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Mat. Zametki, 2009, Volume 86, Issue 5, Pages 664–672 (Mi mz6631)  

Examples of Divergent Fourier Series for Classes of Functions of Bounded $\Lambda$-Variation

A. N. Bakhvalov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The author has shown earlier that the requirement that a continuous function belong to the class $HBV([-\pi,\pi]^m)$ for $m\ge 3$ is not sufficient for the convergence of its Fourier series over rectangles. The author gave examples of functions of three and more variables from the Waterman class which are harmonic in the first variable and significantly narrower in the other variables and whose Fourier series are divergent at some point even on cubes. In the present paper, this assertion is strengthened. The main result is that such an example can be constructed even when the class with respect to the first variable is somewhat narrowed. Also the one-dimensional result due to Waterman is refined.

Keywords: divergent Fourier series, function of bounded $\Lambda$-variation, convergence of Fourier series over rectangles and cubes, Waterman class of functions, harmonic variation

DOI: https://doi.org/10.4213/mzm6631

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English version:
Mathematical Notes, 2009, 86:5, 629–636

Bibliographic databases:

UDC: 517.518
Received: 10.12.2008

Citation: A. N. Bakhvalov, “Examples of Divergent Fourier Series for Classes of Functions of Bounded $\Lambda$-Variation”, Mat. Zametki, 86:5 (2009), 664–672; Math. Notes, 86:5 (2009), 629–636

Citation in format AMSBIB
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