Universality for locally Szegő measures
Journal of Approximation Theory
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Let @m be a measure with compact support. Assume that @x is a Lebesgue point of @m and that @m^' is positive and continuous at @x. Let {A"n} be a sequence of positive numbers with limit ~. We show that one can choose @x"n@?[@x-A"nn,@x+A"nn] such that limn-~K"n(@x"n,@x"n+aK@?"n(@x"n,@x"n))K"n(@x"n,@x"n)=sin@pa@pa, uniformly for a in compact subsets of the plane. Here K"n is the nth reproducing kernel for @m, and K@?"n is its normalized cousin. Thus universality in the bulk holds on a sequence close to @x, without having to assume that @m is a regular measure. Similar results are established for sequences of measures.