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