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Cesàro Means of Subsequences of Partial Sums of Trigonometric Fourier Series

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Abstract

In 1936 Zygmunt Zalcwasser asked, with respect to the trigonometric system, how “rare” can a sequence of strictly monotone increasing integers \((n_j)\) be such that the almost everywhere relation \(\frac{1}{N}\sum _{j=1}^N S_{n_j}f \rightarrow f\) is fulfilled for each integrable function f. In this paper, we give an answer to this question. It follows from the main result that this a.e. relation holds for every integrable function f and lacunary sequence \((n_j)\) of natural numbers.

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Acknowledgements

The author is deeply indebted to the anonymous referees for finding some errors in the first version of the manuscript and for their valuable help.

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Authors and Affiliations

  1. Institute of Mathematics, University of Debrecen, Debrecen, Pf. 400, 4002, Hungary

    György Gát

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  1. György Gát
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Correspondence to György Gát.

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Communicated by Vilmos Totik.

Research supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. K111651 and by project EFOP-3.6.1-16-2016-00022 supported by the European Union, co-financed by the European Social Fund.

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Gát, G. Cesàro Means of Subsequences of Partial Sums of Trigonometric Fourier Series. Constr Approx 49, 59–101 (2019). https://doi.org/10.1007/s00365-018-9438-2

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  • DOI: https://doi.org/10.1007/s00365-018-9438-2

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