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

Given a probability measure μ supported on some compact set K ⊆ C and with orthonormal polynomials {pn(z)}_(n∈N), define the measures dμ_(n) = 1/(n+1) ∑^(n)_(j=0)|p_(j)(z)|^(2)dμ(z) and let ν_n be the normalized zero counting measure for the polynomial p_n. If μ 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 ∈ N the kth moment of ν_(n+1) and μ_n differ by at most O(n^(−1)) as n → ∞