Journal of the European Mathematical Society

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Volume 20, Issue 3, 2018, pp. 747–796
DOI: 10.4171/JEMS/776

Published online: 2018-02-27

Christoffel functions with power type weights

Tivadar Danka[1] and Vilmos Totik[2]

(1) University of Szeged, Hungary
(2) University of Szeged, Hungary and University of South Florida, USA

Precise asymptotics for Christoffel functions are established for power type weights on unions of Jordan curves and arcs. The asymptotics involve the equilibrium measure of the support of the measure. The result at the endpoints of arc components is obtained from the corresponding asymptotics for internal points with respect to a different power weight. On curve components the asymptotic formula is proved via a sharp form of Hilbert's lemniscate theorem while taking polynomial inverse images. The situation is completely different on the arc components, where the local asymptotics is obtained via a discretization of the equilibrium measure with respect to the zeros of an associated Bessel function. The proofs are potential theoretical, and fast decreasing polynomials play an essential role in them.

Keywords: Christoffel functions, asymptotics, power type weights, Jordan curves and arcs, Bessel functions, fast decreasing polynomials, equilibrium measures

Danka Tivadar, Totik Vilmos: Christoffel functions with power type weights. J. Eur. Math. Soc. 20 (2018), 747-796. doi: 10.4171/JEMS/776