Generalized Bounded Variation and Inserting Point Masses

Abstract

Let d μ be a probability measure on the unit circle and d ν be the measure formed by adding a pure point to d μ. We give a formula for the Verblunsky coefficients of d ν following the method of Simon. Then we consider d μ 0, a probability measure on the unit circle with ℓ 2 Verblunsky coefficients (α n (d μ 0)) n=0∞ of bounded variation. We insert m pure points z j into d μ 0, rescale, and form the probability measure d μ m . We use the formula above to prove that the Verblunsky coefficients of d μ m are in the form $\alpha_{n}(d\mu_{0})+\sum_{j=1}^{m}\frac{\overline{z_{j}}^{n}c_{j}}{n}+E_{n}$ , where the c j 's are constants of norm 1 independent of the weights of the pure points and independent of n; the error term E n is in the order of o(1/n). Furthermore, we prove that d μ m is of (m+1)-generalized bounded variation—a notion that we shall introduce in the paper. Then we use this fact to prove that lim  n→∞ φ n *(z,d μ m ) is continuous and is equal to D(z,d μ m )−1 away from the pure points.

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