Differential properties for Sobolev orthogonality on the unit circle

[摘要] The aim of this paper is to study differential properties of the sequence of monic orthogonal polynomials with respect to the following Sobolev inner product: (s) = integral (2 pi)(0) f(e(i theta))g(e(i theta))d mu(theta) + 1/lambda integral (2 pi)(0) f ' (e(i theta))g ' (e(i theta))d theta /2 pi, where mu is a finite positive Borel measure on [0,2 pi] verifying the following conditions: the Caratheodory function associated with mu has an analytic extension outside the unit disk and the induced norm is equivalent to the Lebesgue norm in the space L-2. Here d theta /2 pi is the normalized Lebesgue measure and A is a positive real number. The nonhomogeneous second-order differential equations satisfied by the sequence of monic Sobolev orthogonal polynomials are obtained, Moreover, as an application, a sample of Dirichlet boundary value problem is solved. (C) 2001 Elsevier Science B.V. All rights reserved.

[发布日期] 2001-08-01

[效力级别]  Proceedings Paper

[关键词] orthogonal polynomials;Sobolev inner products;differential operators