Widow factors for the Hilbert norm

Abstract

Given a probability measure µ with non-polar compact support K, we define the n-th Widom factor W2n(µ) as the ratio of the Hilbert norm of the monic n-th orthogonal polynomial and the n-th power of the logarithmic capacity of K. If µ is regular in the Stahl–Totik sense then the sequence (W2n(µ))∞n=0 has subexponential growth. For measures from the Szegő class on [−1, 1] this sequence converges to some proper value. We calculate the corresponding limit for the measure that generates the Jacobi polynomials, analyze the behavior of the corresponding limit as a function of the parameters and review some other examples of measures when Widom factors can be evaluated.