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Geronimus polynomials and weak convergence on a circular arc. (English) Zbl 0965.42016

Geronimus polynomials are orthogonal polynomials on the unit circle with constant reflection coefficients \(\Phi_n(0)=a\), \(0 < |a|< 1\). These polynomials play a similar role for the asymptotic theory of orthogonal polynomials on the unit circle when the reflection coefficients converge to \(a\), as the Chebyshev polynomials of the second kind in the asymptotic theory of orthogonal polynomials on the interval \([-1,1]\). Some properties of these Geronimus polynomials are collected in Section 2 and in Section 3 the weak convergence for Geronimus measures is given. Weak convergence for the López class, in which \(|\Phi_n(0)|\to |a|\) and \(\Phi_{n+1}(0)/\Phi_n(0) \to b\), with \(|b|=1\), is given in Section 4 using operator theory (unitary Hessenberg operators). More refined López classes are defined in Section 5, which allow to obtain some information about the Radon-Nikodým derivative of the orthogonality measure from the weak asymptotics.

MSC:

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