On polynomials orthogonal with respect to Sobolev inner product on the unit circle
In the present paper the authors investigate orthogonal polynomials \(\psi_n\) with respect to an inner product involving a positive definite measure \(\mu\) on the unit circle to which a Sobolev part is added which involves the first order derivatives of the functions at \(m\) points \(z_1,z_2,\ldots,z_m\) in the complex plane. The Sobolev part is of the form \(( f'(z_1)\cdots f'(z_m)) A( g'(z_1) \cdots g'(z_m))^H\), where \(A\) is a complex \(m\times m\) matrix. It is shown that the corresponding orthogonal polynomials exist for large \(n\) whenever \(A\) is non-singular. Algebraic properties expressing these polynomials \(\psi_n\) and their derivative \(\psi_n'\) in terms of the orthogonal polynomials \(\varphi_n\) for the measure \(\mu\) are given. From this the relative asymptotic behavior of the ratio \(\psi_n(z)/\varphi_n(z)\) is deduced. The methods and results are similar to those used to study Sobolev orthogonal polynomials on the real line.
orthogonal polynomials on the unit circle
scientific article; zbMATH DE number 1025782