An Expansion for Polynomials Orthogonal Over an Analytic Jordan Curve |
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Authors: | Erwin Mi?a-Díaz |
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Institution: | (1) Department of Mathematical Sciences, Indiana-Purdue University Fort Wayne, 2101 E. Coliseum Blvd, Fort Wayne, IN 46805-1499, USA |
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Abstract: | We consider polynomials that are orthogonal over an analytic Jordan curve L with respect to a positive analytic weight, and show that each such polynomial of sufficiently large degree can be expanded
in a series of certain integral transforms that converges uniformly in the whole complex plane. This expansion yields, in
particular and simultaneously, Szegő’s classical strong asymptotic formula and a new integral representation for the polynomials
inside L. We further exploit such a representation to derive finer asymptotic results for weights having finitely many singularities
(all of algebraic type) on a thin neighborhood of the orthogonality curve. Our results are a generalization of those previously
obtained in 7] for the case of L being the unit circle.
Dedicated to Prof. Guillermo López Lagomasino on the occasion of his 60th birthday |
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