Basis properties of affine Walsh systems in symmetric spaces

摘要

We study the basis properties of affine Walsh-type systems in symmetric spaces. We show that the ordinary Walsh system is a basis in a separable symmetric space X if and only if the Boyd indices of X are non-trivial, that is, 0 alpha(X) = beta(X) 1. In the more general situation when the generating function f is the sum of a Rademacher series, we find exact conditions for the affine system {f(n)}(n=0)(infinity) to be equivalent to the Walsh system in an arbitrary separable s. s. with non-trivial Boyd indices. We also obtain sufficient conditions for the basis property. In particular, it follows from these conditions that for every p is an element of (1, infinity) there is a function f such that the affine Walsh system {f(n)}(n=0)(infinity) generated by f is a basis in those and only those separable s. s. X that satisfy 1/p alpha(X) = beta(X) 1.