Harmonic Analysis and the Theory of Probability

Contents......Page 3
1.1. Approximation of functions at points......Page 5
1.2. Translation functions......Page 11
1.3. Approximation in norm......Page 13
1.4. Vector-valued functions......Page 15
1.5. Additive set functions......Page 16
1.6. Periodic additive set functions......Page 22
2.1. Fourier integrals......Page 26
2.2. Positive transforms. Plancherel transforms......Page 29
2.3. Fourier series......Page 32
2.4. Poisson summation formula......Page 34
2.5. Summability. Heat and Laplace equations......Page 37
2.6. Theta relations with spherical harmonics......Page 40
2.7. Expansions in spherical harmonies......Page 44
2.8. Zeta integrals......Page 48
2.9. Zeta series......Page 51
3.1. Pseudo-characters and Poisson characters......Page 56
3.2. Pseudo-transforms and positive definite functions......Page 59
3.3. Poisson transforms......Page 63
3.4. Infinitely subdivisible processes......Page 69
3.5. Absolute moments......Page 74
3.6. Locally compact Abelian groups......Page 77
3.7. Random variables......Page 80
3.8. General positivity......Page 84
4.1. Completely monotone functions in one variable......Page 86
4.2. Completely monotone functions in several variables......Page 90
4.3. Subordination of infinitely subdivisible processes......Page 95
4.4. Subordination of Markoff processes......Page 99
4.5. A theorem of Hardy, Littlewood and Paley......Page 103
4.6. Functions of the Laplace operator......Page 106
4.7. Multidimensional time variable......Page 110
4.8. Riemann's functional equation for zeta functions......Page 112
4.9. Summation formulas and Bessel functions in one and several variables......Page 116
5.L Directed sets of probability spaces......Page 122
5.2. Markoff processes......Page 126
5.3. Length of random paths in homogeneous spaces......Page 131
5.4. Euclidean stochastic processes and their characteristic functionals......Page 137
5.5. Random functions......Page 141
5.6. Generating functionals......Page 147
6.1. Basic operations with characteristic functionals......Page 149
6.2. Convergence in probability and integration......Page 154
6.3. Convergence in norm......Page 157
6.4. Expansion in series and integrals......Page 160
6.5. Stationarity and orthogonality......Page 164
6.6. Further statements......Page 167
Notes and References......Page 172
Indexes......Page 177