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A Cohen type inequality for Fourier expansions of orthogonal polynomials with a non‐discrete Gegenbauer‐Sobolev inner product
Author(s) -
Fejzullahu Bujar Xh.
Publication year - 2011
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200810143
Subject(s) - mathematics , orthogonal polynomials , product (mathematics) , sobolev space , gegenbauer polynomials , divergence (linguistics) , fourier transform , type (biology) , sobolev inequality , jacobi polynomials , pure mathematics , mathematical analysis , classical orthogonal polynomials , geometry , ecology , linguistics , philosophy , biology
Let d μ( x ) = (1 − x 2 ) α−1/2 dx ,α> − 1/2, be the Gegenbauer measure on the interval [ − 1, 1] and introduce the non‐discrete Sobolev inner product\documentclass{article}\begin{document}\pagestyle{empty}$$ \langle f,g\rangle = \int_{-1}^{1}f(x) g(x)\, d\mu (x)+ \lambda \int_{-1}^{1}f^{\prime }(x) g^{\prime }(x)\, d\mu (x) $$\end{document} where λ>0. In this paper we will prove a Cohen type inequality for Fourier expansions in terms of the polynomials orthogonal with respect to the above inner product. Results on divergence for Cesàro means of Gegenbauer‐Sobolev expansions are deduced. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim