 3 derivatives
(19)
p+..+p
^
*
a*,
...sxk k
.
0,Px,S
and these are continuous, absolutely integrable, and convergent to zero as x, I + ... + xv   > »
.
§ U.
FOURIER INTEGRAL px = 2 ,
(For the highest derivatives:
21+5
the decreasing to zero part can be
abandoned, and certain discontinuities can also be allowed.) — * For if the expression for the transform E p r pk of the function (i9 ) is formed and integrated partially, there results apart from a constant factor, the function
Pi
(2 0)
a1
Pi,
... «k KE
.
Since the functions (19) are absolutely integrable, the functions (20) are bounded. Hence the function (1 + a1 2 ) ... (1 + ak 2 ) • E is also bounded.
tain for
(2 1 )
Therefore
E
is absolutely integrable.
An application [1 1 4 ].
5. u > 0
“ u / a ? + • • • +ot? e
r !^ ~ • u r
k • —
e ( ih a v xv ) dx
fef"/ It 2
By a Faltung, we find for
By the inverse of (21), §1+3, we ob
"
k£i
•
(u2 +x2+ . ..+xk ) 2
u > 0, u* > 0
(22)e (u+u ') 7 “ i +.. +“ k =r ( ^ n ) u u J' r P(X,,...,xk)e(zxaxxx)dx , where
p (x 1, . . . , x k ) = /
(k + 1 ) / 2 { [ u + y , + . . . + y k ) • [ u ' t ( x r y , ) + . . . t ( x k  y k )2 ] )
By the combination of (21) and (22), with the aid of Theorem 61 which follows, there results
r
TZ?T772 = —
[(u+u')2+x2 +...fX2]
(*T) r : 1W.
uu'
„(k+l)/2
•••»
•
•
CHAPTER IX.
2k6
FUNCTIONS OF SEVERAL VARIABLES
Integrating with respect to
u ! from
u*
to
«,
we obtain
p p p (ki )/2 [(u+u*) +x1+...+xk ]
(23) r i [ =^ r1)i u“ r ____________________ ____________
£+1
dy
) /2
^ 9 1 /2 ^ [u2+y2+ ...+y](k+l)/2[u2+ (xly1 )2+ ... + (xky k )2] (kl)/2.
6.
THEOREM 61.
It (k+l)/2
~
p
p
(k+
J
If two absolutely integrable functions
have the same transform, then they are identical (al most everywhere). PROOF.
The difference of both functions has the null transform,
and by Theorem 60, therefore, this difference likewise has the value zero, Q.E.D. 7 ® In Theorem 58, we have summed the Inverse integral by the method of. the arithmetic mean. A rather meaningful convergence criterion Is arrived at, if one sums the inverse integral in the way that one forms the "spherical" partial sums
(2k)
fR (x1, ..., x k ) =
J
E ( a 1, ..., a k )e(zxa xx x ) da
.
KR Here
KR means the volume p
p
p
a i + ••• + ak < R
•
Inserting (12), there results (25)
fR = c,
J
f(x1 + 11,
xk
+ 5k ) % ( l ), •••, !k ) d
,
where
hR ■ /
e( V x V
da
'
kr
and c1 (later dimension k.
c 2 , cy
...)
If for brevity we set
denotes a constant dependent only on the
§41+.
FOURIER INTEGRAL
+
y*
and
247
k  2
we obtain by Theorem 56 c %
= —
R J
Rl rk//2J ^ ( l r ) dr
= c 2 i~k jT
r ^ + 1 J ^ ( 7 ) dr
0
=v 'V £7 [^ V iH * °a(f )ULiUR) • 0
For fixed x,
we now introduce the function of one variable
(26)
F( ) =
J
f(x1 + IX^
..., xk + Xk ) dm
,
k~ 1 S 2 2 where S means the unit sphere xl+ ... + x^. = 1, and dm denotes the (ki) dimensional element of volume of this sphere, and its volume. F() Is therefore the mean value of f on the sphere of radius  around the fixed point (x1, ..., x^). For example, for k = 2 2 it F(i) = ~
J
f(x1 + I cos cp, x2 +  sin cp) dcp
.
0 If in (25), we integrate for absolutely integrable f, first over the sphere of radius t, and then over the variable t from 0 to », we obtain (27)
fR^x i> •••’ V
= c3
f
” °3 / 0 THEOREM 62. (28)
V j ^ C R t ) dt
p
( h ) t ^ + i(t)dt
In order that the partial integral fp(x^,  , x^)
converge, for absolutely integrable
f,
to a
•
248
CHAPTER IX. finite number as
FUNCTIONS OF SEVERAL VARIABLES R — — > °°,
it is sufficient that
the spherical mean value F (t ) defined by (26) be of the following character. If we denote by V the quantity (k2)/2 when k is even, and the quantity (ki)/2 when k is odd, then the function F(t) is Vtimes continuously differentiable In (0, 00] t and each of the functions g (t) = t xP(t), g '(t), g "(t),
g U ) (t)
has an absolutely integrable derivative in
[0, »].
