D’yachenko, M. I. Two-dimensional Waterman classes and \(u\)-convergence of Fourier series. (English. Russian original) Zbl 0937.42006 Sb. Math. 190, No. 7, 955-972 (1999); translation from Mat. Sb. 190, No. 7, 23-40 (1999). 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 \(\Lambda_{1/2} \text{BV}(T^2)\), where \(\Lambda_a=\{\frac{n^{1/2}}{(\ln(n+1))^a}\}^\infty_{n=1}\), the corresponding \(u(K)\)-partial sums are uniformly bounded, while if \(f(x, y)\in\Lambda_a \text{BV}(T^2)\), where \(a < 1/2\), then the double Fourier series of \(f(x, y)\) is \(u(K)\)-convergent everywhere. Cited in 6 Documents MSC: 42B05 Fourier series and coefficients in several variables 26B30 Absolutely continuous real functions of several variables, functions of bounded variation 42B08 Summability in several variables Keywords:convergence; double Fourier series; Waterman classes PDFBibTeX XMLCite \textit{M. I. D'yachenko}, Sb. Math. 190, No. 7, 955--972 (1999; Zbl 0937.42006); translation from Mat. Sb. 190, No. 7, 23--40 (1999) Full Text: DOI