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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


On the divergence of the $(C,1)$ means
of double Walsh-Fourier series

Author: G. Gát
Journal: Proc. Amer. Math. Soc. 128 (2000), 1711-1720
MSC (1991): Primary 42C10; Secondary 43A75, 40G05, 42B08
Published electronically: October 27, 1999
MathSciNet review: 1657751
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Abstract: In 1992, Móricz, Schipp and Wade proved the a.e. convergence of the double $(C,1)$ means of the Walsh-Fourier series $\sigma _{n}f\to f$ ($\min (n_{1}, n_{2})\to \infty , n=(n_{1},n_{2})\in {\mathbb{N}} ^{2}$) for functions in $L\text{log}^{+} L(I^{2})$ ($I^{2}$ is the unit square). This paper aims to demonstrate the sharpness of this result. Namely, we prove that for all measurable function $\delta :[0,+\infty ) \to [0,+\infty ) , \, \lim _{t\to \infty }\delta (t)=0$ we have a function $f$ such as $f\in L\text{log}^{+} L\delta (L)$ and $\sigma _{n}f$ does not converge to $f$ a.e. (in the Pringsheim sense).

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Additional Information

G. Gát
Affiliation: Department of Mathematics, Bessenyei College, Nyíregyháza, P.O. Box 166., H–4400, Hungary

PII: S 0002-9939(99)05293-4
Keywords: Walsh group, double $(C, 1)$ means, divergence
Received by editor(s): July 13, 1998
Published electronically: October 27, 1999
Additional Notes: Research supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. F020334 and by the Hungarian “Művelődési és Közoktatási Minisztérium", grant no. FKFP 0710/1997.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society

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