Geodesic
Two-dimensional Waterman classes and $u$-convergence of Fourier series
Sbornik. Mathematics, Tome 190 (1999) no. 7, pp. 955-972 Cet article a éte moissonné depuis la source Math-Net.Ru

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New results on the $u$-convergence of the double Fourier series of functions from Waterman classes are obtained. It turns out that none of the Waterman classes wider than $BV(T^2)$ ensures even the uniform boundedness of the $u$-sums of the double Fourier series of functions in this class. On the other hand, the concept of $u(K)$-convergence is introduced (the sums are taken over regions that are forbidden to stretch along coordinate axes) and it is proved that for functions $f(x,y)$ belonging to the class $\Lambda_{1/2}BV(T^2)$, where $\Lambda_a=\biggl\{\dfrac{n^{1/2}}{{(\ln(n+1))}^a}\biggr\}_{n=1}^\infty$, the corresponding $u(K)$-partial sums are uniformly bounded, while if $f(x,y)\in\Lambda_aBV(T^2)$, where $a<\frac12$, then the double Fourier series of $f(x,y)$ is $u(K)$-convergent everywhere. 
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     author = {M. I. Dyachenko},
     title = {Two-dimensional {Waterman} classes and $u$-convergence of {Fourier} series},
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     pages = {955--972},
     year = {1999},
     volume = {190},
     number = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1999_190_7_a1/}
}
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M. I. Dyachenko. Two-dimensional Waterman classes and $u$-convergence of Fourier series. Sbornik. Mathematics, Tome 190 (1999) no. 7, pp. 955-972. http://geodesic.mathdoc.fr/item/SM_1999_190_7_a1/

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