Sobolev-type orthogonal polynomials on the unit circle

摘要

This paper deals with polynomials orthogonal with respect to a Sobolev-type inner product〈f,g〉=∫−ππf(eiθ)g(eiθ)dμ(eiθ)+f(c)A[g(c)]H,where μ is a positive Borel measure supported on [−π,π), A is a nonsingular matrix and |c|>1. We denote f(c)=(f(c),f′(c),…,f(p)(c)) and vH the transposed conjugate of the vector v. We establish the connection of such polynomials with orthogonal polynomials on the unit circle with respect to the measure dν(z)=|z−c|2p+2dμ(z)(z=eiθ,p∈N). Finally, we deduce the relative asymptotics for both families of orthogonal polynomials.