Sobolev-type orthogonal polynomials on the unit circle

Abstract

This paper deals with polynomials orthogonal with respect to a Sobolev-type inner product $$ \langle f,g\rangle =\int_{-\pi} pi f(e i\theta}) \overline{g(e i\theta})} d\mu(e i\theta})\, + \, \bold{f}(c)A (\bold{g}(c)) .$$ where μ is a positive Borel measure supported on [−π,π), A is a nonsingular matrix and 1. We denote f(c)=(f(c),f'(c),\dots,f (p)}(c)) and v 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 [see attached full-text file]. Finally, we deduce the relative asymptotics for both families of orthogonal polynomials.