Szegő ’ s problem on curves ∗

For a system of smooth Jordan curves asymptotics for Christoffel functions is established almost everywhere for measures belonging to Szegő’s class. 1 The result Let μ be a finite Borel-measure on the plane with compact support consisting of infinitely many points. The Christoffel functions associated with μ are defined as λn(z, μ) = inf Pn(z)=1 ∫ |Pn|dμ, where the infimum is taken for all polynomials of degree at most n that take the value 1 at z. Christoffel functions are closely related to orthogonal polynomials (for a survey see [15] by P. Nevai and [22] by B. Simon), to statistical physics (see e.g. [16] by L. Pastur), to universality in random matrix theory (see e.g. the recent breakthrough [11] by D. Lubinsky, as well as [3],[23],[29]), to spectral theory (see e.g. [24], [22] by B. Simon and [1] by Breuer, Last and Simon) and to several other fields in mathematics. For the role and various use of Christoffel functions see [5], [7], [24], and particularly [15] by P. Nevai and [22] by B. Simon. Their asymptotics on the real line and on the unit circle has been thoroughly investigated (see e.g. [11], [12], [13], [24], [23], [21], [26], [28]), but until recently not much has been known on their asymptotic behavior on general curves. In this work we prove AMS Subject Classification 42C05, 30C85, 31A15;