A Cohen-Type Inequality for Jacobi-Sobolev Expansions
© Bujar Xh. Fejzullahu 2007
Received: 21 August 2007
Accepted: 11 December 2007
Published: 6 February 2008
Let be the Jacobi measure supported on the interval [-1, 1]. Let us introduce the Sobolev-type inner product , where . In this paper we prove a Cohen-type inequality for the Fourier expansion in terms of the orthonormal polynomials associated with the above Sobolev inner product. We follow Dreseler and Soardi (1982) and Markett (1983) papers, where such inequalities were proved for classical orthogonal expansions.
- Alfaro M, Marcellán F, Rezola ML: Estimates for Jacobi-Sobolev type orthogonal polynomials. Applicable Analysis 1997,67(1–2):157–174.MathSciNetView ArticleMATHGoogle Scholar
- Rocha IA, Marcellán F, Salto L: Relative asymptotics and Fourier series of orthogonal polynomials with a discrete Sobolev inner product. Journal of Approximation Theory 2003,121(2):336–356. 10.1016/S0021-9045(03)00035-2MathSciNetView ArticleMATHGoogle Scholar
- Zygmund A: Trigonometric Series. Volumes I and II. Cambridge University Press, London, UK; 1968:Vol. I. xiv+383; Vol. II: vii+364.Google Scholar
- Szegő G: Orthogonal Polynomials, American Mathematical Society Colloquium Publications. Volume 23. American Mathematical Society, Providence, RI, USA; 1975.Google Scholar
- Markett C: Cohen type inequalities for Jacobi, Laguerre and Hermite expansions. SIAM Journal on Mathematical Analysis 1983,14(4):819–833. 10.1137/0514063MathSciNetView ArticleMATHGoogle Scholar
- Fejzullahu BXh: Divergent Cesàro means of Jacobi-Sobolev expansions. to appear in Revista Matemática Complutense.Google Scholar
- Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG: Tables of Integral Transforms, Vol. II. McGraw-Hill, New York, NY, USA; 1954:xvi+451.MATHGoogle Scholar
- Nevai P, Erdélyi T, Magnus AP: Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials. SIAM Journal on Mathematical Analysis 1994,25(2):602–614. 10.1137/S0036141092236863MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.