UNCONDITIONAL CONVERGENCE AND UNCONDITIONAL BASES IN HARDY SPACES

Abstract

This paper is devoted to a study of unconditional convergence of series in the Hardy space H1(ℝs) and unconditional bases for H1(ℝs). First, we use quasi-projection operators from approximation theory to give a very general criterion for unconditional convergence in H1. Second, we prove that a system of wavelets forms an unconditional basis of the Hardy space H1, provided the dual wavelet lies in a Lipschitz space of positive order. In particular, for H1(ℝ) we construct an unconditional basis consisting of piecewise constant functions. Third, we demonstrate that our conditions for unconditional bases are sharp by showing that, if the dual refinable function is the characteristic function of the interval [0, 1), then the corresponding system of wavelets does not form an unconditional basis for H1(ℝ), even though the wavelet itself could have arbitrarily high smoothness.