4 research outputs found
An expansion for polynomials orthogonal over an analytic Jordan curve
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.Comment: 15 pages, 1 figur