Self-adjoint difference operators and classical solutions to the Stieltjes--Wigert moment problem

The Stieltjes-Wigert polynomials, which correspond to an indeterminate moment problem on the positive half-line, are eigenfunctions of a second order q-difference operator. We consider the orthogonality measures for which the difference operator is symmetric in the corresponding weighted $L^2$-spaces. These measures are exactly the solutions to the q-Pearson equation.In the case of discrete and absolutely continuous measures the difference operator is essentially self-adjoint, and the corresponding spectral decomposition is given explicitly. In particular, we find an orthogonal set of q-Bessel functions complementing the Stieltjes-Wigert polynomials to an orthogonal basis for $L^2(\mu)$ when $\mu$ is a discrete orthogonality measure solving the q-Pearson equation. To obtain the spectral decomposition of the difference operator in case of an absolutely continuous orthogonality measure we use the results from the discrete case combined with direct integral techniques.Comment: 22 pages; section 2 rewritten, to appear in Journal of Approximation Theor

Finite Gap Jacobi Matrices, I. The Isospectral Torus

Let $\frak{e}\subset\mathbb{R}$ be a finite union of disjoint closed intervals. In the study of OPRL with measures whose essential support is $\frak{e}$, a fundamental role is played by the isospectral torus. In this paper, we use a covering map formalism to define and study this isospectral torus. Our goal is to make a coherent presentation of properties and bounds for this special class as a tool for ourselves and others to study perturbations. One important result is the expression of Jost functions for the torus in terms of theta functions.Comment: 68 pages, 4 figure

A geometric approach to approximating the limit set of eigenvalues for banded Toeplitz matrices

This article is about finding the limit set for banded Toeplitz matrices. Our main result is a new approach to approximate the limit set $\Lambda(b)$ where $b$ is the symbol of the banded Toeplitz matrix. The new approach is geometrical and based on the formula $\Lambda(b) = \cap_{\rho \in (0, \infty)} \text{sp } T(b_\rho)$. We show that the full intersection can be approximated by the intersection for a finite number of $\rho$'s, and that the intersection of polygon approximations for $\text{sp } T(b_\rho)$ yields an approximating polygon for $\Lambda(b)$ that converges to $\Lambda(b)$ in the Hausdorff metric. Further, we show that one can slightly expand the polygon approximations for $\text{sp } T(b_\rho)$ to ensure that they contain $\text{sp } T(b_\rho)$. Then, taking the intersection yields an approximating superset of $\Lambda(b)$ which converges to $\Lambda(b)$ in the Hausdorff metric, and is guaranteed to contain $\Lambda(b)$. We implement the algorithm in Python and test it. It performs on par to and better in some cases than existing algorithms. We argue, but do not prove, that the average time complexity of the algorithm is $O(n^2 + mn\log m)$, where $n$ is the number of $\rho$'s and $m$ is the number of vertices for the polygons approximating $\text{sp } T(b_\rho)$. Further, we argue that the distance from $\Lambda(b)$ to both the approximating polygon and the approximating superset decreases as $O(1/\sqrt{k})$ for most of $\Lambda(b)$, where $k$ is the number of elementary operations required by the algorithm.Comment: 20 pages, 8 figures. Submitted to SIAM Journal on Matrix Analysis and Application

Finite Gap Jacobi Matrices, III. Beyond the Szegő Class

Let e ⊂ R be a finite union of ℓ+1 disjoint closed intervals, and denote by ω_j the harmonic measure of the j left-most bands. The frequency module for e is the set of all integral combinations of ω_1,…,ω_ℓ. Let {a_nb_n}^∞_(n=−∞) be a point in the isospectral torus for e and p_n its orthogonal polynomials. Let {a_nb_n}^∞_(n=1) be a half-line Jacobi matrix with a_n=a_n+δa_n, b_n=b_n+δb_n. Suppose ∑^∞_(n=1)│δan│^2 + │δb_n│^2 < ∞ and ∑^N_n=1^e^(2πiωn), δa_n ∑^N_n=1^e^(2πiωn) δb_n have finite limits as N → ∞ for all ω in the frequency module. If, in addition, these partial sums grow at most subexponentially with respect to ω, then for z∈ℂ∖ℝ, p_n(z)p_n(z) has a limit as n→∞. Moreover, we show that there are non-Szegő class J’s for which this holds