Location: Zoom
https://unm.zoom.us/j/96832154356?pwd=S3BpaDd1TkNCQmN2L3lZdmVDUXdzUT09
Meeting ID: 968 3215 4356
Passcode: 31415926

 

Title: Chebyshev polynomials and Widom factors

 

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).

 

Bio: Prof. Jacob Christiansen is the Head of Algebra, Analysis and Dynamical Systems Division in the Centre for Mathematical Sciences of Lund University, Sweden. His research interests are in spectral theory, mathematical physics, dynamical systems, operator theory, differential and difference equations, complex analysis, orthogonal polynomials, approximation theory, harmonic analysis, special functions, and probability theory.