An expansion for polynomials orthogonal over an analytic Jordan curve

Details An expansion for polynomials orthogonal over an analytic Jordan curve An expansion for polynomials orthogonal over an analytic Jordan curve

2007-12-09

arxiv.org/abs/0712.1366v1

Communications in Mathematical Physics. Vol. 285, 3:1109-1128 (2009)

Mathematics

Classical Analysis and ODEs

15 pages, 1 figure

Scientific paper

10.1007/s00220-008-0541-2

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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