Rational approximation and Sobolev-type orthogonality

[摘要] In this paper, we study the sequence of orthogonal polynomials {Sn}(n=0)(infinity ) with respect to the Sobolev-type inner product < f, g > = integral(1 )(1)f(x)g(x)d mu(x) + Sigma(N )(j=1)eta j f((dj))(cj)g((dj))(cj) where mu, is a finite positive Borel measure whose support supp (mu) subset of [-1.l] contains an infinite set of points, n(j) > 0, N, d(j) is an element of Z+ and {c(1), ..., c(N)} subset of R \ [-1, 1]. Under some restriction of order in the discrete part of <., .>, we prove that for sufficiently large n the zeros of S, are real, simple, n - N of them lie on (-1, 1) and each of the mass points c(j) attracts one of the remaining N zeros. The sequences of associated polynomials {S-n([k])}(n=0)(infinity) are defined for each k is an element of Z(+). If mu, is in the Nevai class M(0, 1), we prove an analogue of Markov's Theorem on rational approximation to Markov type functions and prove that convergence takes place with geometric speed. (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-12-01

[关键词] Rational approximation;Sobolev orthogonality;Markov's theorem;Zero location