The limit of (28) has the value F(0) =
REMARK.
lim F(£) £>0
The following is to be noticed in regard to the ex
istence of the function F( £). The function f (£ 1, ..., £^) is by assumption integrable everywhere in finite intervals. Introducing spheri cal polar coordinates around the point (x^ ..., there results by the theorem of Fubini (Appendix 7 , 10), that the integral (26) exists for almost all positive £, and represents an integrable function in finite intervals. And our convergence condition demands that the function F() after suitable modification on a null set, fulfill the given assumptions. PROOF. By a suitable application of Theorem 6 to (27), we ob tain the result that the limit of (28) exists and has the value c^F(o)
.
That the constant c^, .which is independent of f, has the value obtainable from Theorem 60, since for instance for the function
1
is
(xp...+xp the corresponding value 8.
F(o)
must come out.
We record without details, the following theorem.
THEOREM 63. Theorems 52 and 53 and the definitions underlying them are also valid for functions of kvariables [1151 • We need only replace dx everywhere by dx] ... dx^. The par ticulars to Theorems 52 and 53 were so set up that they can be carried
§45
DIRICHLET INTEGRAL
over to the kdimensional case dy the results of the present chapter. §45. 1. (1)
In
The Dirichlet Integral
the closed 0
"octant" 00)
toi •" ^ki
•
X=1
for
k  1 variables, we
lk
a , we have by 2
0
0,such
.
that for
m = 0 , 1 , 2, 3, ...
a = 2 «m, and
J
< ^ Max ^ a
> a
4t a
.
252
CHAPTER IX. If
FUNCTIONS OF SEVERAL VARIABLES
therefore the function
f
in (i) is nonnegativeand
mono
tonically decreasing, and if (9)
= 2*mx ,
mx = 0, 1, 2, 3, ...
,
then by Theorem 6 4 , for the integral
n
J(a, b) = /
_ sin x xk ) J[ — —  dx
f (x , .•
j
K
a subject to
b K > a R,
k
we have the valuation
0 < J(a, b) < {h~  T T ~ ~ T^
(’O in which
A
denotes a numerical constant and
l
K
a bound ofthefunction
f. Let m
2k
numbers
px , ctx
be given, and nonnegative integers
such that
(11)
p x
> 2mxn,
ax ^ 2mx*,
x = 1, 2, ..., k
.
In the expression J(a, b), set the value (9) for ax, and either px or c?x for bx . In this way there result 2k integrals, all of which satisfy (10). If these integrals are provided with suitable signs, and then added, we obtain the integral
J(p, a).
By assumption
(11), therefore, the valuation IT/ I J
M
(p /
a , l
2 kAK
, £
(in
1'
+ T ' ) ~
7(
n i p " f }
is valid. From this it is recognized without difficulty that the integral
is convergent for each nonnegative, monotonically decreasing function And this convergence is uniform for all functions f which satisfy a common valuation  f ( X l ,
X k
)
< K
Because of this last assertion, the integral
.
f.
§t5
DIRICHEET INTEGRAL
(«VL uniformly in the
nx , nx> c,
with sufficiently large converges, as
px *
253
) ? ^
3
is approximable by an expression
But with
held fixed, this expression
px
to Pv
(1 2 )
ft+ o
,
+ °)ni
f
sin x
o
x
where f( + o, ..., + c) denotes the limit of the function f(x1, ..., x^) as xx > o. We assume the existence of this limit without engaging in the question whether it does not follow out of the monotonic assumption. For arbitrarily large px , the second factor in (12 ) differs arbitrarily little from i. From this we obtain the following theorem. THEOREM 65 [n 6]. If in (1) the function f(x.j, • x^) is nonnegative, bounded and monotonically decreasing, then the relation
Io\k jr f(x,,
lim
(!)
nx>“
'
o
tt sin
..., xk ][X
dx=f(+ 0, ..., + 0)
x
holds. if.
In (1 ) let the function
f
be continuous, convergent to
zero as x,  + x2+ ... + xk 
> 00
and have the derivative
which in (1) is continuous and absolutely integrable. We shall show that in (1 ), f can then be represented as the difference of two functions, each of which in (1) is nonnegative, bounded and monotonically decreasing. We introduce the functions F 1 and F2, the first of which agrees with
CHAPTER IX.
25^ F wherever agrees with
FUNCTIONS OF SEVERAL VARIABLES
F > Q and vanishes otherwise, and the second conversely F wherever F < o and vanishes otherwise. Hence TP1  r2 f = F r
The functions P = 1, 2
fp  y'PpCi,, ..., ik ) as,
,
are nonnegative, bounded and monotonically decreasing, and we wish to show that the function f.2
S
has the value
( i )kf .
(13)
In fact,
g(a1, ..., ak ) =
J
Ft^,
..., !k ) d£
On the other hand h
J
(i1^)
ai
\ • •.
