Analysis and Probability seminar

Abstract:

Let E be an infinite compact set in the complex plane and denote by T_n the minimax (or Chebyshev) polynomials of E, that is, the monic degree n polynomials which minimize the sup-norm on E. A classical result of Szegö states that ||T_n||_E \geq Cap(E)^n for all n, a lower bound that doubles when E is a subset of R. More recently, Totik proved that for real subsets, ||T_n||_E / Cap(E)^n \to 2 if and only if E is an interval.

We shall introduce the so-called Widom factors by W_n(E) := ||T_n||_E / Cap(E)^n and pose the question if there are more subsets of the complex plane for which W_n(E) \to 2. It appears that the answer is indeed affirmative for certain polynomial preimages. Interestingly, our proof relies on properties of the Jacobi orthogonal polynomials due to Bernstein. We shall also settle a conjecture of Widom concerning Jordan arcs and discuss related open problems.

The talk is based on joint work with B. Eichinger (TU Wien) and O. Rubin (Lund).