On stability in the Borg–Hochstadt theorem for periodic Jacobi matrices

On Rakhmanov’s Theorem for Jacobi Matrices

We prove Rakhmanov’s theorem for Jacobi matrices without the additional assumption that the number of bound states is finite. This result solves one of Nevai’s open problems. Consider a measure dμ with bounded support in R. Assume that it has infinitely many growth points. Let {pk} (k = 0, 1, . . .) be the system of polynomials orthonormal with respect to that measure, i.e., ∞ ∫ −∞ pk(x)pm(x)dμ...

Marchenko-Ostrovski mappings for periodic Jacobi matrices

We consider the 1D periodic Jacobi matrices. The spectrum of this operator is purely absolutely continuous and consists of intervals separated by gaps. We solve the inverse problem (including characterization) in terms of vertical slits on the quasimomentum domain . Furthermore, we obtain a priori two-sided estimates for vertical slits in terms of Jacoby matrices.

Titchmarsh theorem for Jacobi Dini-Lipshitz functions

Our aim in this paper is to prove an analog of Younis's Theorem on the image under the Jacobi transform of a class functions satisfying a generalized Dini-Lipschitz condition in the space $mathrm{L}_{(alpha,beta)}^{p}(mathbb{R}^{+})$, $(1< pleq 2)$. It is a version of Titchmarsh's theorem on the description of the image under the Fourier transform of a class of functions satisfying the Dini-Lip...

A Strong Szegő Theorem for Jacobi Matrices

We use a classical result of Gollinski and Ibragimov to prove an analog of the strong Szegő theorem for Jacobi matrices on l2(N). In particular, we consider the class of Jacobi matrices with conditionally summable parameter sequences and find necessary and sufficient conditions on the spectral measure such that ∑ ∞ k=n bk and ∑ ∞ k=n(a 2 k − 1) lie in l2 1, the linearly-weighted l2 space.

Two modified Jacobi methods for M-matrices