J
fx
...xk ^ i >
d  i *'9 d ^k
\
equals a sum (15)
^ + f (g ^, ..., ck )
where each c^ has either the value a^ or the value b^, and the term f(a1, ..., ak ) has the coefficient ( 1 )k .This result is easily ob tained if (1A ) is integrated out. By the assumption concerning f, all components of (15) to within ( 1 )kf (a^
..., a^)
are convergentto zero as bx — —> °°,and on the other convergent to (.13). From this our assertion follows.
hand (1A) is
5 • THEOREM 66. If the function f(x1, ..., xk ) is continuous in the whole space, Is convergent to zero as x, I +
...
+
xk   >
and If it has the derivative
00
then
§1^6.
POISSON SUMMATION FORMULA
255
which is continuous and absolutely integrable in the whole space, then p
f( x ,,
rr sin rui
xk ) =
• • •  * k +' t k ) n  7 ^  d * xk )
Moreover, if f(x1, . tegrable, then for
is absolutely in
“k 5 " j t ) ^ f f i i v
E(cV
•
^ )e(W
x
} d?
the inverse formula f (x1, ..., xk ) = J
(16)
E(a1, ..., ak )e(lx xxax ) da
holds, where the last integral is to be taken as a Cauchy principal value. PROOF. The first part of the theorem follows from k . and Theo rem 65. The second part then follows by considering that the value of the integral n J
E(a1, ..., ak )e(zxaxxx ) da
amounts to
/
s i n .nx (x x  l x ) axil
f (6,,
H6. 1.
• • •>
II
iX
d Xx?x j
The Polsson Summation Formula If the quantity
cp(n1, .. •, nk )
is defined for all ’’lattice
points”, i.e., for all combinations of positive and negative Integers, then the sum (1 )
7 ^ ,
..., nk )
is to extend over all these lattice points. We call the series (1)
256
CHAPTER IX.
FUNCTIONS OF SEVERAL VARIABLES
convergent and denote its sum by
(2 )
\jr,
if the partial sums
1 ••• i , •••; (c)
(8)
y
1>
£k )e (2 jrZ^n^,
) d
•
cp(n1, . . nk )
n' be convergent. REMARK. For example, assumption (a) is then satisfied if f is continuous everywhere, and assumption (b) if there are constants G > 0 and
t)
> 0
such that k ~ 2 " 11
(9)
tf I < G(x^
+
. ..
+ xk )
Concerning assumption (c), it is in a certain sense dispensable. In fact, we shall prove the following. If assumptions (a) and (b) are satisfied, then the series (8 ) is convergent by arithmetic means, and (4 ) holds. Under convergence by arithmetic means, we herewith understand that the series
formed with the partial sums (2) of (8 ), Is convergent as
n^ — > 00. —
If assumption (c) enters, then the assertion of our theorem results from the following fact (Appendix 18). If a series (3 ) on the one hand con verges, and on the other hand is summable by arithmetic means, then both "sums" are equal to each other.
(6)
PROOF. The function F(x1, ..., x ^ ) Is bounded in the interval and continuous at the point (c, ..., 0). The first follows because
the function f is bounded in finite intervals, and the series (5) is uniformly convergent in (6). The second follows because each term of the series (5) is continuous at the point (0, 0) and the series (5) is uniformly convergent. Because of (7), we obtain from (2) and (10), if use is made of elementary formulas of trigonometric sums [118], that
258
CHAPTER I X .
yk ~
)
is also measurable In ^ + 1:“ 5. Sums, differences, and products of measurable functions, and (by the characterization given in b ) the square root of a nonnegative measurable function, are also measurable. We use these results to arrive at the following. If
f
and
g
are measurable functions, then
JF T 7
266
APPENDIX
is also a measurable function (hence in the special case tion f ) is measurable. If
0 )
f1
and
f,(X ,,
f2
•
g = o,
are measurable functions in
•
x k ) • f 2 ( y ,  x 1(
.
.
the func
then
yk  xk )
Is measurable in Let f (x) be measurable on E . For real c , we define fQ (x) as follows. Let fc (x) = f(x) for those points for which f(x) < c and let fQ (x) = c for those points for which f(x) > c. also measurable, as is evident from the relation
Then
fQ (x) is
A similar conclusion Is valid if, in the definition of fQ the ">" sign is replaced by the " b, then f ^ is again measurable. 6.
A definition due to Carath. Is the following.
Two functions
f and cp defined on the same point set are called "equivalent " if they agree almost everywhere. Sums, products and limit functions ofequivalent functions are again equivalent. Equivalent functions are simultaneously measurable.
By 3  5 we arrive at the
f2 defined in 9fk are equivalent to (1) is equivalent to
xk) ■
following.
If the functions
the functions cp1 and
 X,,
yk  xk )
q>2,
f1
and
then
.
SUMMABILITY 7 . Among the functions measurable on A, the ones which are summable play a special role. In the text we have avoided the term "summable", and used instead the more familiar name "integrable". We now, however, make the following observation. If the point set is bounded, then the two terms have exactly the same meaning. But if A Is unbounded, then "summable" and "absolutely integrable" mean the same. Hence the term not absolutely integrable which occurs frequently in the text does not fall under the concept "summabillty" as used by Carath. To preserve the restricted usage of Carath., we shall hereafter in this appendix use the term summa billty. The following theorems are valid (Carath. ^4 4 5 ).
