absolute convergence

On the Representation of Functions by Absolutely Convergent Series by H -system

Authors: Navasardyan K. A.

The paper deals with the representation of absolutely convergent series of functions in spaces of homogeneous type. The definition of a system of Haar type (H-system) associated to a dyadic family on a space of homogeneous type X is given in the Introduction. It is proved that for almost everywhere (a.e.) finite and measurable on a set  X  function f there exists an absolutely convergent series by the system H, which converges to  f  a.e. on  X .

Generalized Absolute Convergence of Series with Respect to Multiplicative Systems of Functions of Generalized Bounded Variation

Authors: Kuznetsova M. A.

A. Zygmund proved that a  2π-periodic function with bounded variation and from any Lipschitz class Lip(α) has absolutely convergent Fourier series. This result was extended to many classes of functions of generalized bounded variation (for example, functions of bounded Jordan-Wiener  p-variation, functions of bounded Λ-variation introduced by D. Waterman et al) and to different spaces defined with the help of moduli of continuity.

On Weighted Analogs of Wiener’s and Levy’s Theorems for Fourier – Vilenkin Series

Authors: Volosivets S. S.

In this paper we find the general form of complex homomorphism for some subalgebras of absolutely convergent Fourier – Vilenkin series algebra. As a corollary, we obtain weighted analogs of Wiener’s and Levy’s theorems for Fourier – Vilenkin series.

Absolute Convergence of Single and Double Fourier Series on Multiplicative Systems

Authors: Volosivets S. S.

Two-dimensional analogs of famous Zygmund and Szasz tests for absolute convergence of Fourier – Vilenkin series are established. Also it is proved that two-dimensional Szasz test is the best possible in the certain sense.