INTRODUCTION 1
1．Plancherel＇s Theorem 1
2．The Fourier Transform of a Function Vanishing Exponentially 3
3．The Fourier Transform of a Function in a Strip 3
4．The Fourier Transform of a Function in a Half-Plane 8
5．Theorems of the Phragmén-Lindelof Type 9
6．Entire Functions of Exponential Type 12
CHAPTER Ⅰ．QUASI-ANALYTIC FUNCTIONS 14
7．The Problem of Quasi-Analytic Functions 14
8．Proof of the Fundamental Theorem on Quasi-Analytic Functions 17
9．Proof of Carleman＇s Theorem 20
10．The Modulus of the Fourier Transform of a Function Vanishing for Large Arguments 24
CHAPTER Ⅱ．SZASZ＇S THEOREM 26
11．Certain Theorems of Closure 26
12．Szász＇s Theorem 32
CHAPTER Ⅲ．CERTAIN INTEGRAL EXPANSIONS 37
13．The Integral Equations of Laplace and Planck 37
14．The Integral Equation of Stieltjes 41
15．An Asymptotic Series 44
16．Watson Transforms 44
CHAPTER Ⅳ．A CLASS OF SINGULAR INTEGRAL EQUATIONS 49
17．The Theory of Hopf and Wiener 49
18．A Note on the Volterra Equation 58
19．A Theorem of Hardy 64
CHAPTER Ⅴ．ENTIRE FUNCTIONS OF THE EXPONENTIAL TYPE 68
20．Classical Theorems Concerning Entire Functions 68
21．A Tauberian Theorem Concerning Entire Functions 70
22．A Condition that the Roots of an Entire Function be Real 75
23．A Theorem on the Riemann Zeta Function 75
24．Some Theorems of Titchmarsh 78
25．A Theorem of Pólya 81
26．Another Theorem of Pólya 83
CHAPTER Ⅵ．THE CLOSURE OF SETS OF COMPLEX EXPONENTIAL FUNCTIONS 86
27．Methods from the Theory of Entire Functions 86
28．The Duality between Closure and Independence 95
CHAPTER Ⅶ．NON-HARMONIC FOURIER SERIES AND A GAP THEOREM 100
29．A Theorem Concerning Closure 100
30．Non-Harmonic Fourier Series 108
31．A New Class of Almost Periodic Functions 116
32．Theorems on Lacunary Series 123
CHAPTER Ⅷ．GENERALIZED HARMONIC ANALYSIS IN THE COMPLEX DOMAIN 128
33．Relevant Theorems of Generalized Harmonic Analysis 128
34．Cauchy＇s Theorem 130
35．Almost Periodic Functions 138
CHAPTER Ⅸ．RANDOM FUNCTIONS 140
36．Random Functions 140
37．The Fundamental Random Function 146
38．The Continuity Properties of a Random Function 157
CHAPTER Ⅹ．THE HARMONIC ANALYSIS OF RANDOM FUNCTIONS 163
39．The Ergodic Theorem 163
40．The Theory of Transformations 163
41．The Harmonic Analysis of Random Functions 170
42．The Zeros of a Random Function in the Complex Plane 172
BIBLIOGRAPHY 179
INDEX 183