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Asymptotic behavior of polynomials orthonormal on a homogeneous set. (English) Zbl 1032.42028
In this paper the asymptotic behavior of orthonormal polynomials with respect to some types of measures \(\sigma\) supported on a homogeneous set \(E\) is established. More concretely, for the positive Borel measures \(\sigma\) such that its absolutely continuous part satisfy the Szegő type condition (\(\log \sigma_{\text{a.c.}}'(z(t))\in L^1\), being \(z(t)\) the universal covering map which maps the interior of the unit circle \(D\) onto \(C\setminus E\). Under these conditions the authors give an asymptotic representation, on and off the support, for the corresponding orthonormal polynomials. As special cases they obtain known results by Widom, Szegő, and Kolmogorov and Krein.

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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