Generalized Bounded Variation and Inserting Point Masses

Abstract. Let dμ be a probability measure on the unit circle and dν be the measure formed by adding a pure point to dμ. We give a simple formula for the Verblunsky coefficients of dν based on a result of Simon. Then we consider dμ0, a probability measure on the unit circle with l Verblunsky coefficients (αn(dμ0)) ∞ n=0 of bounded variation. We insert m pure points to dμ, rescale, and form the probability measure dμm. We use the formula above to prove that the Verblunsky coefficients of dμm are in the form αn(dμ0)+ ∑m j=1 zj cj n +En, where the cj ’s are constants of norm 1 independent of the weights of the pure points and independent of n; the error term En is in the order of o(1/n). Furthermore, we prove that dμm is of (m+1)generalized bounded variation a notion that we shall introduce in the paper. Then we use this fact to prove that limn→∞ φ ∗ n(z, dμm) is continuous and is equal to D(z, dμm) −1 away from the pure points.