On the square and the spherical partial sums of multiple orthogonal series

Abstract

Let Za+ be the set of d-tuples k = (kl, ..., ka) with positive integral coordinates, the tuple (1, ..., 1) is denoted by 1. Z~_ is partially ordered by agreeing that k ~ m iff k j ~ m i for each j. Let (X, d , #) be a positive measure space and let {q~k(X)}={gk(X): kCZa+} be an orthonormal system on (2, ~ , /0 . We shall study the convergence behaviour of the square and the spherical partial sums of the d-multiple orthogonal series