Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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
Published electronically: August 2, 2010
MathSciNet review: 2680053
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. T. Carleman, Über die approximation analytischer funktionen durch lineare aggregate von vorgegebenen potenzen, Archiv. för Math. Atron. och Fysik, 17 (1922) 1-30.
  • 2. Philip J. Davis, The Schwarz function and its applications, The Mathematical Association of America, Buffalo, N. Y., 1974. The Carus Mathematical Monographs, No. 17. MR 0407252
  • 3. P. Dragnev, E. Miña-Díaz, Asymptotic behavior and zero distribution of Carleman orthogonal polynomials, J. Approx. Theory, doi:10.1016/j.jat.2010.05.006
  • 4. Dieter Gaier, Lectures on complex approximation, Birkhäuser Boston, Inc., Boston, MA, 1987. Translated from the German by Renate McLaughlin. MR 894920
  • 5. A. Martínez-Finkelshtein, K. T.-R. McLaughlin, and E. B. Saff, Szegő orthogonal polynomials with respect to an analytic weight: canonical representation and strong asymptotics, Constr. Approx. 24 (2006), no. 3, 319–363. MR 2253965, 10.1007/s00365-005-0617-6
  • 6. Erwin Miña-Díaz, An expansion for polynomials orthogonal over an analytic Jordan curve, Comm. Math. Phys. 285 (2009), no. 3, 1109–1128. MR 2470918, 10.1007/s00220-008-0541-2
  • 7. Erwin Miña-Díaz, An asymptotic integral representation for Carleman orthogonal polynomials, Int. Math. Res. Not. IMRN 16 (2008), Art. ID rnn065, 38. MR 2435755, 10.1093/imrn/rnn065
  • 8. Zeev Nehari, Conformal mapping, McGraw-Hill Book Co., Inc., New York, Toronto, London, 1952. MR 0045823

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30E10, 30E15, 42C05

Retrieve articles in all journals with MSC (2010): 30E10, 30E15, 42C05


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.