The Convolution Transform

279 Pages·1955·13.937 MB·English

by  Isidore Isaac Hirschmann, David Vernon Widder

Contents
CHAPTER I
INTRODUCTION
SEOTION PAGE
1. Introduotion 3
•
2. Convolutions 3
3. Operational calculus . 5
4. Green's functions . 7
5. Operational oalculus continued . 8
6. The generation of kernels · 11
7 . Variation diminishing oonvolutions · 12
8. Outline of program 14
9. Summary . · 16
II
CHAPTER
THE FINITE KERNELS
1. Introduction 17
2. Distribution funotions 17
3. Frequency functions . 19
4. Characteristic functions . · 20
5. Convolutions · 22
6. The finite kernels . · 24
7. Inversion · 28
8. Exponential polynomials . · 30
9. Green's functions . · 32
10. Examples . · 35
11. Summary . · 36
III
CHAPTER
THE NON·F INITE KERNELS
1. Introduction • 37
2. Limits of distribution functions · 38
3. P61ya's olass of entire functions • 42
4. The closure of a class of distribution functions · 48
5. The non-finite kernels · 49
OON TEN TS
PAGE
emOTION'
6. Properties of the non-finite kernels . · 55
7. Inversion .. · 56
8. Green's functions . · 59
9. Examples . • 65
10. Associated kernels · 79
11. Summary . • • 82
CHAPTER IV
VARIATION DIMINISHING TRANSFORMS
1. Introduction . 83
2. Generation of variation diminishing frequency functions . 84
3. Logarithmic convexity . 85
4. Characterization of variation diminishing functions . . 88
5. The changes of sign of G(ft)(t) . 91
6. Intersection properties . 93
7. Generation of totally positive functions. . 95
8. Matrix transformations . . 97
9. Totally positive frequency functions . 103
10. Summary . • 107
CHAPTER V
ASYMPTOTIC BEHAVIOUR OF KERNELS
1. Introduction · 108
2. Asymptotic estimates. · 108
3. Asymptotic estimates continued · III
4. Summary • · 119
CluPTEB VI
REAL INVERSION THEORY
1. Introduction · 120
2. Some preliminary results . · 120
3. Convergence · 123
4. The sequence of kernels . · 125
5. The inversion theorem · 127
6. Stieltjes integrals . · 132
7. Relaxation of continuity conditions · 138
8. Factorization . · 142
9. Summary . · 145
OONTENTS
VII
ClIAPTER
REPRESENTATION THEORY
PAGE
SEO~ION
1. Introduction · 146
•
2. Behaviour at infinity . · 147
3. An elementary representation theorem. · 150
•
4. Determining funotion in L'P · 152
5. Determining funotions of bounded total variation · 156
6. Determining function non-decreasing · 158
7. Representation of products . · 163
8. Summary · 169
VIII
CHAPTER
THE WEIERSTRASS TRANSFORM:
1. Introduction . 170
2. The Weierstrass transform . 173
3. The inversion operator . 179
4. Inversion . 182
5. Tychonoff's uniqueness theorem . 183
6. The Weierstrass theorem of bounded functions . 185
7. Inversion, general case . 188
8. Functions of L'P • 193
9. Weierstrass transforms of funotions in Lf) . • . 195
10. Weierstrass-Stieltjes transforms • . 197
11. Positive temperature functions • . 199
12. Weierstrass-Stieltjes transforms of increasing funotions . 202
13. Transforms of functions vlith presoribed order conditions . 206
14. Summary • . 209
IX
CHAPTER
COMPLEX INVERSION THEORY
1. Introduotion · 210
•
2. Transforms in the complex domain · 212
•
3. Behaviour at infinity . · 217
4. Auxiliary kernels · 218
5. The inversion funotion · 223
6. Application of the inversion operator · 226
7. The inversion theorems . · 230
• •
8. A general representation theorem · 235
9. Determining function non-decreasing · 236
• •
10. Determining function in L'IJ · 238