Table Of ContentFourier Series and Integrals
Probability and Mathematical Statistics
A Series of Monographs and Textbooks
Editors
Z. W. Birnbamn E. Lukaes
University of Washington Catholic University
Seattle, Washington Washington, D.C.
1. Thomas Ferguson. Mathematical Statistics: A Decision Theoretic
Approach. 196'?
2. Howard Tucker. A Graduate Course in Probability. 1967
3. K. R. Parthasarathy. Probability Measures on Metric Spaces. 1967
4. P. Rérésa. The Laws of Large Numbers. 1963
5. H. P. McKean, Jr. Stochastic Integrals. 1969
6. B. V. Gnedenl-to, Yu. K. Belyayev, and A. D. Solovyev. Mathematical
Methods of Reliability Theory. 1969
7. Demetrios A. Kappos. Probability Algebras and Stochastic Spaces.
1969, 1970
3. Ivan N. Pesin. Classical and Modern Integration Theories. 1970
9. S. Vajda. Probabilistic Programming. 1972
10. Sheldon M. Ross. Introduction to Probability Models. 1972
11. Robert B. Ash. Real Analysis and Probability. 1972
12. V. V. Fedorov. Theory of Optimal Experiments. 1972
13. K. V. Mardia. Statistics of Directional Data. I972
14. H. Dym and H. P. McKean. Fourier Series and Integrals. 1972
15. Tatsuo Kawata. Fourier Analysis in Probability Theory. 1972
In Preparation
Fritz Oberhettinger. Fourier Transforms oi Distributions and Their
Inverses: A Collection of Tables
FO URIER SERIES
AND INTEGRALS
H. Dym
DEPARTMENT DF THEORETICAL MATHEMATICS
THE WEIZMANN INSTITUTE OF SCIENCE
REHOIJDT, ISRAEL
H. P. McKetm
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ACADEMIC PRESS New York and London
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Contents
Preface
Historical Introduction
Chapter 1. Fourier Series
1.1 The Lebesgue Integral
1.2 The Geometry of L‘(Q)
1.3 The Geometry of L‘(Q) Continued
1.4 Square Summable Functions onthe Circle and Their Fourier Series
1.5 Summable Functions and Their Fourier Series
l.6* Gibbs‘ Phenomenon
1.7 Miscellaneous Applications
1.3 Applications to the Partial Differential Equations of One-Dimensional
Mathematical Physics
l.9* More General Eigenfunction Expansions
I.l0 Several-Dimensional Fourier Series
Chapter 2. Fourier Integrals
2.1 Fourier Integrals
2.2 Fourier Integrals for C"f(R‘)
2.3 Fourier Integrals for Li(R‘): First Method
2.4* Fourier Integrals for Li(R'}: Second Method
2.$* Fourier Integrals for Li(R'): Third Method
2.6 Fourier Integrals for L‘ (R')
2.7 Miscellaneous Applications
2.B* I-leisenberg‘s Inequality
2.9* Bancl- and Time-Limited Functions
2.10 Several-Dimensional Fourier Integrals
2.1 l Miscellaneous Applications of Several-Dimensional Fourier Integrals
vii
viii contents
Chapter 3. Fourier Integrals and Complex Function Theory
3.1 A Short Course in Function Theory 144
3.2 I-Iardy’s Theorem 156
3.3 The Paley-Wiener Theorem 158
3.4 Hardy Functions ltil
3.5* Hardy Functions and Filters l'l0
3.6* Wiener-I-lopf Factorization: Milne’s Equation 1T6
17* Spitter‘s Identity 134
3.8* Hardy Functions in the Disk and Szegiifs Theorem I37
3.9* Polynomial Approximation: The Szasz-Miintz Theorem I94
3.l0* The Prime Number Theorem 196
Chapter 4. Fourier Series and Integrals on Groups
4.1 Groups 203
4.2 Fourier Series on the Circle 206
4.3 Fourier Integrals on the Line 209
4.4 Finite Commutative Groups 214
4.5 Fourier Series on a Finite Commutative Group 217
4.6* Gauss‘ Law of Quadratic Reciprocity 222
4.7 Noncommutative Groups 226
4.8 The Rotation Group 223
4.9 Three Convolution Algebras 237
4.10 Homomorphisrns of L‘(K/G/K) 239
4.1] Spherical Functions Are Eigenfunctions of the Laplacian 242
4.12 Spherical Functions Are Legendre Polynomials 245
4.13 Spherical Harmonics 249
4.l4* Representations of SO(3) 255
4.l$* The Euclidean Motion Group 26!
4.l6* SL(2,R) and the Hyperbolic Plane 273
Additional Reading 282
Bibliography 233
Index 289
Preface
Score. The purpose of this book is to give a mathematical account of Fourier
ideas on the circle and the line. on finite commutative groups, and on a few
important noncommutative groups. Purely technical aspects of the subject,
such as the pointwise convergence of Fourier series of wild functions, will be
avoided. The emphasis is placed instead on the extraordinary power and flexi-
bility of Fourieris basic series and integrals and on the astonishing variety
ofapplications in which it is the chicftool. A brielsample ofsuch applications
will sufiice here:
(l) summation of series (1.7)
(2) the isoperimetric problem (1.7)
(3) Jacobiis identity for the theta function (1.7, 2.7, 2.1 I)
(4) heat flow (1.8, 2.7)
(5) wave motion, especially Huygens’ principle (1.8, 2.7, 2.11)
(6) random walks (1.10, 2.11)
(7) the Poisson summation formula (2.7, 2.11, 4.5)
(8) electrical circuits and filters (2.7, 3.5)
(9) the central limit theorem (2.7, 2.1 1)
(10) I-leisenberg‘s inequality (2.8)
(l l) band- and time-limited signals (2.9)
(12) Minkowskiis theorem in the geometry of numbers (2.1 I)
(13) steady-state radiation in stars (3.6)
(14) Spitzer's identity (3.7)
(15) polynomial approximation (1.7, 3.3, 3.9)
(16) the distribution of primes (3.10)
(17) Gauss‘ law of quadratic reciprocity (4.6)
(18) the Zeeman effect in hydrogen (4.13)
(19) representations of the rotation group SO(3) (4.14)
PREREQUISITES. The level of preparation expected is a thorough knowledge
of advanced calculus. Courant-John [I965] would be ideal. To this must be
added a willingness to believe in (or to study up on) the Lebesgue integral,
ix
