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A Fejér type theorem to determine jumps in terms of the Abel-Poisson mean of double Fourier series

Authors: Mónika Bagota and Ferenc Móricz
Journal: Proc. Amer. Math. Soc. 130 (2002), 2617-2623
MSC (2000): Primary 42B05, 42A16
Published electronically: March 25, 2002
MathSciNet review: 1900869
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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 [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. Móricz, Extension of a theorem of Fejér to double Fourier-Stieltjes series. J. Fourier Anal. Appl. 7 (2001), 601-614. CMP 2002:03
  • 3. A. Zygmund, Trigonometric series, Vol. 1, Cambridge Univ. Press, 1959. MR 21:6498

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Additional Information

Mónika Bagota
Affiliation: Department of Mathematics, Gyula Juhász College, University of Szeged, Boldogasszony Sgt. 4, 6720 Szeged, Hungary

Ferenc Móricz
Affiliation: Bolyai Institute, University of Szeged, Aradi Vértanúk Tere 1, 6720 Szeged, Hungary

Keywords: Fej\'{e}r's theorem, formally differentiated Fourier series, first arithmetic mean, simple discontinuity, Abel-Poisson mean, generalized jump, smoothness of function in two variables, Zygmund classes $\lambda_{*}$ and $\Lambda _{*}$ on $\mathcal{T} ^{2}$
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
Article copyright: © Copyright 2002 American Mathematical Society