TWO-SIDED ESTIMATES OF THE LEBESGUE CONSTANTS WITH RESPECT TO VILENKIN SYSTEMS AND APPLICATIONS

What is it about?

In this paper we derive two-sided estimates of the Lebesgue constants for bounded Vilenkin systems, we also present some applications of importance e.g. we obtain a characterization for the boundedness of a subsequence of partial sums with respect to Vilenkin-Fourier series of H 1 martingales in terms of n , s variation.

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Why is it important?

In this paper we improve the upper bound of Lukomskii and also prove a new similar lower bound by using a completely different new method. We also derive a relationship of the ratio of convergence of the partial sum of the Vilenkin series with the modulus of continuity of a martingale. The conditions given in the paper are in a sense necessary and sufficient.

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