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

 

Wiener type theorems for Jacobi series with nonnegative coefficients


Authors: H. N. Mhaskar and S. Tikhonov
Journal: Proc. Amer. Math. Soc. 140 (2012), 977-986
MSC (2010): Primary 33C45, 42C10; Secondary 46E30
Published electronically: August 31, 2011
MathSciNet review: 2869082
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Abstract: This paper gives three theorems regarding functions integrable on $ [-1,1]$ with respect to Jacobi weights and having nonnegative coefficients in their (Fourier-)Jacobi expansions. We show that the $ L^p$-integrability (with respect to the Jacobi weight) on an interval near $ 1$ implies the $ L^p$-integrability on the whole interval if $ p$ is an even integer. The Jacobi expansion of a function locally in $ L^\infty$ near $ 1$ is shown to converge uniformly and absolutely on $ [-1,1]$; in particular, such a function is shown to be continuous on $ [-1,1]$. Similar results are obtained for functions in local Besov approximation spaces.


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Additional Information

H. N. Mhaskar
Affiliation: Department of Mathematics, California State University, Los Angeles, California 90032
Email: hmhaska@calstatela.edu

S. Tikhonov
Affiliation: ICREA and Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain
Email: stikhonov@crm.cat

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10964-X
PII: S 0002-9939(2011)10964-X
Keywords: Fourier–Jacobi expansion, non-negative coefficients, Besov spaces
Received by editor(s): February 27, 2010
Received by editor(s) in revised form: December 19, 2010
Published electronically: August 31, 2011
Additional Notes: The research of the first author was supported in part by grant DMS-0908037 from the National Science Foundation and grant W911NF-09-1-0465 from the U.S. Army Research Office.
The research of the second author was supported in part by grants MTM2008-05561-C02-02/MTM, 2009 SGR 1303, RFFI 09-01-00175, and NSH3252.2010.1.
Communicated by: Walter Van Assche
Article copyright: © Copyright 2011 American Mathematical Society