Unconditional Convergence Of Series Research Articles

Unconditional Convergence of Functional Series in Problems of Probability Theory

We study the unconditional convergence of series in Banach spaces. We consider series of special type (Hadamard series), obtain the condition of their unconditional convergence, and discuss some of their applications. Further, we examine the almost sure unconditional convergence of random series in Banach spaces and, in the case of Gaussian series, we establish the relationship between the almost sure unconditional convergence and the geometry of Banach spaces. We also consider some probabilistic problems related to the convergence of series.

On the unconditional convergence of series with respect to the Faber-Schauder system

We prove that for every 0 1−ɛ, such that the Faber-Schauder system is an unconditional convergence system for the representation of functions of the class C(E).

UNCONDITIONAL CONVERGENCE AND UNCONDITIONAL BASES IN HARDY SPACES

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.

On absolute and unconditional convergence of series in the general Franklin system

We prove that, for any admissible sequence, the corresponding general Franklin system possesses the following property. A series is absolutely convergent almost everywhere on a set if and only if it is unconditionally convergent almost everywhere on .

Weyl multipliers for the unconditional convergence of series in a Franklin system

Weyl multipliers for the unconditional convergence of series in a Franklin system

Series in the Haar system

We obtain some results concerning the unconditional convergence of series in the Haar system in the metric of L(0, 1).

On bases and unconditional convergence of series in Banach spaces

On bases and unconditional convergence of series in Banach spaces