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

 

Gibbs' phenomenon and surface area


Authors: L. de Michele and D. Roux
Journal: Proc. Amer. Math. Soc. 134 (2006), 3561-3566
MSC (2000): Primary 42B99
Published electronically: May 31, 2006
MathSciNet review: 2240668
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Abstract: If a function $ f$ is of bounded variation on $ T^N\ (N\geq 1)$ and $ \{{\varphi}_n\}$ is a positive approximate identity, we prove that the area of the graph of $ f*{\varphi}_n$ converges from below to the relaxed area of the graph of $ f$. Moreover we give asymptotic estimates for the area of the graph of the square partial sums of multiple Fourier series of functions with suitable discontinuities.


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Additional Information

L. de Michele
Affiliation: Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via R. Cozzi 53, 20126 Milano, Italia

D. Roux
Affiliation: Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via R. Cozzi 53, 20126 Milano, Italia

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08639-4
PII: S 0002-9939(06)08639-4
Keywords: Gibbs phenomenon, Fourier series, approximate identity.
Received by editor(s): June 21, 2005
Published electronically: May 31, 2006
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.