Authors:
Ferenc Móricz Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., 6720 Szeged, Hungary

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Antal Veres Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., 6720 Szeged, Hungary

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Abstract

We consider the double Walsh orthonormal system
ea
on the unit square , where {wm(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
eb
where {amn} 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.
  • [1] Clarkson, J. A., Adams, C. R. 1933 On definitions of bounded variation for functions of two variables Trans. Amer. Math. Soc. 35 824854 .

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  • [2] Gogoladze, L., Meskhia, R. 2006 On the absolute convergence of trigonometric Fourier series Proc. Razmadze Math. Inst. 141 2940.

  • [3] Schipp, F., Wade, W. R., Simon, P., Pál, J. 1990 Walsh Series: an Introduction to Dyadic Harmonic Analysis Adam Hilger Bristol and New York.

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  • [4] Stein, E. M., Weiss, G. 1971 Introduction to Fourier Analyis on Euclidean Spaces Princeton University Press Princeton.

  • [5] Zygmund, A. 1959 Trigonometric Series Cambridge University Press Cambridge.

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Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation
1950
Volumes
per Year
3
Issues
per Year
6
Founder Magyar Tudományos Akadémia  
Founder's
Address
H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

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