Orthogonal Polynomials with Recursion Coefficients of Generalized Bounded Variation

Abstract

We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an ${\ell }^p$ condition 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., ${\sum }_{n=0}^{\infty }|e^{i{\varphi }_l}{\beta }_{n+1}^{(l)}-{\beta }_n^{(l)}|$ is finite for some ${\varphi }_l$. This includes discrete Schrödinger operators on a half-line or line with finite linear combinations of Wigner--von Neumann type potentials. For the real line, we prove that in the Lebesgue decomposition $d\mu =fdm+d{\mu }_s$ of such measures, the intersection of (-2,2) with the support of $d{\mu }_s$ 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.

arXiv comment28 pages, no figures

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