Asymptotic behaviour of some families of orthonormal polynomials and an associated Hilbert space

We characterise asymptotic behaviour of families of symmetric orthonormal polynomials whose recursion coefficients satisfy certain conditions, satisfied for example by the (normalised) Hermite polynomials. More generally, these conditions are satisfied by the recursion coefficients of the form $c(n+1)^p$ for $00$, as well as by recursion coefficients which correspond to polynomials orthonormal with respect to the exponential weight $W(x)=\exp(-|x|^\beta)$ for $\beta>1$. We use these results to show that, in a Hilbert space defined in a natural way by such a family of orthonormal polynomials, every two complex exponentials $e_{\omega}(t)={e}^{i \omega t}$ and $e_{\sigma}(t)={e}^{i \sigma t}$ of distinct frequencies $\omega,\sigma$ are mutually orthogonal. We finally formulate a surprising conjecture for the corresponding families of non-symmetric orthonormal polynomials; extensive numerical tests indicate that such a conjecture appears to be true.Comment: a conjecture added, simplifier proof of Lemma 29, a few remaining typos corrected, this is the final version, I promis

Periodic perturbations of unbounded Jacobi matrices II: Formulas for density

We give formulas for the density of the measure of orthogonality for orthonormal polynomials with unbounded recurrence coefficients. The formulas involve limits of appropriately scaled Tur\'an determinants or Christoffel functions. Exact asymptotics of the polynomials and numerical examples are also provided.Comment: 21 pages, 2 figures, 2 table

Periodic perturbations of unbounded Jacobi matrices I: Asymptotics of generalized eigenvectors

We study asymptotics of generalized eigenvectors associated with Jacobi matrices. Under weak conditions on the coefficients we identify when the matrices are self-adjoint and show that they satisfy strong non-subordinacy condition.Comment: 34 page

Spectral properties of block Jacobi matrices

We study the spectral properties of bounded and unbounded Jacobi matrices whose entries are bounded operators on a complex Hilbert space. In particular, we formulate conditions assuring that the spectrum of the studied operators is continuous. Uniform asymptotics of generalised eigenvectors and conditions implying complete indeterminacy are also provided.Comment: 27 page

Asymptotic behaviour of Christoffel-Darboux kernel via three-term recurrence relation II

We study orthogonal polynomials with periodically modulated Jacobi parameters in the case when $0$ lies on the soft edge of the spectrum of the corresponding periodic Jacobi matrix. We determine when the orthogonality measure is absolutely continuous and we provide a constructive formula for it in terms of the limit of Tur\'an determinants. We next consider asymptotics of the solutions of associated second order difference equation. Finally, we study scaling limits of the Christoffel--Darboux kernel.Comment: 40 page