《American Mathematical Society Colloquium Publications Volume XIX Fourier Transforms in The Complex D》求取 ⇩

INTRODUCTION1

1．Plancherel＇s Theorem1

2．The Fourier Transform of a Function Vanishing Exponentially3

3．The Fourier Transform of a Function in a Strip3

4．The Fourier Transform of a Function in a Half-Plane8

5．Theorems of the Phragmén-Lindelof Type9

6．Entire Functions of Exponential Type12

CHAPTER Ⅰ．QUASI-ANALYTIC FUNCTIONS14

7．The Problem of Quasi-Analytic Functions14

8．Proof of the Fundamental Theorem on Quasi-Analytic Functions17

9．Proof of Carleman＇s Theorem20

10．The Modulus of the Fourier Transform of a Function Vanishing for Large Arguments24

CHAPTER Ⅱ．SZASZ＇S THEOREM26

11．Certain Theorems of Closure26

12．Szász＇s Theorem32

CHAPTER Ⅲ．CERTAIN INTEGRAL EXPANSIONS37

13．The Integral Equations of Laplace and Planck37

14．The Integral Equation of Stieltjes41

15．An Asymptotic Series44

16．Watson Transforms44

CHAPTER Ⅳ．A CLASS OF SINGULAR INTEGRAL EQUATIONS49

17．The Theory of Hopf and Wiener49

18．A Note on the Volterra Equation58

19．A Theorem of Hardy64

CHAPTER Ⅴ．ENTIRE FUNCTIONS OF THE EXPONENTIAL TYPE68

20．Classical Theorems Concerning Entire Functions68

21．A Tauberian Theorem Concerning Entire Functions70

22．A Condition that the Roots of an Entire Function be Real75

23．A Theorem on the Riemann Zeta Function75

24．Some Theorems of Titchmarsh78

25．A Theorem of Pólya81

26．Another Theorem of Pólya83

CHAPTER Ⅵ．THE CLOSURE OF SETS OF COMPLEX EXPONENTIAL FUNCTIONS86

27．Methods from the Theory of Entire Functions86

28．The Duality between Closure and Independence95

CHAPTER Ⅶ．NON-HARMONIC FOURIER SERIES AND A GAP THEOREM100

29．A Theorem Concerning Closure100

30．Non-Harmonic Fourier Series108

31．A New Class of Almost Periodic Functions116

32．Theorems on Lacunary Series123

CHAPTER Ⅷ．GENERALIZED HARMONIC ANALYSIS IN THE COMPLEX DOMAIN128

33．Relevant Theorems of Generalized Harmonic Analysis128

34．Cauchy＇s Theorem130

35．Almost Periodic Functions138

CHAPTER Ⅸ．RANDOM FUNCTIONS140

36．Random Functions140

37．The Fundamental Random Function146

38．The Continuity Properties of a Random Function157

CHAPTER Ⅹ．THE HARMONIC ANALYSIS OF RANDOM FUNCTIONS163

39．The Ergodic Theorem163

40．The Theory of Transformations163

41．The Harmonic Analysis of Random Functions170

42．The Zeros of a Random Function in the Complex Plane172

BIBLIOGRAPHY179

INDEX183