Title:

Mixing conditions and weight functions on the real line

The thesis in concerned with two problems from the prediction theory of continuous parameter stationary stochastic processes, and the related questions concerning the measurent on the real line which is associated with the process via Bochner's theorem. In Section 1 of Chapter 1, we describe briefly the background required from the theory of Hardy spaces in the upper halfplane, and some facts about entire functions of exponential type are given. Then, in Section 2, we discuss stationary processes and describe the main problems, motivating their study by a brief description of the classical prediction problems of Wiener and Kolmogorov and the work of Nelson, Sarason and Szegb. Chapter 2 is devoted to the proof of two representation theorems for weight functions satisfying the strong mixing condition pl  0 and the positive angle criterion pl < 1. The proof uses a result on analytic continuation and a characterisation of the algebra H4 + BUC. These results generalise the known results for discrete parameter processes. Chapter 3 consists of a discussion of the spaces BMO and VMO and their relationship to the strong mixing condition; and the HeleonSzegb condition of Chapter 2. We prove a result characterising those positive functions f on R for which log f E.VMO, and derive a connection between BMO, the condition pl t 1,and the boundedness of the con3ugat+enoperator on a subset of L2(p), depending on A. This generalises the discrete, version which is due to Nelson and Szegb. In Chapter 4, we consider the mixing conditions for a multivariate stationary process. The main result is an example of the mitian 2 x 2 matrix G, all of whose entries are real VMD functions, which is such that exp G does not satisfy the strong mixing condition pl '110. The proof depends on the construction of a VMO function which goes off to infinity at the origin, and the fact that no VMO function can have a jump discontinuity.
