Lebesgue perturbation of a quasi-definite Hermitian functional. The positive definite case

Abstract

In this work we study the problem of orthogonality with respect to a sum of measures or functionals. First we consider the case where one of the functionals is arbitrary and quasidefinite and the other one is the Lebesgue normalized functional. Next we study the sum of two positive measures. The first one is arbitrary and the second one is the Lebesgue normalized measure and we obtain some relevant properties concerning the new measure. Finally we consider the sum of a Bernstein-Szeg" o measure and the Lebesgue measure. In this case we obtain more simple explicit algebraic relations as well as the relation between the corresponding Szeg" o's functions.

Orthogonal Laurent polynomials on the unit circle, extended CMV ordering and 2D Toda type integrable hierarchies

“…Proof of Proposition 3 Expressions (16), (17), (19) and (20) are obtained expressing the factorization problem as a system of equations. From (17) we deduce (S 1 ) lk = (S 2 ) ll ((g [l+1] ) −1 ) l,k = (S 2 ) ll det g [l+1] (−1) l+k M (l+1) k,l , so that ϕ (l)…”

On Laguerre–Hahn affine orthogonal polynomials on the unit circle from matrix Sylvester equations

“…Well-known families of such polynomials include the semi-classical orthogonal polynomials on the unit circle -characterized through a rational logarithmic derivative for the weight function, equivalently, through an differential equation (ODE) (1.1) with a specific polynomial D [1][2][3] -as well as some of their perturbations, for instance, the ones studied in [4][5][6]. The analysis of relations between differential properties of sequences of orthogonal polynomials and differential properties of the corresponding Carathéodory function through ODEs (1.1) is an often encountered problem in the literature of Orthogonal Polynomials.…”

“…Well-known examples of such families of orthogonal polynomials include the so-called generalized Jacobi polynomials [20, Sections 3, 4] (see also [8,15]), as well as some of its perturbations related to the ones studied in [4,5].…”

Orthogonal polynomials with respect to the sum of an arbitrary measure and a Bernstein–Szegö measure

“…In particular, in [1] this problem was studied for the sum of a measure µ supported on the unit circle and the normalized Lebesgue measure. The translation in terms of the entries of the resulting Toeplitz matrix means that we only perturb the main diagonal of the Toeplitz matrix associated with µ by adding a constant term.…”

“…The generalization of the problem studied in [1] can be done in two different ways. The first one consists in the perturbation of the Toeplitz matrix T (µ) associated with µ by adding a finite number of moments of the second measure to the corresponding moments of T (µ).…”