Verblunsky coefficients and Nehari sequences
نویسندگان
چکیده
منابع مشابه
Circular Jacobi Ensembles and Deformed Verblunsky Coefficients
Using spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble:
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We also consider the monic orthogonal polynomials Φn(z). They obey the Szegő recursion Φn+1(z) = zΦn(z)− αnΦn(z), where Φn(z) = z Φn(1/z). The αn are called Verblunsky coefficients and they belong to the unit disk D = {z ∈ C : |z| < 1}. Conversely, every α ∈ ×n=0D corresponds to a unique measure. See [14, 15, 16] for background material on orthogonal polynomials on the unit circle (OPUC). In th...
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/" f2" ei~ dO) (1.3) D(z) = e x p / t ~ log(w(0)) U~ \ J 0 e --z Not only does w determine D, but D determines w, since limr,1 D(rei~ i~ exists for a.e. 0 and ~(o) = ID(e {~ I ~. (1.4) Indeed, D is the unique function analytic on D = { z l l z I <1} with D ( 0 ) > 0 and D nonvanishing on D so that (1.4) holds. Given d#, we let ~n be the monic orthogonal polynomial and {,~:~n/llOnllL2(du). The ~...
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We prove that the Szegő function, D(z), of a measure on the unit circle is entire meromorphic if and only if the Verblunsky coefficients have an asymptotic expansion in exponentials. We relate the positions of the poles of D(z)−1 to the exponential rates in the asymptotic expansion. Basically, either set is contained in the sets generated from the other by considering products of the form, z1 ....
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Relations between halfand full-lattice CMV operators with scalarand matrix-valued Verblunsky coefficients are investigated. In particular, the decoupling of full-lattice CMV operators into a direct sum of two half-lattice CMV operators by a perturbation of minimal rank is studied. Contrary to the Jacobi case, decoupling a full-lattice CMV matrix by changing one of the Verblunsky coefficients re...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2013
ISSN: 0002-9947,1088-6850
DOI: 10.1090/s0002-9947-2013-05874-6