Generalized Bounded Variation and Inserting point masses

Abstract

Let $d\mu$ be a probability measure on the unit circle and $d\nu$ be the measure formed by adding a pure point to $d\mu$. We give a simple formula for the Verblunsky coefficients of $d\nu$ based on a result of Simon. Then we consider $d{\mu }_0$, a probability measure on the unit circle with ${\ell }^2$ Verblunsky coefficients $({\alpha }_n(d{\mu }_0){)}_{n=0}^{\infty }$ of bounded variation. We insert $m$ pure points to $d\mu$, rescale, and form the probability measure $d{\mu }_m$. We use the formula above to prove that the Verblunsky coefficients of $d{\mu }_m$ are in the form $α_n(dμ_0) + \sum_{j=1}^m \frac{\ol{z_j}^{n} c_j}{n} + E_n$, 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 $d{\mu }_m$ is of $(m+1)$-generalized bounded variation - a notion that we shall introduce in the paper. Then we use this fact to prove that $\lim_{n \to \infty} \vp_n^*(z, dμ_m)$ is continuous and is equal to $D(z,d{\mu }_m{)}^{-1}$ away from the pure points.

To appear in Constructive Approximation

References in corpus (1)

• Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrodinger operators