Szegö theorem, Carathéodory domains, and boundedness of calculating functionals (Q2387821)

Let \(G\) be a bounded, simply connected, plane domain with boundary \(\Gamma\), \(\mu = \mu_a + \mu_s\) be a finite Borel measure with support in \(\Gamma\) and \(\omega\) be the harmonic measure of a point \(z_0 \in G\) on \(\Gamma\). Inspired by the classical Szegö-Kolmogorov-Krein theorem [\textit{G. Szegö}, Math. Zeitschr. 6, 167--202 (1920; JFM 47.0391.04)], the authors study the condition \[ \int \ln \biggl( \frac{d\mu_a}{d\omega} \biggr) \,d\omega = -\infty. \tag \(*\) \] In particular they describe the geometry of \(G\) (in terms of Carathéodory domains and the connectedness of \(\mathbb C \setminus \overline G\)) when \((\ast)\) is equivalent to the completeness of the polynomials in \(L^t(\mu)\) or to the unboundedness of the calculating functional \(p \mapsto p(z_0)\) for polynomials in \(L^t(\mu)\). For measures \(\mu\) not necessarily with support on \(\Gamma\), a complete such characterization is not yet known. In the present paper a condition is presented for when the calculating functional is unbounded for the set of polyomials in \(L^2(\mu)\), where \(\mu\) has arbitrary compact support.