Abstract
Let G be a bounded Jordan domain in ℂ and let w n = 0 be an analytic function on G such that tS G s¦ω¦2 dm < ∞, where dm is the area measure. We investigate the zero distribution of the sequence of polynomials that are orthogonal on G with respect to ¦ω¦2 dm. We find that such a distribution depends on the location of the singularities of the reproducing kernel K w(z, ζ) of the space L ω 2(G):= f analytic on G: ∫G ¦ f ¦2¦ω¦2 dm < ∞. A fundamental theorem is given for the case when Kω(·, ζ) has a singularity on ∂G for at least some ζ ∈ G. To investigate the opposite case, we consider two examples in detail: first when G is the unit disk and ω is meromorphic, and second when G is a lens-shaped domain and ω is entire. Our analysis can also be applied for ω ≡ 1 in the case when G is a rectangle or a special triangle. We also provide formulas for K ω(·, ζ) that are of help for the determination of its singularities.
Similar content being viewed by others
References
D. Gaier, Lectures on Complex Approximation, Boston, Birkhäuser, 1987 (translated from German by Renate McLaughlin).
L. I. Hedberg, Weighted mean approximation in Carathéodory regions, Math. Scand. 23 (1968), 113–122.
L. I. Hedberg, Weighted mean square approximation in plane regions and generators of an algebra of analytic functions, Ark. Mat. 5 (1965), 541–552.
H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman Spaces, in: Grad. Texts in Math., Vol. 199, Springer-Verlag, Berlin, New York, 2000.
A. L. Levin, E. B. Saff and N. S. Stylianopoulos, Zero distribution of Bergman orthogonal polynomials for certain planar domains, Const. Approx. 19 (2003), 411–435.
V. Maymeskul and E. B. Saff, Zeros of polynomials orthogonal over regular N-gons, J. Approx. Theory 122 (2003), 129–140.
S. N. Mergelyan, On the completeness of systems of analytic functions, Amer. Math. Soc. Transl. (2) 19 (1962), 109–166.
H. N. Mhaskar and E. B. Saff, On the distribution of zeros of polynomials orthogonal on the unit circle, J. Approx. Theory 63 (1990), 30–38.
E. Miña-Díaz, Doctoral Dissertation, Vanderbilt University (to appear).
Z. Nehari, Conformal Mapping, McGraw-Hill Book Company, Inc., 1952.
N. Papamichael, E. B. Saff and J. Gong, Asymptotic behaviour of zeros of Bieberbach polynomials, J. Comput. Appl. Math. 34 (1991), 325–342.
T. Ransford, Potential Theory in the Complex Plane, London Math. Soc. Student Texts 28, Cambridge University Press, 1995.
E. B. Saff, Orthogonal polynomials from a complex perspective, in: P. Nevai. Ed., Orthogonal Polynomials: Theory and Practice, Kluwer, Dordrecht, 1990, 363–393.
E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Berlin, Springer-Verlag, 1997.
H. Stahl and V. Totik, General Orthogonal Polynomials, Cambridge, Cambridge University Press, 1992.
P. K. Suetin, Polynomials Orthogonal over a Region and Bieberbach Polynomials, Proceedings of the Steklov Institute of Mathematics, Amer. Math. Soc., Providence, Rhode Island, 1975.
J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, Amer. Math. Soc. Colloq. Publ. 20, Amer. Math. Soc., Providence, RI, 5th ed., 1969.
—, Selected Papers, T. J. Rivlin and E. B. Saff (eds.), Springer-Verlag, 1991.
J. L. Walsh, Overconvergence, degree of convergence, and zeros of sequences of analytic functions, Duke Math J. 13 (1946), 195–234 (also in [18], 545–584).
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of E. B. Saff was supported, in part, by the U. S. National Science Foundation under grant DMS-0296026.
Rights and permissions
About this article
Cite this article
Miña-Díaz, E., Saff, E.B. & Stylianopoulos, N.S. Zero Distributions for Polynomials Orthogonal with Weights over Certain Planar Regions. Comput. Methods Funct. Theory 5, 185–221 (2005). https://doi.org/10.1007/BF03321094
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03321094