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Almost Everywhere Convergence of Greedy Algorithm with Respect to Vilenkin System

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Abstract

In this paper, we prove that for any ε ∈ (0, 1) there exists ameasurable set E ∈ [0, 1) with measure |E| > 1 − ε such that for any function f ∈ L1[0, 1), it is possible to construct a function \(\tilde f \in {L^1}[0,1]\) coinciding with f on E and satisfying \(\int_0^1 {|\tilde f(x) - f(x)|dx < \varepsilon } \), such that both the Fourier series and the greedy algorithm of \(\tilde f\) with respect to a bounded Vilenkin system are almost everywhere convergent on [0, 1).

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Authors and Affiliations

  1. Yerevan State Umiversity, Yerevan, Armenia

    M. G. Grigoryan & S. A. Sargsyan

Authors
  1. M. G. Grigoryan
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  2. S. A. Sargsyan
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Correspondence to M. G. Grigoryan.

Additional information

Original Russian Text © M. G. Grigoryan, S. A. Sargsyan, 2018, published in Izvestiya Natsional’noi Akademii Nauk Armenii, Matematika, 2018, No. 6, pp. 13–32.

This work was supported by the RAMES State Committee of Science, in the frames of the research project 18T-1A148

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Grigoryan, M.G., Sargsyan, S.A. Almost Everywhere Convergence of Greedy Algorithm with Respect to Vilenkin System. J. Contemp. Mathemat. Anal. 53, 331–345 (2018). https://doi.org/10.3103/S1068362318060043

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  • DOI: https://doi.org/10.3103/S1068362318060043

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