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Constructive Approximation

, Volume 49, Issue 1, pp 59–101 | Cite as

Cesàro Means of Subsequences of Partial Sums of Trigonometric Fourier Series

  • György Gát
Article
  • 154 Downloads

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.

Keywords

Cesàro means Subsequences of partial sums Trigonometric Fourier series a.e. convergence Zalcwasser’s problem 

Mathematics Subject Classification

42A24 

Notes

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. 1.Institute of MathematicsUniversity of DebrecenDebrecenHungary

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