Generalized Bounded Variation and Inserting point masses
| dc.creator | Wong, Manwah Lilian | |
| dc.date | 2007-07-10 | |
| dc.date | 2008-09-28 | |
| dc.date.accessioned | 2026-07-07T10:05:29Z | |
| dc.date.available | 2026-07-07T10:05:29Z | |
| dc.description | 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 $dμ_0$, a probability measure on the unit circle with $\ell^2$ Verblunsky coefficients $(α_n (dμ_0))_{n=0}^{\infty}$ of bounded variation. We insert $m$ pure points to $dμ$, rescale, and form the probability measure $dμ_m$. We use the formula above to prove that the Verblunsky coefficients of $dμ_m$ 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 $dμ_m$ 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 $D(z, dμ_m)^{-1}$ away from the pure points. | |
| dc.description | To appear in Constructive Approximation | |
| dc.identifier | https://arxiv.org/abs/0707.1368 | |
| dc.identifier | http://arxiv.org/abs/0707.1368 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/170021 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.title | Generalized Bounded Variation and Inserting point masses | |
| dc.type | text |