Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle

Abstract

The principal objective here is to look at some algebraic properties of the orthogonal polynomials Ψn (b,s,t) n with respect to the Sobolev inner product on the unit circle <f,g>S (b,s,t) = (1 − t) <f,g>μ(b) + t f(1) g(1) + s <f', g'>μ(b+1), where <f, g> μ(b) = τ(b)/2π∫2π 0 f(eiθ) g(eiθ) (eπ−θ)Im(b)(sin2(θ/2))Re(b)dθ. Here, Re(b) > −1/2, 0 ≤ t < 1, s > 0 and τ(b) is taken to be such that <1, 1>μ(b) = 1. We show that, for example, the monic Sobolev orthogonal polynomials Ψ(b,s,t) n satisfy the recurrence Ψ(b,s,t) n (z)−β(b,s,t) n Ψ(b,s,t) n−1 (z) = Φ(b,t) n (z), n ≥ 1, where Φ(b,t) n are the monic orthogonal polynomials with respect to the inner product <f, g>μ(b,t) = (1 − t) <f, g> μ(b) + t f(1) g(1). Some related bounds and asymptotic properties are also given.