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Orthogonal polynomials on infinite gap sets

Jacob Stordal Christiansen
(Copenhagen University)
Analyseseminar
Torsdag, 24 februar, 2011, at 16:15-17:15, in Aud. D3 (1531-215)
Abstrakt:

In the talk, I'll discuss orthogonal polynomials on infinite gap sets $E$ of Parreau-Widom type. This notion covers a wide class of compact sets that trivially contains all finite unions of disjoint intervals but also includes Cantor sets of positive measure. For probability measures $\mathrm{d}m=f(t)\mathrm{d}t+\mathrm{d}m_s$ with essential support $E$, we shall concentrate on the Szego condition

\[ \int_E \log f(t) \mathrm{d}m_E(t) > -\infty, \]

where $\mathrm{d}m_E$ is the equilibrium measure of $E$. Under certain assumptions on the mass points of $\mathrm{d}m_s$ outside $E$, we show that this condition is equivalent to boundedness of the leading coefficients in the associated orthonormal polynomials $P_n$ (when $\mathrm{Cap}(E)=1$). We then consider the large $n$ behaviour of $P_n$ with the aim of establishing Szego asymptotics, which is stronger than root and ratio asymptotics. The set-up is based on potential theory and our techniques rely on a covering space formalism introduced into spectral theory by Sodin-Yuditskii.

Kontaktperson: Jacob Schach Møller