Zeros of orthogonal polynomials and some matrix inequalities

Abstract: The main aim of this work is to apply the matrix approach of ortho-gonal polynomials associated with infinite Hermitian definite positive matrices in relation with an important question regarding the location of zeros of Sobolev orthogonal polynomials via the study of the boundedness of multiplication operator. We apply the notion of bounded point evaluations of a measure, and more generally to infinite HPD matrices, to the problem of boundedness of multiplication operator. Moreover, we introduce certain Wirtinger-type inequalities, relating the norm of the polynomials with the norm of their derivatives, in order to provide new examples of Sobolev polynomials for which we may ensure that the zeros of Sobolev polynomials are uniformly bounded. In particular, we consider the case of Lebesgue measures supported on circles. With these techniques, we may obtain many examples of vectorial measures such that the zeros of Sobolev orthogonal polynomials are bounded, and nevertheless they are not sequentially dominated, not even matrix sequentially dominated.