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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Ferenc Lukács type theorems in terms of the Abel-Poisson mean of conjugate series


Author: Ferenc Móricz
Journal: Proc. Amer. Math. Soc. 131 (2003), 1243-1250
MSC (2000): Primary 42A50, 42A16
Published electronically: September 5, 2002
MathSciNet review: 1948116
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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 [Enhancements On Off] (What's this?)

  • 1. L. Fejér, Über die Bestimmung des Sprunges der Funktion aus ihrer Fourierreihe, J. reine angew. Math. 142 (1913), 165-188.
  • 2. F. Lukács, Über die Bestimmung des Sprunges einer Funktion aus ihrer Fourierreihe, J. reine angew. Math. 150 (1920), 107-112.
  • 3. A. Zygmund, Smooth functions, Duke Math. J. 12 (1945), 47–76. MR 0012691 (7,60b)
  • 4. A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776 (21 #6498)

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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

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06669-8
PII: S 0002-9939(02)06669-8
Keywords: Function of bounded variation, induced Borel measure, Fourier series, theorem of Fej\'er, conjugate series, generalized jump, theorem of Ferenc Luk\'acs, Abel-Poisson mean, smoothness, Zygmund classes $\lambda_*$ and $\Lambda_*$.
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
Article copyright: © Copyright 2002 American Mathematical Society