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Báo cáo hóa học: " Research Article A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner Product"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner Product | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 128746 22 pages doi 10.1155 2010 128746 Research Article A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner Product Bujar Xh. Fejzullahu1 and Francisco Marcellan2 1 Department of Mathematics Faculty of Mathematics and Natural Sciences University of Prishtina Mother Teresa 5 Prishtine 10000 Kosovo 2 Departamento de Matematicas Escuela Politecnica Superior Universidad Carlos III de Madrid Avenida de la Universidad 30 28911 Leganes Spain Correspondence should be addressed to Francisco Marcellan pacomarc@ing.uc3m.es Received 5 May 2010 Accepted 24 August 2010 Academic Editor J ozef Banas Copyright 2010 B. Xh. Fejzullahu and F. Marcellan. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Let Qn f x n 0 denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product f g f-1 f x g x dpaf x A. f-1 f x g x dpa 1 P x where A 0 and dpa f x 1 - x 1 x fdx with a -1 f -1. In this paper we prove a Cohen type inequality for the Fourier expansion in terms of the orthogonal polynomials Qn f x n. Necessary conditions for the norm convergence of such a Fourier expansion are given. Finally the failure of almost everywhere convergence of the Fourier expansion of a function in terms of the orthogonal polynomials associated with the above Sobolev inner product is proved. 1. Introduction Let dpaf x 1 - x 1 x fdx with a f -1 be the Jacobi measure supported on the interval -1 1 . We say that f e L dpaf if f is measurable on -1 1 and f Lp d TO where IIf WcptUmf if 1 p TO esssuplf x l t -1 x 1 if p TO. 1.1 2 Journal of Inequalities and Applications Let us introduce the Sobolev-type spaces see for instance 1 Chapter III in a more general .