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12:00 pm Monday, January 10, 2011 Stulken Geometry-Analysis Seminar: Orthogonal Polynomials with Recursion Coefficients of Generalized Bounded Variationby
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.
Submitted by damanik@rice.edu |