Böttcher, Albrecht ; Dörfler, Peter : Weighted Markov-type inequalities, norms of Volterra operators, and zeros of Bessel functions

Author(s):

Böttcher, Albrecht
Dörfler, Peter

Title:

Weighted Markov-type inequalities, norms of Volterra operators, and zeros of Bessel functions

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 1, 2009

Mathematics Subject Classification:

41A10			[ Approximation by polynomials ]
15A18			[ Eigenvalues, singular values, and eigenvectors ]
26D10			[ Inequalities involving derivatives and differential and integral operators ]
41A44			[ Best constants ]
45D05			[ Volterra integral equations ]
47G10			[ Integral operators ]

Abstract:

The first term of the asymptotics of the best constants in Markov-type inequalities for higher derivatives of polynomials is determined in the two cases where the underlying norm is the $L^2$ norm with Laguerre weight or the $L^2$ norm with Gegenbauer weight. The coefficient in this term is shown to be the norm of a certain Volterra integral operator which depends on the weight and the order of the derivative. For first order derivatives, the norms of the Volterra operators are expressed in terms of the zeros of Bessel functions. The asymptotic behavior of the coefficients is studied and tight bounds for them are given.

Keywords:

Markov-type inequality, orthogonal polynomials, Volterra integral operators, Bessel function, singular values

Language:

English

Publication time:

1 / 2009