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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn065, 38 pages, doi:10.1093/imrn/rnn065 published on June 20, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

An Asymptotic Integral Representation for Carleman Orthogonal Polynomials

Erwin Miña-Díaz

Indiana University–Purdue University Fort Wayne, Department of Mathematical Sciences, 2101 E. Coliseum Boulevard, Fort Wayne, IN 46805-1499, USA

Correspondence: Correspondence to be sent to: IPFW, Department of Mathematical Sciences, 2101 E. Coliseum Boulevard, Fort Wayne, IN 46805-1499, USA. e-mail: minae{at}ipfw.edu

In this paper, we investigate the asymptotic behavior of polynomials that are orthonormal over the interior domain of an analytic Jordan curve L with respect to area measure. We prove that, inside L, these polynomials behave asymptotically like a sequence of certain integrals involving the canonical conformal map of the exterior of L onto the exterior of the unit circle and certain meromorphic kernel function defined in terms of a conformal map of the interior of L onto the unit disk. We then use this result to obtain more precise asymptotic formulas for the polynomials under certain additional geometric considerations. These formulas yield, in turn, fine results on the location, limiting distribution, and accumulation points of the zeros of the polynomials.

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