Harmonic Analysis and the Theory of

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

1

(6)

1.2. Translation functions

7

(2)

1.3. Approximation in norm

9

(2)

1.4. Vector-valued functions

11

(1)

1.5. Additive set functions

12

(6)

1.6. Periodic additive set functions

18

(4)

Chapter 2. Fourier Expansions

2.1. Fourier integrals

22

(3)

2.2. Positive transforms. Plancherel transforms

25

(3)

2.3. Fourier series

28

(2)

2.4. Poisson summation formula

30

(3)

2.5. Summability. Heat and Laplace equations

33

(3)

2.6. Theta relations with spherical harmonics

36

(4)

2.7. Expansions in spherical harmonics

40

(4)

2.8. Zeta integrals

44

(3)

2.9. Zeta series

47

(5)

Chapter 3. Closure Properties of Fourier Transforms

3.1. Pseudo-characters and Poisson characters

52

(3)

3.2. Pseudo-transforms and positive definite functions

55

(4)

3.3. Poisson transforms

59

(6)

3.4. Infinitely subdivisible processes

65

(5)

3.5. Absolute moments

70

(3)

3.6. Locally compact Abelian groups

73

(3)

3.7. Random variables

76

(4)

3.8. General positivity

80

(2)

Chapter 4 Laplace and Mellin Transforms

4.1. Completely monotone functions in one variable

82

(4)

4.2. Completely monotone functions in several variables

86

(5)

4.3. Subordination of infinitely subdivisible processes

91

(4)

4.4. Subordination of Markoff processes

95

(4)

4.5. A theorem of Hardy, Littlewood and Paley

99

(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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