11 research outputs found
Wall Polynomials on the Real Line: a classical approach to OPRL Khrushchevâs formula
The standard proof of Khrushchevâs formula for orthogonal polynomials on the unit circle given in Khrushchev (J Approx Theory 108:161â248, 2001, J Approx Theory 116:268â342, 2002) combines ideas from continued fractions and complex analysis, depending heavily on the theory of Wall polynomials. Using operator theoretic tools instead, Khrushchevâs formula has been recently extended to the setting of orthogonal polynomials on the real line in the determinate case (GrĂŒnbaum and VelĂĄzquez in Adv Math 326:352â464, 2018). This paper develops a theory of Wall polynomials on the real line, which serves as a means to prove Khrushchevâs formula for any sequence of orthogonal polynomials on the real line. This real line version of Khrushchevâs formula is used to rederive the characterization given in Simon (J Approx Theory 126:198â217, 2004) for the weak convergence of pn2d”, where pn are the orthonormal polynomials with respect to a measure ” supported on a bounded subset of the real line (Theorem 8.1). The generality and simplicity of such a Khrushchevâs formula also permits the analysis of the unbounded case. Among other results, we use this tool to prove that no measure ” supported on an unbounded subset of the real line yields a weakly convergent sequence pn2d” (Corollary 8.10), but there exist instances such that pn2d” becomes vaguely convergent (Example 8.5 and Theorem 8.6). Some other asymptoptic results related to the convergence of pn2d” in the unbounded case are obtained via Khrushchevâs formula (Theorems 8.3, 8.7, 8.8, Proposition 8.4, Corollary 8.9). In the bounded case, we include a simple diagrammatic proof of Khrushchevâs formula on the real line which sheds light on its graph theoretical meaning, linked to PĂłlyaâs recurrence theory for classical random walks. © 2022, The Author(s)
CMV matrices in random matrix theory and integrable systems: a survey
We present a survey of recent results concerning a remarkable class of
unitary matrices, the CMV matrices. We are particularly interested in the role
they play in the theory of random matrices and integrable systems. Throughout
the paper we also emphasize the analogies and connections to Jacobi matrices.Comment: Based on a talk given at the Short Program on Random Matrices, Random
Processes and Integrable Systems, CRM, Universite de Montreal, 200