An Expansion for Polynomials Orthogonal Over an Analytic Jordan Curve
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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  for the case of L being the unit circle.
KeywordsUnit Circle Analytic Continuation Extremal Polynomial Expansion Yield Algebraic Type
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