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Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points. (English) Zbl 0976.42012
Szegő theory gives the strong asymptotics of orthogonal polynomials on the unit circle under the condition that the absolutely continuous part of the measure has an integrable logarithm (Szegő’s condition). This result can then easily be transformed to give strong asymptotics of orthogonal polynomials on an interval (usually \([-1,1]\), but the authors consider \([-2,2]\)). The authors investigate orthogonal polynomials with respect to a positive measure \(\mu\), for which the absolutely continuous part satisfies Szegő’s condition on the interval \([-2,2]\), but in addition they allow \(\mu\) to have a discrete part outside \([-2,2]\), concentrated at mass points \(x_k\). They assume a Blaschke condition \(\sum \sqrt{x_k^2-4} < \infty\) and prove the strong asymptotics for the orthonormal polynomials and their leading coefficients. The result involves the Szegő function for the absolutely continuous part and a Blaschke product of the points \(\zeta_k=(x_k-\sqrt{x_k^2-4})/2\) in the open unit disk.

MSC:
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
30D50 Blaschke products, etc. (MSC2000)
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