Random Fields on the Sphere : Representation, Limit Theorems and Cosmological Applications.

880-01 Cover; Title; Copyright; Contents; Dedication; Preface; 1 Introduction; 1.1 Overview; 1.2 Cosmological motivations; 1.3 Mathematical framework; 1.4 Plan of the book; 2 Background Results in Representation Theory; 2.1 Introduction; 2.2 Preliminary remarks; 2.3 Groups: basic definitions; 2.3.1 First definitions and examples; 2.3.2 Cosets and quotients; 2.3.3 Actions; 2.4 Representations of compact groups; 2.4.1 Basic definitions; 2.4.2 Group representations and Schur Lemma; 2.4.3 Direct sum and tensor product representations; 2.4.4 Orthogonality relations; 2.5 The Peter-Weyl Theorem.

3 Representations of SO(3) and Harmonic Analysis on S23.1 Introduction; 3.2 Euler angles; 3.2.1 Euler angles for SU(2); 3.2.2 Euler angles for SO(3); 3.3 Wigner's D matrices; 3.3.1 A family of unitary representations of SU(2); 3.3.2 Expressions in terms of Euler angles and irreducibility; 3.3.3 Further properties; 3.3.4 The dual of SO(3); 3.4 Spherical harmonics and Fourier analysis on S2; 3.4.1 Spherical harmonics and Wigner's Dl matrices; 3.4.2 Some properties of spherical harmonics; 3.4.3 An alternative characterization of spherical harmonics; 3.5 The Clebsch-Gordan coefficients.

3.5.1 Clebsch-Gordan matrices3.5.2 Integrals of multiple spherical harmonics; 3.5.3 Wigner 3 j coefficients; 4 Background Results in Probability and Graphical Methods; 4.1 Introduction; 4.2 Brownian motion and stochastic calculus; 4.3 Moments, cumulants and diagram formulae; 4.4 The simplified method of moments on Wiener chaos; 4.4.1 Real kernels; 4.4.2 Further results on complex kernels; 4.5 The graphical method for Wigner coefficients; 4.5.1 From diagrams to graphs; 4.5.2 Further notation; 4.5.3 First example: sums of squares; 4.5.4 Cliques and Wigner 6 j coefficients.

4.5.5 Rule n. 1: loops are zero4.5.6 Rule n. 2: paired sums are one; 4.5.7 Rule n. 3: 2-loops can be cut, and leave a factor; 4.5.8 Rule n. 4: three-loops can be cut, and leave a clique; 5 Spectral Representations; 5.1 Introduction; 5.2 The Stochastic Peter-Weyl Theorem; 5.2.1 General statements; 5.2.2 Decompositions on the sphere; 5.3 Weakly stationary random fields in Rm; 5.4 Stationarity and weak isotropy in R3; 6 Characterizations of Isotropy; 6.1 Introduction; 6.2 First example: the cyclic group; 6.3 The spherical harmonics coefficients; 6.4 Group representations and polyspectra.

6.5 Angular polyspectra and the structure of?l1 ... ln6.5.1 Spectra of strongly isotropic fields; 6.5.2 The structure of?l1 ... ln; 6.6 Reduced polyspectra of arbitrary orders; 6.7 Some examples; 7 Limit Theorems for Gaussian Subordinated Random Fields; 7.1 Introduction; 7.2 First example: the circle; 7.3 Preliminaries on Gaussian-subordinated fields; 7.4 High-frequency CLTs; 7.4.1 Hermite subordination; 7.5 Convolutions and random walks; 7.5.1 Convolutions on?SO (3); 7.5.2 The cases q = 2 and q = 3; 7.5.3 The case of a general q: results and conjectures; 7.6 Further remarks.

7.6.1 Convolutions as mixed states.

Reviews recent developments in the analysis of isotropic spherical random fields, with a view towards applications in cosmology.

Print version record.

Includes bibliographical references (pages 326-337) and index.

eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - Worldwide