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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 dm=f(t)dt+dm_s with essential support E, we shall concentrate on the Szego condition


\int_E  log f(t)  dm_E(t) > -\infty,


where dm_E is the equilibrium measure of E. Under certain assumptions on the mass points of dm_s outside E, we show that this condition is equivalent to boundedness of the leading coefficients in the associated orthonormal polynomials P_n (when 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