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.