267
APPENDIX
f
i ) To each function sponds a finite number
which is sumtnable on
/
E,. there corre
f dx
E
which is called the integral of f over E. (If under the integral sign is to be replaced by dx, measurable and bounded (for example if f = c), and measure* then
f
then the dx dxk .) If f is has a finite
is summable and
fo
of
dx =
E
mE
dx = c
E
2) Equivalent functions are simultaneously summable, and yield the same values for the integral. Among the functions equivalent to a summable one, there are always functions which are finite (at each point of E).
that
f
3) In order that f be summable, it is necessary and sufficient Itself be measurable and f be summable. Hence
(2 )
/ f
k)
If for the summable function
whose values lie in the interval (3)
< J ' f  dx
dx
(g, G),
f,
an equivalent one exists
then
g * m E < y ' f dx < G* mE , E
(Carath. §ko6 , Theorem 3) • (k)
In particular if J
t
dx
>
0
f ^ 0
and measurable, then
.
E
and
x2
5) If f 1 and f2are finite and summable functions, and \ arefinite constants, then x1f 1 + x2f2 Is summable. Further / ( x lfl + x2f2 ) dx = x, / f , E E
to + x2 f E
f2 dx
If in particular, f2 « f f , then in conjunction with (4 ), results. If f, < f and f ] and f are summable, then
.
the following
APPENDIX
268
jTf, dx < f r dx
Hence it
A similar conclusion is valid if " almost every f  9
almost
'
G^ results, namely that if
Gh dx < e
E then J
(23)
f  cp dx < e
E Because of (10), the quantity
e
in (22) can be taken arbitrarily small.
Hence (23) holds for e = 0, and f and cp are therefore equal to each other almost everywhere on E. In a similar manner, one proves the last assertion in 11. For the proof of 10. one first introduces the function fn (x) which vanishes outside the interval  n < x^ < n , has the value n where f(x) > n, with
has the value
f(x).
 n
where
f(x) <  n,
and otherwise agrees
It is evident that If(x)  fn (x) < (f
,
and on the other hand that lim f(x)  fn (x) = 0 n >00 11
,
276
APPENDIX
for almost all
x.
By 7 ,7 )>therefore, for each (2*)
/
We now form the functions functions
e
there is an n such
If  fn l dx < £
Fnh
.
in accordance with (16) and with them, the
Gh  lfn  Fi*l The functions
G^
that
h < ho
•
vanish outside of an interval independent of
h.
In
side these intervals they are uniformly hounded, and moreover, for almost all x, lim G, = 0 h >o h By 7 , 8 ), there is therefore a suitable J %
(25)
h
.
such that
dx < 
.
Prom (2*0 and (25), it follows for the function
* = Pr*
'
continuous and vanishing outside of a finite interval, that
J
f  q> dx < 
.
COMPLEX VALUED FUNCTIONS 1 2 . Let xk ) = f 1 (x^
f(x^
be a given function on E where f 1 fsummable ormeasurable,if f and
..., xk ) + if2 (x1, ..., xk ) and fg are real valued. We call f2 aresummable ormeasurable.
Prom
ifi < It follows that
f
J t* +f < if,i + if2i
Is also summable or measurable.
The observations of 1 to 1 , 3) carry over fairly easily.
Only
APPENDIX the relation (26)
is not trivial.
f f dx d
+00. in addition to the representation by Hobson in regard to Ces&rosummability for arbitrary exponents, cf. the work of M. Jacob, Bulletin International de l !Acad6mie Polonaise de Cracovie, Series A: Sciences mathdmatlques, p. 4 0 7 4 , 1926, and Mathematische Zeitschrlft 29, 2033, 1928; S. Pollard, Proceedings of
286
REMARKS  QUOTATIONS
t h e C a m b rid g e P h i l o s o p h i c a l S o c i e t y , X X I I I , 3 7 3  3 8 2 , 43. p . 2 0 5 2 0 8 . 44.
1927*
F o rm u la ( 2 4 ) s te m s fr o m C a u c h y , c f . M e y e r c i t e d i n £ 3 5 1 , B y E . C a ta la n , B u rk h a rd t, p . 1 1 0 5 .
45. F o rm u la ( 2 9 ) an d a g e n e r a l i z a t i o n o f i t h a s b e e n fo u n d b y D i r ic h le t , c f . B u rk h a rd t, p . 1114. 1* 6 . p . 850 to 8 5 1 .
47.
F o rm u la s ( 3 0 ) ,
( 3 1 ) an d
( 3 2 ) ste m fr o m C a u c h y , B u r k h a r d t ,
C f . M e y e r, c i t e d I n £ 3 5 ] , p* 3 1 0  3 1 3 *
48. F o rm u la ( i 4 ) ste m s fr o m 0. S c h lo m il c h an d A . d e M o rgan , c f . B u r k h a r d t , p . 1 1 4 5  4 7 *  Our p r o o f i s i n p r i n c i p l e t h e on e b y G . H. H a rd y , P r o c e e d in g s o f t h e L o n d o n M a t h e m a t ic a l S o c i e t y , S e r i e s ( 1 ) , 3 5 , 8 1  1 0 3 , 1 9 0 2 / 0 3 , e s p e c ia lly p . 9 6 .  F o r a fu n c tio n th e o ry p ro o f b y L . K r o n e c k e r , s e e h i s V o r le s u n g e n , V o l . 1 , B e s tim m te I n t e g r a l e , 1 8 9 4 , p . 19 9  2 1 4 .
