Document Type
Lecture
Publication Date
2-3-2010
Abstract
Consider the sequence {��ₙ(��)}ₙ₌₀ of polynomials of a complex variable ��, where ��ₙ(��) is of degree �� and has a positive leading coefficient. These polynomials are orthonormal over the unit disk �� = {�� : |��| < 1} with respect to a weight of the form |ℎ(��)|², where ℎ(��) is a polynomial without zeros in ��. That is, they satisfy ∫�� ��ₙ(��)��ₘ(��) |ℎ(��)|² dx dy = δₙ,ₘ. We establish the behavior of ��ₙ(��) as �� → ∞ at every point of the complex plane. We will also discuss the behavior these formulas impose on the zeros of the polynomials ��ₙ. A comparison with similar known results for polynomials orthogonal over the unit circle will be made, and future research problems/extensions will be discussed. Students are welcome.
Relational Format
presentation
Recommended Citation
Miña-Díaz, Erwin, "Asymptotics of Polynomials Orthogonal Over the Complex Unit Disk with Respect to a Positive Polynomial Weight, Part I" (2010). Analysis Seminar. 29.
https://egrove.olemiss.edu/math_analysis/29
