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On a series representation for Carleman orthogonal polynomials


Authors: Peter Dragnev and Erwin Miña-Díaz
Journal: Proc. Amer. Math. Soc. 138 (2010), 4271-4279
MSC (2010): Primary 30E10, 30E15, 42C05
DOI: https://doi.org/10.1090/S0002-9939-2010-10583-X
Published electronically: August 2, 2010
MathSciNet review: 2680053
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Abstract: Let $ \{p_n(z)\}_{n=0}^\infty$ be a sequence of complex polynomials ($ p_n$ of degree $ n$) that are orthonormal with respect to the area measure over the interior domain of an analytic Jordan curve. We prove that each $ p_n$ of sufficiently large degree has a primitive that can be expanded in a series of functions recursively generated by a couple of integral transforms whose kernels are defined in terms of the degree $ n$ and the interior and exterior conformal maps associated with the curve. In particular, this series representation unifies and provides a new proof for two important known results: the classical theorem by Carleman establishing the strong asymptotic behavior of the polynomials $ p_n$ in the exterior of the curve, and an integral representation that has played a key role in determining their behavior in the interior of the curve.


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Additional Information

Peter Dragnev
Affiliation: Department of Mathematical Sciences, Indiana-Purdue University Fort Wayne, 2101 E. Coliseum Boulevard, Fort Wayne, Indiana 46805-1499
Email: dragnevp@ipfw.edu

Erwin Miña-Díaz
Affiliation: Department of Mathematics, Hume Hall 305, University of Mississippi, P.O. Box 1848, University, Mississippi 38677-1848
Email: minadiaz@olemiss.edu

DOI: https://doi.org/10.1090/S0002-9939-2010-10583-X
Received by editor(s): November 28, 2009
Published electronically: August 2, 2010
Communicated by: Walter Van Assche
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.