Rational approximation and Sobolev-type orthogonality

Abstract

In this paper, we study the sequence of orthogonal polynomials {Sn}n=0∞ with respect to the Sobolev-type inner product 〈f,g〉=∫−11f(x)g(x)dμ(x)+∑j=1Nηjf(dj)(cj)g(dj)(cj)where μ is a finite positive Borel measure whose support suppμ⊂[−1,1] contains an infinite set of points, ηj>0, N,dj∈Z+ and {c1,…,cN}⊂R∖[−1,1]. Under some restriction of order in the discrete part of 〈⋅,⋅〉, we prove that for sufficiently large n the zeros of Sn are real, simple, n−N of them lie on (−1,1) and each of the mass points cj “attracts” one of the remaining N zeros.The sequences of associated polynomials {Sn[k]}n=0∞ are defined for each k∈Z+. If μ 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.