Abstract
For every system of OPRL or OPUC, we construct Sobolev orthogonal polynomials
, with explicit integral representations involving
. Two concrete families of Sobolev orthogonal polynomials (depending on an arbitrary number of complex parameters) which are generalized eigenvalues of a difference operator (in n) and generalized eigenvalues of a differential operator (in z) are given. We define suitable Sobolev spaces with matrix weights and consider measurable factorizations of weights. Applications of a general connection between Sobolev orthogonal polynomials and orthogonal systems of functions in the direct sum of scalar
spaces are discussed.
2010 Mathematics Subject Classification:
Acknowledgements
The author thanks the referees for their valuable comments and suggestions. One of the referees proposed to replace the eigenvalue in (Equation2(2)
(2) ) by a function of z, and then Problem 3.1 appeared.
Disclosure statement
No potential conflict of interest was reported by the author(s).