Harmonic Analysis and the Theory of Probability

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ISBN: 9780486446202 | 0486446204
Cover: Paperback
Copyright: 8/27/2005

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This classic text emphasizes the stochastic processes and the interchange of stimuli between probability and analysis. Non-probabilistic topics include Fourier series and integrals in many variables; the Bochner integral; and the transforms of Plancherel, Laplace, Poisson, and Mellin. Most notable is the systematic presentation of Bochner's characteristic functional. 1955 edition.

Chapter 1. Approximations

1.1. Approximation of functions at points

(6)

1.2. Translation functions

(2)

1.3. Approximation in norm

(2)

1.4. Vector-valued functions

(1)

1.5. Additive set functions

(6)

1.6. Periodic additive set functions

(4)

Chapter 2. Fourier Expansions

2.1. Fourier integrals

(3)

2.2. Positive transforms. Plancherel transforms

(3)

2.3. Fourier series

(2)

2.4. Poisson summation formula

(3)

2.5. Summability. Heat and Laplace equations

(3)

2.6. Theta relations with spherical harmonics

(4)

2.7. Expansions in spherical harmonics

(4)

2.8. Zeta integrals

(3)

2.9. Zeta series

(5)

Chapter 3. Closure Properties of Fourier Transforms

3.1. Pseudo-characters and Poisson characters

(3)

3.2. Pseudo-transforms and positive definite functions

(4)

3.3. Poisson transforms

(6)

3.4. Infinitely subdivisible processes

(5)

3.5. Absolute moments

(3)

3.6. Locally compact Abelian groups

(3)

3.7. Random variables

(4)

3.8. General positivity

(2)

Chapter 4 Laplace and Mellin Transforms

4.1. Completely monotone functions in one variable

(4)

4.2. Completely monotone functions in several variables

(5)

4.3. Subordination of infinitely subdivisible processes

(4)

4.4. Subordination of Markoff processes

(4)

4.5. A theorem of Hardy, Littlewood and Paley

(3)

4.6. Functions of the Laplace operator

102

(4)

4.7. Multidimensional time variable

106

(2)

4.8. Rimann's functional equation for zeta functions

108

(4)

4.9. Summation formulas and Bessel functions in one and several variables

112

(6)

Chapter 5 Stochastic Processes and Characteristic functionals

5.1. Directed sets of probability spaces

118

(4)

5.2. Markoff processes

122

(5)

5.3. Length of random paths in homogeneous spaces

127

(6)

5.4. Euclidean stochastic processes and their characteristic functionals

133

(4)

5.5. Random functions

137

(6)

5.6. Generating functionals

143

(2)

Chapter 6. Analysis of Stochastic Processes

6.1. Basic operations with characteristic functionals

145

(5)

6.2. Convergence in probability and integration

150

(3)

6.3. Convergence in norm

153

(3)

6.4. Expansion in series and integrals

156

(4)

6.5. Stationarity and orthogonality

160

(3)

6.6. Further statements

163

(5)

Notes and References

168

(5)

Indexes

173

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Harmonic Analysis and the Theory of Probability

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Table of Contents

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