The Convolution Transform [1 ed.] 0691653089, 9780691653082

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Contents CHAPTER

I

INTRODUCTION PAGE

SEOTION

1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduotion • Convolutions Operational calculus . Green's functions . Operational oalculus continued . The generation of kernels Variation diminishing oonvolutions Outline of program Summary . CHAPTER

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3 3 5 7 8 11 12 14 16

II

THE FINITE KERNELS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

17

Introduction Distribution funotions Frequency functions . Characteristic functions . Convolutions The finite kernels . Inversion Exponential polynomials . Green's functions . Examples . Summary . CHAPTER

17 19

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20 22

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24 28

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35 36

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42

III

THE NON· FINITE KERNELS 1. 2. 3. 4. 5.

Introduction Limits of distribution functions P61ya's olass of entire functions The closure of a class of distribution functions The non-finite kernels

· 48 · 49

OON TEN TS PAGE

emOTION'

6. 7. 8. 9. 10. 11.

Properties of the non-finite kernels . Inversion . Green's functions . Examples . Associated kernels Summary .

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65 79 82

CHAPTER IV VARIATION DIMINISHING TRANSFORMS Introduction Generation of variation diminishing frequency functions Logarithmic convexity Characterization of variation diminishing functions . The changes of sign of G(ft)(t) Intersection properties Generation of totally positive functions. Matrix transformations . Totally positive frequency functions

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Summary

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. 83 . 84 . 85 . 88 . 91 . 93 . 95 . 97 . 103 • 107

CHAPTER V ASYMPTOTIC BEHAVIOUR OF KERNELS 1. Introduction 2. Asymptotic estimates. 3. Asymptotic estimates continued 4. Summary • CluPTEB

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108 108 III 119

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120 120 123 125 127 132 138 142 145

VI

REAL INVERSION THEORY 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction Some preliminary results . Convergence The sequence of kernels . The inversion theorem Stieltjes integrals . Relaxation of continuity conditions Factorization . Summary .

OONTENTS ClIAPTER

VII

REPRESENTATION THEORY PAGE

SEO~ION

1. 2. 3. 4. 5. 6. 7. 8.

Introduction Behaviour at infinity . An elementary representation theorem. Determining funotion in L'P Determining funotions of bounded total variation Determining function non-decreasing Representation of products . Summary CHAPTER

1. 2. 3.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

• •

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146 147 150 152 156 158 163 169

VIII

THE WEIERSTRASS TRANSFORM: Introduction . 170 The Weierstrass transform . 173 The inversion operator . 179 Inversion . 182 Tychonoff's uniqueness theorem . 183 The Weierstrass theorem of bounded functions . 185 Inversion, general case . 188 Functions of L'P • 193 Weierstrass transforms of funotions in Lf) . • . 195 Weierstrass-Stieltjes transforms • . 197 Positive temperature functions • . 199 Weierstrass-Stieltjes transforms of increasing funotions . 202 Transforms of functions vlith presoribed order conditions . 206 Summary • . 209 CHAPTER

IX

COMPLEX INVERSION THEORY

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Introduotion • Transforms in the complex domain Behaviour at infinity . Auxiliary kernels The inversion funotion Application of the inversion operator The inversion theorems . A general representation theorem Determining function non-decreasing Determining function in L'IJ

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· 212

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217 218 223 226 230 235 236 238

OON TEN TS

CHAPTER

X

:MISCELLANEOUS TOPICS PA.GE

SEOTION

1. 2. 3. 4. 5.

Introduction . • . . • • • . • Bernstein polynomia.ls. . . • . Behaviour at infinity. . . . . . . The analytic character of kernels of olasses I Quasi-analyticity . •.

BIBLIOGRAPHY.

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SYMBOLS AND NOTATIONS . INDEX

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240 246 256 259

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