Jacob Stordal Christiansen: Large Widom factors

Abstract: For any compact set in the complex plane, the Chebyshev polynomial is the unique monic polynomial of a given degree that minimizes the maximum absolute value on the set. While it is well known that the n-th root of this minimum value converges to the logarithmic capacity of the set, the finer asymptotic behavior is captured by the so-called Widom factors: the ratio of the polynomial norm to the capacity raised to the power of the degree. It is known that for sets with sufficiently smooth boundaries, these factors remain uniformly bounded. However, it remains a major open question whether there exists any compact connected set for which the Widom factors are unbounded. In this talk, I will discuss the search for such a set. We will examine the theoretical lower bounds for Chebyshev norms and explore the specific pathological or fractal geometries that would be required to force the Widom factors to grow indefinitely, effectively characterizing the potential worst-case scenario in polynomial approximation.