A new approach to relative asymptotic behavior for discrete Sobolev-type orthogonal polynomials on the unit circle

This paper deals with polynomials orthogonal with respect to a Sobolev type inner product hf ; gi 1⁄4 Z p p f ðeÞgðeihÞdlðeÞ þ fðcÞA 1⁄2gðcÞ H ; where l is a positive Borel measure supported on 1⁄2 p; pÞ, A is a nonsingular matrix and jcj > 1. We denote fðcÞ 1⁄4 ðf ðcÞ; f 0ðcÞ; . . . ; f ðpÞðcÞÞ and vH the transposed conjugate of the vector v. We establish the connection of such polynomials with orthogonal poly nomials on the unit circle with respect to the measure dmðzÞ 1⁄4 jz cj dlðzÞ ðz 1⁄4 eih; p 2 NÞ. Finally, we deduce the relative asymptotics for both families of or thogonal polynomials. 2002 Elsevier Science Inc. All rights reserved.