Geodesic
Examples of Divergent Fourier Series for Classes of Functions of Bounded $\Lambda$-Variation
Matematičeskie zametki, Tome 86 (2009) no. 5, pp. 664-672

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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.
Mots-clés : divergent Fourier series, harmonic variation.
Keywords: function of bounded $\Lambda$-variation, convergence of Fourier series over rectangles and cubes, Waterman class of functions 
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     author = {A. N. Bakhvalov},
     title = {Examples of {Divergent} {Fourier} {Series} for {Classes} of {Functions} of {Bounded} $\Lambda${-Variation}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {664--672},
     publisher = {mathdoc},
     volume = {86},
     number = {5},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2009_86_5_a2/}
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A. N. Bakhvalov. Examples of Divergent Fourier Series for Classes of Functions of Bounded $\Lambda$-Variation. Matematičeskie zametki, Tome 86 (2009) no. 5, pp. 664-672. http://geodesic.mathdoc.fr/item/MZM_2009_86_5_a2/