2005/04/28 by Maria J. Cantero, Maŕıa José Cantero, Cantero, Maria J. +5
Computer Science · Mathematics · #42C05 #47B36 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.CA #math.SP #msc:42C05 #msc:47B36
paper · pdf · doi:10.48550/arxiv.math/0504580
53 pages, 12 figures
arxiv created 2005/04/28 · openalex publication_date 2005/04/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. Among other results, we obtain a natural generalization of the known result for the Lopez class. 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.