Seminar Event Detail

Complex Analysis, Dynamics and Geometry

Date:  Monday, February 05, 2018
Location:  3096 East Hall (4:00 PM to 5:00 PM)

Title:  On the maximum modulus of weighted polynomials in the plane, zero distribution and zero location of weighted extremal polynomials.

Abstract:   For a closed, strongly regular subset of the complex plane E of positive capacity and a non negative continuous weight w on E satisfying that the set where w is positive is of positive capacity, we prove that any weighted polynomial P_nw^n of degree at most n, n at least 1 satisfies that all points for which it attains its maximum on E live in the support of the weight w. If E is unbounded, we assume that w is of sufficient fast decrease with large argument. Examples are given to show that our requirements on E cannot in general be relaxed. As a consequence of this result we show that If E is a real interval of positive length, and p is a fixed positive number, we prove a necessary and sufficient condition which ensures that the Lp norm of P_nw^n on E is in an $n$th root sense, controlled by a corresponding discrete Lp Holder norm of P_nw^n on a certain well separated admissible triangular scheme of points $E_n$, n at least 1 of $E$. When P_nw^n is extremal on E_n, this condition implies results on zero distribution and zero location of P_nw^n on E.


Speaker:  Steven Damelin
Institution:  Math Reviews

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