Absolute convergence of double Walsh–Fourier series and related results

We consider the double Walsh orthonormal system on the unit square , where { w m ( x )} is the ordinary Walsh system on the unit interval in the Paley enumeration. Our aim is to give sufficient conditions for the absolute convergence of the double Walsh–Fourier series of a function for some 1< p...

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Vydáno v:	Acta mathematica Hungarica Ročník 131; číslo 1-2; s. 122 - 137
Hlavní autoři:	Móricz, Ferenc, Veres, Antal
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Dordrecht Springer Netherlands 01.04.2011
Témata:	Mathematics Mathematics and Statistics 42C10 dyadic modulus of continuity double Walsh–Fourier series 26A16 dyadic functions of modulus of continuity 26A15 bounded fluctuation absolute convergence dyadic Lipschitz classes of functions in two variables
ISSN:	0236-5294, 1588-2632
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Popis
Shrnutí:	We consider the double Walsh orthonormal system on the unit square , where { w m ( x )} is the ordinary Walsh system on the unit interval in the Paley enumeration. Our aim is to give sufficient conditions for the absolute convergence of the double Walsh–Fourier series of a function for some 1< p ≦2. More generally, we give best possible sufficient conditions for the finiteness of the double series where { a mn } is a given double sequence of nonnegative real numbers satisfying a mild assumption and 0< r <2. These sufficient conditions are formulated in terms of (either global or local) dyadic moduli of continuity of f .
ISSN:	0236-5294 1588-2632
DOI:	10.1007/s10474-010-0065-z