4 9 . C f . , f o r e x a m p le , G . H. H a rd y , T r a n s a c t i o n s o f t h e C a m b rid g e P h ilo s o p h ic a l S o c ie t y , 2 1 , 39 86, 1 9 1 2 . 50.
B y L a p la c e ; B u rk h a rd t, p . 1 1 2 6  1 1 2 7 .
51*
B y C a u c h y ; B u r k h a r d t , p . 1 1 2 8 , fo r m u la ( 9 5 1 ) •
52.
A f t e r A . M. L e g e n d r e ; B u r k h a r d t , p . 1 1 4 2 —4 3 
53*
A fte r
0
. S c h lo m il c h ; B u r k h a r d t , p .
1154.
54. F o r fo r m u la s ( 1 3 ) an d ( 1 4 ) c f . M e y e r , c i t e d i n £ 3 5 3 , p . 2 8 6  2 8 9 an d p . 3 1 9  3 2 4 . 55* B y E . C . J . v o n Lom m el; W a ts o n , p . 4 8 . The i n v e r s i o n i n t e g r a l i s c o n t a in e d i n a g e n e r a l i n t e g r a l fo r m u la o f H. W eber an d P . S c h a f h e i t l i n ; W a ts o n , p . 3 8 9  4 1 5 . 55a .
B y H. W e b e r; W a ts o n , p . 4 0 5 .
56.
B y L . G e g e n b a u e r ; W a ts o n , p . 5 0 , fo r m u la ( 3 ) *
57*
B y N. J . S o n in e an d G e g e n b a u e r ; W a ts o n , p .
58.
B y E . G. G a l l o p ; W a ts o n , p .
59*
C f . W a ts o n , p . 1 5 0 , fo r m u la ( i ) .
4
1 5 , fo r m u la ( 1 ) .
422.
60. F o rm u la ( 1 3 ) i s c o n t a in e d i n a g e n e r a l fo r m u la o f S . R am an u jan ; W a ts o n , p . 4 4 9 * N um erous o t h e r (n o t n e c e s s a r i l y r e l a t e d t o B e s s e l f u n c t i o n s ) F o u r i e r i n t e g r a l s an d i n v e r s i o n s o f s u c h i n t e g r a l s ste m fr o m R a m a n u ja n . T h e s e a r e d i s t r i b u t e d i n v a r i o u s p a p e r s o f h i s c o l l e c t e d w o rk s ( C o l l e c t e d P a p e r s o f S . R am an u jan , C a m b rid g e , 1 9 2 7 ) * We s h a l l m eet se v e ra l la t e r in §35
REMARKS  QUOTATIONS 61.
By F. G. Mehler and Sonine; Watson, p. 169170.
62.
By A. B. Basset; Watson, p. 172.
63.
Formulas (19) and (20) go back to 0 . Heavisidie; Watson,
287
p. 388. 6k . Dirichlet’s application of the discontinuity factors forms the source of this theorem, cf. §**, k ; for the evaluation of certain definite integrals, cf. Burkhardt, p. 13211322. Formula (7) of our theorem stems from Dirichlet, the general formula stems from Schlomilch, cited In Cl], p. 160181.
65. Concerning this, cf. say 0 . Perron, Die Lehre von den Kettenbriichen, 2nd Ed., 1929, p. 362367. 66.
Cf. P. L 235241, 1927* 101. M. Plancherel, E. C . Titchmarsh among others; cf. [31], and Hobson, p. 748751, gave criteria for the convergence of integrals in the usual sense. 102.
f(x) dx
Cf. G. H. Hardy and E. C. Titchmarsh, cited in [7 7 l*
103. I. C. Burkhill studied the case where the differential was replaced in the integral (1 5 T) by the more general dtp(x),
cf. citation in [86]; also similarly in the case of the Hankel integral. For this, cf. S. Izumi, The Tohoku Mathematical Journal, 29, 2 6 6 2 7 7 , 1928. 1 0 4 . E. C. Titchmarsh, Proceedings of the London Mathematical Society* Series (2), 23, 279281, 1925 and Journal of the London Mathe matical Society, 2, 148150, 1927; A. C. Berry, Annals of Mathematics, Series
32, 227238 and 830838, 1931.  Cf. also Hobson, p.7 4 2 7 4 7 *
2, 105.
Forgeneralizations, cf. G. H. Hardy and E. C.
Titchmarsh,
Journal of the London Mathematical Society, 6, 4 4 4 8 , 1930. 106. Cf. [100] and I. C. Burkhill, cited in [103]. For litera ture concerning the topic prior to Plancherel cf. Watson, cited in [92]. For generalizations of the Fourier integral.theorems based on other than Hankel kernels, cf. M. H. Stone, Mathematische Zeitschrift, 28, 654676, 1928. 107.
