American Mathematical Society Colloquium Publications Volume XIX Fourier Transforms in The Complex DPDF电子书下载

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