Conformal Maps and Orthogonal Polynomials for Planar Regions with Analytic Boundaries
Document Type
Lecture
Publication Date
9-10-2008
Abstract
For an arbitrary analytic Jordan curve �� in the complex plane, whose interior domain is denoted by ��, we shall look at the sequence of polynomials ��ₙ(��), �� = 0,1,2,... (��ₙ of exact degree ��) that are orthonormal over �� with respect to area measure, that is, ∫₉ ��ₙ(��)��ₘ(��) d��(��) = 0 if �� ≠ �� and 1 if �� = ��, where d�� is the two-dimensional Lebesgue (area) measure. Specifically, we want to understand how these polynomials and their zeros behave as �� → ∞. We shall give a quite complete and satisfactory answer to this question, which required us to gain a good understanding of the meromorphic continuation properties of the interior and exterior canonical conformal maps associated with the analytic curve ��. The results will be illustrated with some concrete (far from trivial) examples and numerical computations. This is joint work with Dr. P. Dragnev of Indiana-Purdue University Fort Wayne.
Relational Format
presentation
Recommended Citation
Miña-Díaz, Erwin, "Conformal Maps and Orthogonal Polynomials for Planar Regions with Analytic Boundaries" (2008). Analysis Seminar. 44.
https://egrove.olemiss.edu/math_analysis/44