Cf. Watson, cited in [92], p. 4 06, formula (8).
108.
This proof by E. Czuber, Monatshefte fiir Mathematik und
Physik, 2, 119 1 2 4 , 1891.
1173.
109. For k = 2, 3, by Poisson and Cauchy, Burkhardt, p. 1165The theorem is also not new for k arbitrary. 110.
By Poisson and Cauchy, Burkhardt, p. 116 4 .
111.
Cf. Watson, cited in [92], Chapt. 1 3 *4 7 *
112.
Cf. [17]*
113. Cf. Ch. H. Miintz, Mathematische Annalen, 90, 2 7 9  291, 1923 and Sitzungsberichte der Berliner Mathematischen Gesellschaft 1925, p* 8193. 114  W. Thomson Lord Kelvin, Papers on Electrostatics and
REMARKS  QUOTATIONS
291
Magnetism, Second Edition, 18 8 4 , p. 112125. 115°
Of. A. C. Berry, cited in [1 Ok] .
116. Our definition of monotone functions of several variables should amount essentially to that by G. H. Hardy, Quarterly Journal of Mathematics, 3 7 , 5560, 1905/06. For literature concerning the subject and for analogues to our Theorem 65 in the case of repeated Fourier series, cf. H. Hahn, Theorie der reellen Funktionen, Bd. I, Berlin 1921, p. 5 3 9 5 ^7 , Hobson, p. 702711, and Tonelli, Chapter IX. 117*
The Polsson summation formula in several variables was
first studied by Ch. H. Miintz, cited in [113 3 > then by L. I. Mordell, cf. [120]. For the present criteria and generalizations, cf. my note in Mathematische Annalen, 106, 5663, 1932. 118.
For this formula, including formula (13), cf. say Tonelli,
p. 1+87. 1 1 9  This is a theorem concerning Fourier series in several variables. Cf. Tonelli, p. 1+86500. 120. Considerable further generalizations than (19) play of late an important role in analytic number theory. For literature, cf. L. I. Mordell, Proceedings of the London Mathematical Society, Series 2, 32, 501556,
1931 . 121.
C. L. Siegel, Mathematische Annalen, 87, 3638, 1922.
AUTHOR’S SUPPLEMENT
MONOTONIC FUNCTIONS, STIELTJES INTEGRALS AND HARMONIC ANALYSIS INTRODUCTION Consider the totality of all monotonically increasing1 functions of one variable. It Is known that for this function class the following notions of equality and convergence are very appropriate. Two functions are called "essentially equal" if they agree at their points of continuity. A sequence of functions fn (x) is called "essentially convergent" if there exists a function fQ (x) such that at each point of continuity of fQ (x) the sequence of numbers fR (x) converges to the number f (x). The im portance of these concepts of equality and convergence consists in the fact that the following compactness theorem is then valid. Each infinite set of 2 uniformly bounded functions contains an essentially convergent subsequence , and the limit function of an essentially convergent sequence remains un changed if each function of the sequence is replaced by an essentially equal one. The compactness theorem can be formulated more profitably if the point functions f(x) are replaced by interval functions f (x; y) = f(y) f (x). An interval is called a continuity interval of the function f (x; y) if small changes of the boundaries of the Interval produce little change in the function value itself. Correspondingly one defines the concepts "essentially equal" and "essentially equivalent." The compactness theorem will then read as follows. Each infinite set of uniformly bounded (nonnegative, additive) interval functions contains an essentially convergent subsequence, and the limit function of an essentially convergent sequence remains unchanged if each function of the sequence is replaced by an essentially equal one. The basic advantage of this approach Is that the compactness or monotonically decreasing. 2 This theorem is usually named after E. Helly, Sitzungsberichte der Wiener Akademie [Proceedings of the Vienna Academy] 121 (1921 ), p. 265297. The terms "essentially equal" and "essentially convergent" are due to A. WIntner, Spextraltheorie der unendlichen Matrizen, 1929, p* 7 7 . 292
S T IE L T JE S INTEGRALS
th e o re m and o t h e r f a c t s a b o u t i n t e r v a l f u n c t i o n s an d t h e R i e m a n  S t i e l t j e s i n t e g r a l s ^ c o n n e c t e d w i t h them c a n b e c a r r i e d o v e r v e r b a t i m f o r f u n c t i o n s o f s e v e r a l v a r i a b l e s , a lt h o u g h t h e d i s c o n t i n u i t y c h a r a c t e r o f t h e i n t e r v a l f u n c t i o n s i n t h e c a s e o f m ore v a r i a b l e s i s c o n s i d e r a b l y m ore c o m p lic a t e d t h a n i n t h e c a s e o f one v a r i a b l e . I t w i l l b e e v i d e n t fr o m t h e r e s u l t s o f t h i s w o rk t h a t t h i s f o r m u la t i o n i s a n a p p r o p r i a t e o n e . A th e o re m o f F o u r i e r  S t i e l t j e s i n t e g r a l s u s e f u l i n p r o b a b i l i t y t h e o r y i s t h e f o l l o w i n g . I f f o r a s e q u e n c e o f m o n o to n ic f u n c t i o n s V n (cr), th e c o rre sp o n d in g " c h a r a c t e r i s t i c fu n c t io n s "
f
f n (x) =
00
e l x a dVn (ct)
—OO
c o n v e r g e u n i f o r m ly i n e a c h f i n i t e x  I n t e r v a l , t h e n t h e f u n c t i o n s Vn ( a ) a r e e s s e n t i a l l y c o n v e r g e n t .^ We s h a l l p r o v e t h i s th e o re m n o t o n ly f o r s e v e r a l v a r i a b l e s , b u t s h a l l a l s o f r e e i t fr o m t h e r e q u ir e m e n t o f u n ifo r m c o n v e r g e n c e . I t w i l l be s u f f i c i e n t th a t th e fu n c tio n s f n ( x ) c o n v e r g e f o r a lm o s t a l l x an d a t t h e o r i g i n . I n a d d i t i o n we s h a l l g i v e a n e c e s s a r y an d s u f f i c i e n t c o n d i t i o n t h a t a f u n c t i o n c a n b e w r i t t e n * i n t h e fo rm oo
J
f(x) =
e l x “ d V (a )
.
