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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| Published in: | Acta mathematica Hungarica Vol. 131; no. 1-2; pp. 122 - 137 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Dordrecht
Springer Netherlands
01.04.2011
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| Subjects: | |
| ISSN: | 0236-5294, 1588-2632 |
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| Abstract | 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
. |
|---|---|
| AbstractList | 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
. |
| Author | Veres, Antal Móricz, Ferenc |
| Author_xml | – sequence: 1 givenname: Ferenc surname: Móricz fullname: Móricz, Ferenc email: moricz@math.u-szeged.hu organization: Bolyai Institute, University of Szeged – sequence: 2 givenname: Antal surname: Veres fullname: Veres, Antal organization: Bolyai Institute, University of Szeged |
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| Cites_doi | 10.1090/S0002-9947-1933-1501718-2 |
| ContentType | Journal Article |
| Copyright | Akadémiai Kiadó, Budapest, Hungary 2011 |
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| Keywords | 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 |
| Language | English |
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| References | Zygmund (CR5) 1959 Schipp, Wade, Simon, Pál (CR3) 1990 Clarkson, Adams (CR1) 1933; 35 Gogoladze, Meskhia (CR2) 2006; 141 Stein, Weiss (CR4) 1971 A. Zygmund (65_CR5) 1959 L. Gogoladze (65_CR2) 2006; 141 E. M. Stein (65_CR4) 1971 F. Schipp (65_CR3) 1990 J. A. Clarkson (65_CR1) 1933; 35 |
| References_xml | – volume: 141 start-page: 29 year: 2006 end-page: 40 ident: CR2 article-title: On the absolute convergence of trigonometric Fourier series publication-title: Proc. Razmadze Math. Inst. – year: 1971 ident: CR4 publication-title: Introduction to Fourier Analyis on Euclidean Spaces – year: 1990 ident: CR3 publication-title: Walsh Series: an Introduction to Dyadic Harmonic Analysis – volume: 35 start-page: 824 year: 1933 end-page: 854 ident: CR1 article-title: On definitions of bounded variation for functions of two variables publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-1933-1501718-2 – year: 1959 ident: CR5 publication-title: Trigonometric Series – volume: 141 start-page: 29 year: 2006 ident: 65_CR2 publication-title: Proc. Razmadze Math. Inst. – volume-title: Walsh Series: an Introduction to Dyadic Harmonic Analysis year: 1990 ident: 65_CR3 – volume: 35 start-page: 824 year: 1933 ident: 65_CR1 publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-1933-1501718-2 – volume-title: Introduction to Fourier Analyis on Euclidean Spaces year: 1971 ident: 65_CR4 – volume-title: Trigonometric Series year: 1959 ident: 65_CR5 |
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| Snippet | 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... |
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| SubjectTerms | Mathematics Mathematics and Statistics |
| Title | Absolute convergence of double Walsh–Fourier series and related results |
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