Eigenvalue Problem for Discrete Jacobi–Sobolev Orthogonal Polynomials

• Main Authors: Juan F. Mañas-Mañas, Juan J. Moreno-Balcázar, Richard Wellman
• Format: Article
• Language: English
• Published: MDPI AG 2020-02-01
• Series: Mathematics
• Subjects: sobolev orthogonal polynomials; jacobi weight; asymptotics
• Online Access: https://www.mdpi.com/2227-7390/8/2/182

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.