Abstract
With the aid of Havin’s Lemma (which we generalize) we prove that polynomials orthogonal over the unit disk with respect to certain weighted area measures (Bergman polynomials) cannot satisfy a finite-term recurrence relation unless the weight is radial, in which case the polynomials are simply monomials. For polynomials orthogonal over the unit circle (Szegő polynomials) we provide a simple argument to show that the existence of a finite-term recurrence implies that the weight must be the reciprocal of the square modulus of a polynomial.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, second ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003.
K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Mathematical Series, vol. 48, Princeton University Press, Princeton, NJ, 2009.
D. Barrios Rolanía and G. López Lagomasino, Ratio asymptotics for polynomials orthogonal on arcs of the unit circle, Constr. Approx. 15 no.1 (1999), 1–31.
J. B. Conway, A Course in Operator Theory, Graduate Studies in Mmathematics, 21, Amer. Math. Soc., Providence, RI, 2000.
Ya. L. Geronimus, On polynomials orthogonal on the circle, on the trigonometric moment problem, and on allied Carathéodory and Schur functions, Mat. Sb. 15 (1944), 99–130.
H. Hedenmalm, The dual of a Bergman space on simply connected domains, J. Anal. Math. 88 (2002), 311–335 (dedicated to the memory of Tom Wolff).
D. Khavinson and N. Stylianopoulos, Recurrence Relations for Orthogonal Polynomials and Algebraicity of Solutions of the Dirichlet Problem, Around the Research of Vladimir Maz’ya II, International Mathematical Series, vol. 12, Springer, New York, 2009, pp. 219–228.
M. Putinar and N. Stylianopoulos, Finite-term relations for planar orthogonal polynomials, Complex Anal. Oper. Theory 1 no. 3 (2007), 447–456.
B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005.
G. Szegő, Orthogonal Polynomials, fourth ed., American Mathematical Society, Providence, R.I., 1975, American Mathematical Society, Colloquium Publications, Vol. XXIII.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Nicolas Papamichael, a great mentor, collaborator, and friend.
The work of the second author was supported, in part, from the U.S. National Science Foundation grant DMS-0808093.
Rights and permissions
About this article
Cite this article
Baratchart, L., Saff, E.B. & Stylianopoulos, N.S. On Finite-Term Recurrence Relations for Bergman and Szegő Polynomials. Comput. Methods Funct. Theory 12, 393–402 (2012). https://doi.org/10.1007/BF03321834
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03321834