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, 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.
- 4.Davis, P.J.: The Schwarz function and its applications. The Carus Mathematical Monographs Vol. 17. Washington, DC: The Mathematical Association of America, 1974Google Scholar
- 8.Miña-Díaz, E.: An asymptotic integral representation for Carleman orthogonal polynomials. Int. Math. Res. Not. vol. 2008, rnn065, 38pp. (2008) doi:10.1093/imrn/rnn065
- 14.Rudin, W.: Real and complex analysis. 3rd ed., New York, McGraw-Hill, 1986Google Scholar
- 19.Szegő, G.: Orthogonal Polynomials. Amer. Math. Soc. Colloq. Publ. Vol. 23, Providence, RI, Amer. Math. Soc., 4th ed., 1975Google Scholar