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Transactions of the American Mathematical Society

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Convergence of double Fourier series and $W$-classes

Authors: M. I. Dyachenko and D. Waterman
Journal: Trans. Amer. Math. Soc. 357 (2005), 397-407
MSC (2000): Primary 42B05, 26B30; Secondary 26B05
Published electronically: July 22, 2004
MathSciNet review: 2098101
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Abstract: The double Fourier series of functions of the generalized bounded variation class $\{n/\ln (n+1)\}^{\ast }BV$ are shown to be Pringsheim convergent everywhere. In a certain sense, this result cannot be improved. In general, functions of class $\Lambda ^{\ast }BV,$ defined here, have quadrant limits at every point and, for $f\in \Lambda ^{\ast }BV,$ there exist at most countable sets $P$ and $Q$ such that, for $x\notin P$ and $y\notin Q,$ $f$is continuous at $(x,y)$. It is shown that the previously studied class $ \Lambda BV$ contains essentially discontinuous functions unless the sequence $\Lambda $ satisfies a strong condition.

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Additional Information

M. I. Dyachenko
Affiliation: Professor of the Chair of Theory of Functions and Functional Analysis, Department of Mathematics and Mechanics, Moscow State University, Vorobyevi Gori, GZ, Moscow, Russia 119992

D. Waterman
Affiliation: Research Professor, Florida Atlantic University (Professor Emeritus, Syracuse University), 7739 Majestic Palm Drive, Boynton Beach, Florida 33437

Keywords: Multiple Fourier series, generalized bounded variation, Waterman classes
Received by editor(s): March 17, 2003
Received by editor(s) in revised form: September 29, 2003
Published electronically: July 22, 2004
Additional Notes: The first author gratefully acknowledges the support of RFFI grant N03-01-00080
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.