A note on the asymptotics of orthogonal polynomials on a complex arc: The case of a measure with a discrete part
Orthogonal polynomials with respect to a measure \(\mu= \mu_1+ \mu_2\) in the complex plane are studied, where \(\mu_1\) has support on a complex rectifiable are \(E\) and is absolutely continuous with respect to arc length, and \(\mu_2\) has finite support on \(N\) points in the exterior of \(E\). For \(E= [- 1,1]\) the asymptotic behaviour of the orthogonal polynomials is fairly well-known, but for a complex arc there is no recurrence relation and there are no nice properties of the zeros, so that the analysis is much harder. Also for a closed curve there are nice results, but the case of an arc is quite different. The author proposes to study the orthogonal polynomials using the extremal property, an approach previously developed by H. Widom in 1969. The main result is that for a sufficiently smooth arc and for a measure \(\mu_1\) satisfying a Szegö condition on \(E\), one can describe the strong asymptotic behaviour of the \(L_2(\mu)\)-norm of the monic orthogonal polynomials in terms of the capacity of \(E\), the asymptotic behaviour of the orthogonal polynomials for the measure \(\mu\) in \(L_2(\mu)\) and the strong asymptotics in the exterior of \(E\) with a limit function that vanishes at the \(N\) points in the support of \(\mu_2\).
scientific article; zbMATH DE number 720671