Event Details

Speaker: Jacob S. Christiansen (Lund University, Sweden) Title: Chebyshev polynomials and Widom factors

Abstract:Let E⊂ℂ be an infinite compact set. We consider the monic polynomials Tₙ of degree n that minimize the supremum norm on E. These are the classical Chebyshev (or minimax) polynomials of the set E.

A foundational result by Szegő states that the minimal norm is bounded below by the logarithmic capacity of the set: ‖Tₙ‖_E ≥ Cap(E)ⁿ. This lower bound doubles when E⊂ℝ, a result proven by Schiefermayr. More recently, Totik showed that for real subsets, ‖Tₙ‖_E/Cap(E)ⁿ → 2 if and only if E is a single interval.

In this talk, we study the associated Widom factors, defined as Wₙ(E) := ‖Tₙ‖_E/Cap(E)ⁿ, and address the question of which other subsets of ℂ exhibit the limiting behaviour Wₙ(E)→2. We show that this is indeed the case for certain polynomial preimages, which form balanced, tree-like structures in the complex plane.

Our proof connects this geometric problem to the classical theory of weighted orthogonal polynomials on the real line, building on seminal results of Bernstein. We will discuss the symmetry properties underlying this phenomenon and explore related open problems in the field.

This talk is based on joint work with B. Eichinger (Lancaster) and O. Rubin (Lund).