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Ferenc Lukács type theorems in terms of the Abel-Poisson mean of conjugate series
Author(s):
Ferenc
Móricz
Journal:
Proc. Amer. Math. Soc.
131
(2003),
1243-1250.
MSC (2000):
Primary 42A50, 42A16
Posted:
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  in terms of the partial sum of the conjugate series to the Fourier series of  . 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  at those points at which  is smooth. Finally, we reveal the intimate relation between these two results.
References:
-
- 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 J. 12 (1945), 47-76. MR 7:60b
- 4.
- A. Zygmund, Trigonometric series, Vol. 1, Cambridge Univ. Press, Cambridge, UK, 1959. MR 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:
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
Posted:
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 of article:
Copyright
2002,
American Mathematical Society
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