A Fejér type theorem to determine jumps in terms of the Abel-Poisson mean of double Fourier series
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- by Mónika Bagota and Ferenc Móricz PDF
- Proc. Amer. Math. Soc. 130 (2002), 2617-2623 Request permission
Abstract:
We extend from single to double Fourier series a theorem of Zygmund to determine the generalized jumps of a periodic integrable function at a simple discontinuity point. As a by-product of the proof, we obtain an estimate of the fourth mixed partial derivative of the Abel-Poisson mean of any integrable function $F(x,y)$ at such a point where $F$ is smooth. We also consider the extension of the Zygmund classes $\lambda _{*}$ and $\Lambda _{*}$ to the two-dimensional torus $\mathcal {T} ^{2}$.References
- L. Fejér, Über die Bestimmung des Sprunges der Funktion aus ihrer Fourierreihe, J. reine angew. Math. 142 (1913), 165-188.
- F. Móricz, Extension of a theorem of Fejér to double Fourier-Stieltjes series. J. Fourier Anal. Appl. 7 (2001), 601–614.
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Additional Information
- Mónika Bagota
- Affiliation: Department of Mathematics, Gyula Juhász College, University of Szeged, Boldogasszony Sgt. 4, 6720 Szeged, Hungary
- Email: bagota@jgytf.u-szeged.hu
- 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): March 29, 2001
- Published electronically: March 25, 2002
- Additional Notes: This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant T 029094.
- Communicated by: Andreas Seeger
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 2617-2623
- MSC (2000): Primary 42B05, 42A16
- DOI: https://doi.org/10.1090/S0002-9939-02-06347-5
- MathSciNet review: 1900869