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arXiv · 0712.1366

An expansion for polynomials orthogonal over an analytic Jordan curve

Erwin Miña-Díaz
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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, Szego'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.

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Erwin Miña-Díaz. 2007-12-09. An expansion for polynomials orthogonal over an analytic Jordan curve. https://doi.org/10.1007/s00220-008-0541-2

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