—OO
B y r e a s o n o f I t , we s h a l l t h e n b e a b l e t o fo r m u la t e an d p r o v e , i n a n e s p e c i a l l y l u c i d m a n n e r, a th e o re m o f N o r b e r t W ie n e r o n s p e c t r a l a n a l y  . s i s o f a r a t h e r g e n e r a l k in d o f f u n c t i o n . L e t g ( x ) b e a s q u a r e i n t e g r a b l e f u n c t i o n ( o f one v a r i a b l e ) . I f t h i s f u n c t i o n I s a l s o s q u a r e i n t e g r a b l e i n t h e i n f i n i t e r e g i o n , t h e n P l a n c h e r e l h a s show n t h a t i t c a n b e r e p r e s e n t e d b y a F o u r ie r in t e g r a l g(x) ~ J
elxctr(cc) d a
.
I f , h o w e v e r , i t i s n o t s q u a r e i n t e g r a b l e i n th e i n f i n i t e r e g i o n , b u t i n s t e a d i s p e r i o d i c o r m ore g e n e r a l l y a lm o s t p e r i o d i c i n I t 3 e n t i r e d o m a in , th e n I t p o s s e s s e s a F o u r ie r s e r ie s 3 T h e t h e o r y o f S t i e l t j e s i n t e g r a l s i n s e v e r a l v a r i a b l e s w as f i r s t s y s t e m a t i c a l l y i n v e s t i g a t e d b y J . R a d o n , W ie n e r B e r i c h t e 1 2 2 ( 1 9 1 3 )* C f . a l s o A. K o lm o g o r o f f, U n te r s u c h u n g e n u b e r d e n I n t e g r a l b e g r i f f, Math. A n n a le n 10 3 ( 1 9 3 0 ) , p . 6 5 ^ 6 9 6 . ^ F o r f u n c t i o n s o f on e v a r i a b l e i t h a s a l r e a d y b e e n g i v e n i n t h e a u t h o r ’ s V o r l e s u n g e n u b e r F o u r i e r s c h e I n t e g r a l e , L e i p z i g , A k a d e m is c h e V e r l a g s g e s e l l s c h a f t , 1 9 3 2 , i n p a r t i c u l a r C h a p t e r I V . I n w h a t f o l l o w s we s h a l l q u o te t h i s book b y "F o u r ie r I n t e g r a l s " .
2 9k
STIELTJES INTEGRALS
Z
ix a
e
vr ( a v )
.
V
T he " s p e c t r a l i n t e n s i t y " o f g ( x ) i s t h a t i n t e r v a l f u n c t i o n w h ic h i n t h e one c a s e b e lo n g s t o t h e (m o n o to n ic ) f u n c t i o n a
Via)
=
f
 r ( a ) 2 dor
,
o and i n t h e o t h e r t o t h e f u n c t i o n Of £ r (c r v )2 QfV =0
Via) =
.
I n b o th c a s e s , I t i s a b o u n d e d , m o n o to n ic f u n c t i o n . tro d u c e s th e fu n c tio n
I f i n a d d i t i o n one i n
00 (0,1)
f
f(x) 
e^dvia )
,
—OO t h e n i n t h e on e c a s e , we h a v e 00
f(x) = J
g(x + I ) g U ) d
,
“00
an d i n t h e o t h e r T
( 0, 2 )
f(x) =
li m T ~>»
Jm f
g(x
I)gTT T
d£
.
