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On the asymptotics of polynomials orthogonal on a system of curves with respect to a measure with discrete part. (English. Russian original) Zbl 1197.42014

St. Petersbg. Math. J. 21, No. 2, 217-230 (2010); translation from Algebra Anal. 21, No. 2, 71-91 (2009).
Let \(E=\bigcup_{k=1}^p E_k\) be a union of complex, mutually exterior and rectifiable Jordan curves and arcs, each of which is assumed to be of \(C^{2+}\) class. Let \(\Omega\) denote the connected component of \(\mathbb{C}\backslash E\), which contains infinity, and \(\rho\) be a weight function on \(E\). The main result of the paper provides the explicit strong asymptotic formula for the monic orthogonal polynomials \(Q_n(z,\sigma)\) for the measure \(\sigma=\rho(t)|dt|+\alpha\), where \(\rho\) satisfies the Szegő condition, and \(\alpha\) is a discrete measure with masses \(\lambda_j>0\) at the points \(z_1,\ldots,z_n\) in \(\Omega\). The key ingredient is a certain extremal problem in a class of multivalued functions.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
30E15 Asymptotic representations in the complex plane
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