Zero distributions for polynomials orthogonal with weights over certain planar regions (Q816263)

Let \(G\) be the interior of a closed Jordan curve \(L\). Let \(\mu_L\) be the equilibrium measure of \(L\). Let \(w\) be a holomorphic function on \(G\) such that \(\int_G| w(z)| ^2dm \,<\,\infty\), where \(m\) is the Lebesgue measure on \(\mathbb C\). Put \(\mathcal L^2_w(G):=\{f\in\mathcal O(G)\); \(\int_G| f(z)| ^2| w(z)| ^2 dm\,<\,\infty\}\), and let \(K_w (z\,,\,\zeta)\) be the Bergman kernel of \(\mathcal L^2_w(G)\). Let \(P_n(z)\equiv P_n(z,w)\) \((n\geq 0)\) be the sequence of orthonormal polynomials in the space \(\mathcal L^2_w(G)\). If \(\{z_{n1},\cdots,z_{nn}\}\) are zeros (counted with their multiplicities) of \(P_n\) we put \(\nu_n\,:=\,\sum_{j=1}^n\delta_{z_{nj}}\), where \(\delta_z\) is the Dirac measure concentrated at \(z\). {Main result}: If (a) there exists a subsequence \(\mathcal N\subset \mathbb N\) such that \(\nu_n\) tends weakly to \(\mu_L\) as \(n\rightarrow \infty\) \((n\in \mathcal N)\) then (b) there exists a point \(\zeta\in G\) such that \(K(\cdot\, .\zeta)\) has a singularity on the boundary of \(G\). Moreover, if \(w\) is such that the set of all polynomials is dense in \(\mathcal L^2_w(G)\) then \((a)\Leftrightarrow (b)\).