12:00 pm Monday, January 10, 2011
Stulken Geometry-Analysis Seminar: Orthogonal Polynomials with Recursion Coefficients of Generalized Bounded Variation
by
Milivoje Lukic (California Institute of Technology) in HB 227
We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an $\ell^p$ condition ($1\le p < \infty$) and a generalized bounded variation condition. This latter condition requires that a sequence can be expressed as a sum of sequences $\beta^{(l)}$, each of which has rotated bounded variation, i.e. \begin{equation*} \sum_{n=0}^\infty \lvert e^{i\phi_l} \beta_{n+1}^{(l)} - \beta_n^{(l)} \rvert < \infty \end{equation*} for some $\phi_l \in\mathbb{R}$. This includes discrete Schr\"odinger operators on a half-line or line with finite linear combinations of Wigner--von Neumann potentials $\cos(n\phi+\alpha)/n^\gamma$, where $\gamma>0$. For the real line, our results state that in the Lebesgue decomposition $d\mu = f dm + d\mu_s$ of such measures, $\operatorname{supp}(d\mu_s) \cap (-2,2)$ is contained in an explicit finite set $S$ (thus, $d\mu$ has no singular continuous part), and $f$ is continuous and non-vanishing on $(-2,2) \setminus S$. The results for the unit circle are analogous, with $(-2,2)$ replaced by the unit circle. Host Department: Rice University-Mathematics
Submitted by damanik@rice.edu |
