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Divergence of Spherical General Terms of Double Fourier Series

Authors:

Rostom Getsadze

Institution:

1. Department of Mathematics and Mathematical Statistics, University of Ume?, S-901 87, Ume?, Sweden

Abstract:

We prove the following theorem: For arbitrary $\epsilon > 0$ there exists a nonnegative function $g \in L0, 1]2$ such that ${\rm supp} g \subset 0, \epsilon]2$ and

$\lim \sup_{ R\rightarrow \infty}\vert\sum_{ i^2+j^2=R^2}a_{i,j} (g)w_i (x)w_j (y)\vert =\infty$

almost everywhere on $0, 1]^2,$

where $\{w_i (x)w_j (y)\}^{\infty}_{ i,j=1}$ is the double Walsh-Paley system. This statement remains true also for the double trigonometric system.

Keywords:

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