Christoffel functions for doubling measures on quasismooth curves and arcs

Abstract

Matching two-sided estimates are given for Christoffel functions associated with a doubling measure ν over a quasismooth curve or arc. The size of the the n-th Christoffel function at a point z is given by the ν-measure of the largest disk about z which lies within the 1/n-level line of the Green’s function. The main theorem contains as special case all previously known weak asymptotics for Christoffel functions, and it also gives their size in explicit form about smooth corners. Applications are given for estimating the size of orthonormal polynomials and for Nikolskii-type inequalities.