Eigenvalue Problem for Discrete Jacobi–Sobolev Orthogonal Polynomials

Juan F. Mañas-Mañas, Juan J. Moreno-Balcázar and Richard Wellman
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Juan F. Mañas-Mañas: Departamento de Matemáticas, Universidad de Almería, 04120 Almería, Spain
Juan J. Moreno-Balcázar: Departamento de Matemáticas, Universidad de Almería, 04120 Almería, Spain
Richard Wellman: Department of Mathematics & Computer Science, Colorado College, CO 80903, USA

Mathematics, 2020, vol. 8, issue 2, 1-19

Abstract: In this paper, we consider a discrete Sobolev inner product involving the Jacobi weight with a twofold objective. On the one hand, since the orthonormal polynomials with respect to this inner product are eigenfunctions of a certain differential operator, we are interested in the corresponding eigenvalues, more exactly, in their asymptotic behavior. Thus, we can determine a limit value which links this asymptotic behavior and the uniform norm of the orthonormal polynomials in a logarithmic scale. This value appears in the theory of reproducing kernel Hilbert spaces. On the other hand, we tackle a more general case than the one considered in the literature previously.

Keywords: Sobolev orthogonal polynomials; Jacobi weight; asymptotics (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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