21
Now N . W ie n e r^ h a s p r o v e d t h e f o l l o w i n g r e s u l t (a n d a l s o g e n e r a l i z e d i t t o s e v e r a l v a r ia b le s ) . I f g ( x ) I s an a r b i t r a r y sq u are in te g r a b le fu n c tio n in t h e f i n i t e r e g i o n , r e g a r d i n g w h o se b e h a v i o r I n t h e i n f i n i t e r e g i o n i t i s known o n ly t h a t t h e ( f i n i t e ) l i m i t (o , 2 ) e x i s t s f o r a l l x , t h e n one c a n a s s i g n t o i t i n a m e a n in g fu l m a n n e r, a bo u n d ed m o n o to n ic f u n c t i o n V(of) a s i t s s p e c t r a l f u n c t i o n . M ore p r e c i s e l y s t a t e d , t h i s a s s ig n m e n t m eans t h a t t h e r e l a t i o n ( o , 1 ) h o ld s f o r a lm o s t a l l x (a n d c o r r e s p o n d i n g ly f o r m ore v a r i a b l e s ) b e tw e e n t h e f a l t u n g s  f u n c t i o n f ( x ) an d t h e s p e c t r a l f u n c t i o n V (o r). Due t o a v e r y o r i g i n a l (u n p u b lis h e d ) i d e a o f M. R i e s z , we s h a l l r e p l a c e t h e c o n v e r g e n c e d e n o m in a to r 2 T i n t h e i n t e g r a l ( 0 , 2 ) b y a n a r b i t r a r y 5 It
W ie n e r , A c t a M a th e m a tic a
55
0 93° )
, pp.
1 17  2 5 8 .
STIELTJES INTEGRALS
p o s i t i v e m o n o t o n i c a l ly i n c r e a s i n g f u n c t i o n lim T ~>oo
PtTT
p(t) =
295
f o r w h ic h
1
T h a t i s , we s h a l l p r o v e t h e e x i s t e n c e o f t h e s p e c t r a l f u n c t i o n t h e m ore g e n e r a l ^ a s s u m p t io n t h a t t h e l i m i t
V(a)
under
T J
f(x) = ^lim
g(x + I )gTT) d
e x i s t s f o r a l l x . E v e n t h e e x i s t e n c e o f t h i s l i m i t w i l l h e n e e d e d o n ly f o r a d i s c r e t e s e q u e n c e o f v a l u e s T 1 , T 2 , T ^ , . . . . T h i s v e r y c o m p re h e n s iv e f o r m u l a t i o n o f t h e W ie n e r th e o re m h a s t h e m e r i t t h a t i t a l s o i n c l u d e s t h e P l a n c h e r e l c a s e w hen p ( T ) = 1 . I t t h e r e f o r e c a n b e lo o k e d a t , i n i t s w a y , a s a n a c t u a l g e n e r a l i z a t i o n o f t h e F o u r i e r i n t e g r a l fo r m u la . I . §1.
MONOTONIC FUNCTIONS
D E FIN IT IO N OF THE MONOTONIC FUNCTIONS
1. 1. We t a k e a s a b a s i s a E u c l i d e a n s p a c e o f s e v e r a l d im e n s io n s . We s h a l l d e n o te i t s d im e n s io n num ber b y k , an d a n in d e x w h ic h t a k e s on t h e v a l u e s 1 t o k , b y x . A p o i n t o f t h i s s p a c e w i l l b e d e n o te d b y a , 3, . . . , a 1 , . . . , x, y, — e t c . , an d I t s i n d i v i d u a l co m p o n e n ts b y u p p e r a c c e n t s . H ence
«p(cr),
^ A b e g i n n i n g o f t h i s g e n e r a l i z a t i o n w h ic h g o e s b a c k t o M. J a c o b , c a n a l r e a d y b e fo u n d i n N . W ie n e r , o p . c i t . ^ I f a l e f t i n t e r v a l b o u n d a ry I s  » , t h e n t h e s i g n < I n ( 1 , 11 ) i s t o be r e p la c e d b y < .
STIELTJES INTEGRALS
b y t h e sym b o l
8
(a ; p ).
B e s i d e s t h e i n t e r v a l s , we s h a l l c o n s i d e r , w hen t h e c o n t r a r y i s n o t e x p r e s s ly s t r e s s e d , o n ly su ch p o in t s e t s w h ic h c a n b e e x p r e s s e d a s t h e sum o f f i n i t e l y m any ( d i s j o i n t ) i n t e r v a l s . B y a p o i n t s e t , we s h a l l s im p ly m ean a s e t c o n s t i t u t e d i n s u c h a m a n n e r. I t i s n o t d i f f i c u l t t o s e e t h a t sums an d d i f f e r e n c e s , u n io n an d i n t e r s e c t i o n o f tw o p o i n t s e t s ( o f o u r k in d ) a r e a g a in p o in t s e t s ( o f o u r k in d ) . 1.2 . w e l l d e te r m in e d num ber
I.
To e a c h b o u n d ed i n t e r v a l 8 , I t i s p o s s i b l e t o a s s i g n a £ M the limit (1 .3 1 )
lim q>ftin )
exists. Indeed it has the important property that for bounded it agrees with the original value (a; 0 )
or also by (1, 53)
(a). V
If for a bounded interval, none of the 2 corners falls in an exceptional plane, or what amounts to the same thing, if no end point of the interval falls in an exceptional plane, then to each q > 0, one can determine a boundary layer © € of the Interval, for which ^ cp^
and
and
cp2,
and
cp2
and