Norm convergence of double Fourier series on unbounded Vilenkin groups
V. I. Shcherbakov [Anal. Math. 10, 133--150 (1984; Zbl 0563.42015)] proved an analogue of the Dini-Lipshitz test for the convergence of Fourier series with respect to multiplicative Vilenkin systems. It looks like \( \omega(1/M_k,f)=o(1/\log M_{k+1}) \) as \( k\to\infty. \) Here the authors extend this result to the convergence of rectangular partial sums of double Fourier series with respect to multiplicative Vilenkin systems. In particular, they prove the convergence (in \( C \) and \( L_1 \)) of those sums if \(\omega_1(1/M_k,f)=o(1/\log M_{k+1}) \) and \( \omega_2(1/M_k,f)=o(1/\log M_{k+1}) \) as \( k\to\infty. \) The result is sharp in the sense that those sums may diverge if both moduli of continuity (in each of the variables) satisfy the conditions from above with \( O \) instead of \( o. \)
scientific article; zbMATH DE number 6837908
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