5 - Spectral Representations

Summary

Introduction

In this chapter, we present a number of representation results for random fields defined on ℝ³, or on the sphere S². We shall focus on random fields whose law verifies some invariance properties, namely isotropy (i.e., invariance with respect to rotations) and/or stationarity (i.e., invariance with respect to translations).

Section 5.2 contains facts that are implicitly used throughout the rest of this book, connecting strong isotropy to group representations and the Peter-Weyl Theorem – with special emphasis on spherical random fields, that is, random fields indexed by the unit sphere. In Sections 5.3 and 5.4 we focus on weak versions of stationarity and isotropy and the associated spectral representations. In particular, the content of Section 5.4 provides alternate proofs (with no group theory involved) of some of the main findings of Section 5.2. Note that Section 5.3 and Section 5.4 are not used in the rest of the book, so that some technical details have been skipped in order to keep the presentation as concise and to the point as possible. Adequate references are given below: in general, for a textbook treatment of stationary and isotropic fields on ℝ^m, one can consult Yadrenko [201], Adler [2], Ivanov and Leonenko [104], Leonenko [125], Adler and Taylor [3] and the references therein.

The Stochastic Peter-Weyl Theorem

General statements

We will now use notions and definitions that have been introduced in Chapter 2. Let G be a topological compact group, and let dg be the associated Haar measure with unit mass.