Radom Fourier Series on Compact Groups

Abstract

In these lectures I shall present some aspects of the theory of random Fourier series on compact topological groups. The results presented here are taken largely from the papers of S. Helgason [8], A. Figà-Talamanca [4], [5], A. Figà-Talamanca and D. Rider [6], [7] and D. Rider [14]. For the classical case these results are contained, in the treatise of A. Zygmund [19] and in the monograph of J. P. Kahane [11] as well as in the original papers which are quoted there. There is some overlap between the exposition given here and that given in the second volume of [9] by E. Hewitt and K. Ross. However the point of view is different and in some cases, aiming at a less complete exposition of the subject matter, I have been able to shorten some of the proofs.

In Chapter I, I give an account, without proofs, of some of the classical results. In Chapter II I begin with a preliminary section in which I state the results from the general theory of compact groups, that I need in the sequel. In the succeeding sections I present extensions to the noncommutative situation of the results stated in Chapter I. Basically very little knowledge of probability theory is needed to understand these lectures. I do not define the notions of probability space, random variable and independence. In order to treat the case of L^∞ in section II.5 some more sophistication in probabilistic reasonings is required. Chapter I of [11] is a convenient, concise reference for the probability methods needed in the theory of Fourier series.