Widom factors for the Hilbert norm
Let \(\mu\) be a probability Borel measure with compact support \(K\) on the complex plane and let (\(q_{n}\)) be the sequence of monic orthogonal polynomials with respect to \(\mu\). The \(n\)-th Widom-Hilbert factor of \(\mu\) is defined by NEWLINE\[NEWLINEW_{n}^{2}(\mu)=\frac{\|q_{n}\|_{L^{2}(\mu)}}{\mathrm{Cap}^{n}(K)},NEWLINE\]NEWLINE where \(\mathrm{Cap}(K)\) denotes the logarithmic capacity of \(K\).NEWLINENEWLINEIn the paper under review, the authors calculate \(\lim_{n\to\infty}W_{n}^{2}(\mu)\) for the measure \(\mu\) that generates the Jacobi polynomials, they examine the behavior of the Widom-Hilbert factors of the measure that generates the Pollaczek polynomials and of some irregular measures and they prove some extremal properties of the equilibrium measure of the interval \([-1,1]\) among the measures of the Szegö class. Also, they analyze the behavior of the Widom-Hilbert factors of the equilibrium measures of Julia sets.NEWLINENEWLINEFor the entire collection see [Zbl 1337.41001].
scientific article; zbMATH DE number 6556699