Measures on the unit circle and unitary truncations of unitary operators

• Authors: M. J. Cantero;L. Moral;L. Velázquez
• Venue: Journal of Approximation Theory
• Year: 2006

In this paper, we obtain new results about the orthogonality measure of orthogonal polynomials on the unit circle, through the study of unitary truncations of the corresponding unitary multiplication operator, and the use of the five-diagonal representation of this operator. Unitary truncations on subspaces with finite co-dimension give information about the derived set of the support of the measure under very general assumptions for the related Schur parameters (a"n). Among other cases, we study the derived set of the support of the measure when lim"n|a"n"+"1/a"n|=1, obtaining a natural generalization of the known result for the Lopez class lim"na"n"+"1/a"n@?T, lim"n|a"n|@?(0,1). On the other hand, unitary truncations on subspaces with finite dimension provide sequences of unitary five-diagonal matrices whose spectra asymptotically approach the support of the measure. This answers a conjecture of L. Golinskii concerning the relation between the support of the measure and the strong limit points of the zeros of the para-orthogonal polynomials. Finally, we use the previous results to discuss the domain of convergence of rational approximants of Caratheodory functions, including the convergence on the unit circle.