Widom factors for the Hilbert norm
Volume 107 / 2015
Banach Center Publications 107 (2015), 11-18
MSC: 42C05, 33C45, 30C85, 47B36.
DOI: 10.4064/bc107-0-1
Abstract
Given a probability measure $\mu$ with non-polar compact support $K$, we define the $n$-th Widom factor $W^2_n(\mu)$ 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 $\mu$ is regular in the Stahl–Totik sense then the sequence $(W^2_n(\mu))_{n=0}^{\infty}$ 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.
