Asymptotics of polynomials orthogonal over the unit disk with respect to a polynomial weight without zeros on the unit circle
The paper is devoted to obtain asymptotic formulas for the orthogonal polynomials over the unit disk with respect to a weight of the form \(| w(z)|^2\) with \(w(z)\) a polynomial without zeros on \( | z |=1\). The formulas describe the asymptotic behavior at every point of the complex plane. The following situation is considered. If \(w(z)=\Pi_{k=1}^s(z-a_k)^{\alpha_k}\Pi_{k=1}^u(z-b_k)^{\beta_k}\), where \(a_1,\dotsc, a_s\) are the zeros of \(w\) on \(| z | <1\), and \(b_1,\dotsc, b_u\) denote those lying on \(| z | >1\), then the polynomial \(g(z)= \Pi_{k=1}^u(z-\frac{1}{\overline{b_k}})^{\beta_k}\Pi_{k=1}^s(z-a_k)^{\alpha_k+1}\) is defined, which can be written in the form \(g(z)=\Pi_{k=1}^v(z-a_k)^{m_k}\), where the \(a_k\) are pairwise distinct, and \(m_k\) is the multiplicity of \(a_k\), so that \(\max\{s,u\} \leq v \leq s+u.\) If \(\rho= \max_{1\leq k \leq v}| a_k | \), then the exterior asymptotics are obtained for \( | z | > \rho\) and it is remarked that, if all the zeros lie on the open unit disk, the error terms decay geometrically. When \( | z | < \rho\), the asymptotic behavior of the polynomials is connected with a reproducing kernel of the \(| w|^2\)-weighted Bergman space of the unit disk. The behavior is determined only by the points \(a_k\) closer to the unit circle and with largest multiplicity. Finally, the behavior at the critical points \(a_k\) is described, which depends on the fact whether \(a_k\) is a zero of \(w\) or not.
scientific article; zbMATH DE number 6136916