Martingale theory and pointwise convergence of certain orthogonal series

Abstract

The purpose of this paper is to point out the relevance of martingale theory in the study of some classical problems concerning the pointwise convergence of orthogonal series. Particular attention is given to the Haar and the Walsh systems. The special character of their convergence properties has been the focus of some recent investigations, (see for example [13], [14], [15]) but the usefulness of ideas from martingale theory does not seem to have been recognized. We take advantage of the structure of these systems together with standard martingale techniques to obtain some new convergence theorems for a class of orthonormal systems which include the Haar and the Walsh functions as special cases. In the first section, the connection between the Haar and Walsh systems is described, and the martingale properties of these systems are isolated. This leads to the definition of a class of orthonormal systems, called H-systems, that are generalizations of the Haar system. Our interest in H-systems stems from two facts: (a) it is shown that every complete orthonormal system of martingale differences is an H-system; and (b) optional stopping and skipping transformations are especially simple when the martingales in question come from H-systems. The construction of the Haar system from the Rademacher functions suggests a specialization of the notion of an H-system: an H*-system is an H-system constructed from a sequence of independent binomial functions in the manner the Rademacher functions generate the Haar system. Consequently, any H*system has an associated parameter sequence {Pk}k= , the sequence of probabilities associated with the independent binomial functions. (In this way, the Haar system is associated with the sequence Pk= 2, k-1,2,.). The influence of the parameter sequence on the convergence properties of H*-systems is studied in the second and third sections. In the second section of the paper, we consider H-systems in relation to the following question: Given a complete orthonormal system {Uk}lk? 1 and a measurable function f, does there exist a series k 1 that converges to f