The Convolution Transform

In studies of general operators of the same nature, general convolution transforms are immediately encountered as the objects of inversion. The relation between differential operators and integral transforms is the basic theme of this work, which is geared toward upper-level undergraduates and graduate students. It may be read easily by anyone with a working knowledge of real and complex variable theory. Topics include the finite and non-finite kernels, variation diminishing transforms, asymptotic behavior of kernels, real inversion theory, representation theory, the Weierstrass transform, and complex inversion theory.

CHAPTER I INTRODUCTION

1. Introduction

3

(1)

2. Convolutions

3

(2)

3. Operational calculus

5

(2)

4. Green's functions

7

(1)

5. Operational calculus continued

8

(3)

6. The generation of kernels

11

(1)

7. Variation diminishing convolutions

12

(2)

8. Outline of program

14

(2)

9. Summary

16

(1)

CHAPTER II THE FINITE KERNELS

1. Introduction

17

(1)

2. Distribution functions

17

(2)

3. Frequency functions

19

(1)

4. Characteristic functions

20

(2)

5. Convolutions

22

(2)

6. The finite kernels

24

(4)

7. Inversion

28

(2)

8. Exponential polynomials

30

(2)

9. Green's functions

32

(3)

10. Examples

35

(1)

11. Summary

36

(1)

CHAPTER III THE NON-FINITE KERNELS

1. Introduction

37

(1)

2. Limits of distribution functions

38

(4)

3. Pólya's class of entire functions

42

(6)

4. The closure of a class of distribution functions

48

(1)

5. The non-finite kernels

49

(6)

6. Properties of the non-finite kernels

55

(1)

7. Inversion

56

(3)

8. Green's functions

59

(6)

9. Examples

65

(14)

10. Associated kernels

79

(3)

11. Summary

82

(1)

CHAPTER IV VARIATION DIMINISHING TRANSFORMS

1. Introduction

83

(1)

2. Generation of variation diminishing frequency functions

84

(1)

3. Logarithmic convexity

85

(3)

4. Characterization of variation diminishing functions

88

(3)

5. The changes of sign of G(n)(t)

91

(2)

6. Intersection properties

93

(2)

7. Generation of totally positive functions

95

(2)

8. Matrix transformations

97

(6)

9. Totally positive frequency functions

103

(4)

10. Summary

107

(1)

CHAPTER V ASYMPTOTIC BEHAVIOUR OF KERNELS

1. Introduction

108

(1)

2. Asymptotic estimates

108

(3)

3. Asymptotic estimates continued

111

(8)

4. Summary

119

(1)

CHAPTER VI REAL INVERSION THEORY

1. Introduction

120

(1)

2. Some preliminary results

120

(3)

3. Convergence

123

(2)

4. The sequence of kernels

125

(2)

5. The inversion theorem

127

(5)

6. Stieltjes integrals

132

(6)

7. Relaxation of continuity conditions

138

(4)

8. Factorization

142

(3)

9. Summary

145

(1)

CHAPTER VII REPRESENTATION THEORY

1. Introduction

146

(1)

2. Behaviour at infinity

147

(3)

3. An elementary representation theorem

150

(2)

4. Determining function in Lp

152

(4)

5. Determining functions of bounded total variation

156

(2)

6. Determining function non-decreasing

158

(5)

7. Representation of products

163

(6)

8. Summary

169

(1)

CHAPTER VIII THE WEIERSTRASS TRANSFORM

1. Introduction

170

(3)

2. The Weierstrass transform

173

(6)

3. The inversion operator

179

(3)

4. Inversion

182

(1)

5. Tychonoff's uniqueness theorem

183

(2)

6. The Weierstrass theorem of bounded functions

185

(3)

7. Inversion, general case

188

(5)

8. Functions of Lp

193

(2)

9. Weierstrass transforms of functions in Lp

195

(2)

10. Weierstrass-Stieltjes transforms

197

(2)

11. Positive temperature functions

199

(3)

12. Weierstrass-Stieltjes transforms of increasing functions

202

(4)

13. Transforms of functions with prescribed order conditions

206

(3)

14. Summary

209

(1)

CHAPTER IX COMPLEX INVERSION THEORY

1. Introduction

210

(2)

2. Transforms in the complex domain

212

(5)

3. Behaviour at infinity

217

(1)

4. Auxiliary kernels

218

(5)

5. The inversion function

223

(3)

6. Application of the inversion operator

226

(4)

7. The inversion theorems

230

(5)

8. A general representation theorem

235

(1)

9. Determining function non-decreasing

236

(2)

10. Determining function in Lp

238

(2)

CHAPTER X MISCELLANEOUS TOPICS

1. Introduction

240

(1)

2. Bernstein polynomials

240

(6)

3. Behaviour at infinity

246

(10)

4. The analytic character of kernels of classes I and II

256

(3)

5. Quasi-analyticity

259

(2)

BIBLIOGRAPHY

261

(4)

SYMBOLS AND NOTATIONS

265

(1)

INDEX

266

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The Convolution Transform

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