55 research outputs found
Weak Convergence of CD Kernels: A New Approach on the Circle and Real Line
Let m be a probability measure supported on some infinite and compact set K
in the complex plane and let p_n(z) be the corresponding degree n orthonormal
polynomial with positive leading coefficient. Let v_n be the normalized zero
counting measure for the polynomial p_n and let u_n be the probability measure
given by (n+1)u_n=K_n(z,z)m, where K_n(z,w) is the reproducing kernel for
polynomials of degree at most n. If m is supported on a compact subset of the
real line or the unit circle, we provide a new proof of a 2009 theorem due to
Simon, that for any fixed natural number k, the k^{th} moment of u_n and
v_{n+1} differ by at most O(1/n) as n tends to infinity.Comment: 6 pages; Version 2 includes minor changes to the exposition and
corrects minor typo
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