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Matrix Uvarov transformation on the unit circle: asymptotic properties. (English) Zbl 1470.33008

This paper extends to the matrix case some algebraic and analytic properties of orthogonal polynomials on the unit circle associated with the Uvarov matrix transformation of a Hermitian matrix measure \(\sigma\), i.e. \[ d\sigma(z)+\sum_{j=1}^mM_j\delta(z-\zeta_j), \] where \(\zeta_i\neq\zeta_j\), and \(M_j\) are positive definite matrices. These properties include connection formulas, ratio and relative asymptotics when the measure belongs to the Nevai class and properties of their zeros.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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