Convergence of double Fourier series and -classes

Authors:
M. I. Dyachenko and D. Waterman

Journal:
Trans. Amer. Math. Soc. **357** (2005), 397-407

MSC (2000):
Primary 42B05, 26B30; Secondary 26B05

DOI:
https://doi.org/10.1090/S0002-9947-04-03525-1

Published electronically:
July 22, 2004

MathSciNet review:
2098101

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Abstract | References | Similar Articles | Additional Information

Abstract: The double Fourier series of functions of the generalized bounded variation class are shown to be Pringsheim convergent everywhere. In a certain sense, this result cannot be improved. In general, functions of class defined here, have quadrant limits at every point and, for there exist at most countable sets and such that, for and is continuous at . It is shown that the previously studied class contains essentially discontinuous functions unless the sequence satisfies a strong condition.

**1.**M. Avdispahic, Concepts of generalized bounded variation and the theory of Fourier series. Internat. J. Math. Math. Sci.**9**(1986), no. 2, 223-244. MR**88c:42001****2.**L. A. D'Antonio, D. Waterman, A summability method for Fourier series of functions of generalized bounded variation. Analysis**17**(1997), no. 2-3, 287-299. MR**99h:42008****3.**M. I. Dyachenko, Waterman classes and spherical partial sums of double Fourier series. Anal. Math.**21**(1995), no. 1, 3-21. MR**97m:42009****4.**C. Goffman, D. Waterman, The localization principle for double Fourier series. Studia Math.**69**(1980/81), no. 1, 41-57. MR**82d:42010****5.**A. A. Saakyan, On the convergence of double Fourier series of functions of bounded harmonic variation. (Russian) Izv. Akad. Nauk Armyan. SSR Ser. Mat.**21**(1986), no. 6, 517-529; English transl., Soviet J. Contemp. Math. Anal.**21**(1986), no. 6, 1-13. MR**88j:42017****6.**A. I. Sablin, -variation and Fourier series. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 1987, no. 10, 66-68; English transl., Soviet Math. (Iz. VUZ) 31 (1987), no. 10, 87-90. MR**89c:42008****7.**D. Waterman, On convergence of Fourier series of functions of generalized bounded variation. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity. II. Studia Math.**44**(1972), 107-117. MR**46:9623****8.**-, On the summability of Fourier series of functions of -bounded variation. Studia Math.**54**(1975/76), no. 1, 87-95 MR**53:6212****9.**-, On -bounded variation. Studia Math.**57**(1976), no. 1, 33-45. MR**54:5408****10.**-, Fourier series of functions of -bounded variation. Proc. Amer. Math. Soc.**74**(1979), no. 1, 119-123. MR**80j:42010****11.**-, On some high-indices theorems. II. J. London Math. Soc. (2)**59**(1999), no. 3, 978-986. MR**2000k:40006**

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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

Email:
dyach@mail.ru

**D. Waterman**

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

Email:
fourier@adelphia.net

DOI:
https://doi.org/10.1090/S0002-9947-04-03525-1

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.