Weak convergence of CD kernels: A new approach on the circle and real line

Given a probability measure μμ supported on some compact set K⊆CK⊆C and with orthonormal polynomials {pn(z)}n∈N{pn(z)}n∈N, define the measures dμn(z)=1n+1∑j=0n|pj(z)|2dμ(z) and let νnνn be the normalized zero counting measure for the polynomial pnpn. If μμ is supported on a compact subset of the real line or on the unit circle, we provide a new proof of a 2009 theorem due to Simon that for any fixed k∈Nk∈N the kkth moment of νn+1νn+1 and μnμn differ by at most O(n−1)O(n−1) as n→∞n→∞.