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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Ferenc Lukács type theorems in terms of the Abel-Poisson mean of conjugate series
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Proc. Amer. Math. Soc. 131 (2003), 1243-1250 Request permission

Abstract:

A theorem of Ferenc Lukács determines the generalized jumps of a periodic, Lebesgue integrable function $f$ in terms of the partial sum of the conjugate series to the Fourier series of $f$. The main aim of this paper is to prove an analogous theorem in terms of the Abel-Poisson mean. We also prove an estimate of the partial derivative (with respect to the angle) of the Abel-Poisson mean of an integrable function $F$ at those points at which $F$ is smooth. Finally, we reveal the intimate relation between these two results.
References
  • L. Fejér, Über die Bestimmung des Sprunges der Funktion aus ihrer Fourierreihe, J. reine angew. Math. 142 (1913), 165-188.
  • F. Lukács, Über die Bestimmung des Sprunges einer Funktion aus ihrer Fourierreihe, J. reine angew. Math. 150 (1920), 107-112.
  • P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
  • A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
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Additional Information
  • Ferenc Móricz
  • Affiliation: Bolyai Institute, University of Szeged, Aradi Vértanúk Tere 1, 6720 Szeged, Hungary
  • Email: moricz@math.u-szeged.hu
  • Received by editor(s): June 21, 2001
  • Received by editor(s) in revised form: December 3, 2001
  • Published electronically: September 5, 2002
  • Additional Notes: This research was started during the author’s visit to the Université de Paris-Sud, Orsay, in May 2001, and it was partially supported by the Hungarian National Foundation for Scientific Research under Grant T 029 094
  • Communicated by: Andreas Seeger
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 1243-1250
  • MSC (2000): Primary 42A50, 42A16
  • DOI: https://doi.org/10.1090/S0002-9939-02-06669-8
  • MathSciNet review: 1948116