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Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case

Grzegorz Świderski, Bartosz Trojan

Abstract

We study orthogonal polynomials with periodically modulated recurrence coefficients when $0$ lies on the hard edge of the spectrum of the corresponding periodic Jacobi matrix. In particular, we show that their orthogonality measure is purely absolutely continuous on a real half-line and purely discrete on its complement. Additionally, we provide the constructive formula for the density in terms of Turán determinants. Moreover, we determine the exact asymptotic behavior of the orthogonal polynomials. Finally, we study scaling limits of the Christoffel-Darboux kernel.

Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case

Abstract

We study orthogonal polynomials with periodically modulated recurrence coefficients when lies on the hard edge of the spectrum of the corresponding periodic Jacobi matrix. In particular, we show that their orthogonality measure is purely absolutely continuous on a real half-line and purely discrete on its complement. Additionally, we provide the constructive formula for the density in terms of Turán determinants. Moreover, we determine the exact asymptotic behavior of the orthogonal polynomials. Finally, we study scaling limits of the Christoffel-Darboux kernel.

Paper Structure

This paper contains 24 sections, 32 theorems, 543 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stolz class
  4. Finite matrices
  5. Jacobi matrices
  6. Periodic Jacobi parameters
  7. Periodic modulations
  8. The shifted conjugation
  9. Essential spectrum
  10. Generalized Turán determinants
  11. Asymptotics of the generalized eigenvectors
  12. Approximation procedure
  13. Turán determinants
  14. Asymptotics of the polynomials
  15. The Christoffel--Darboux kernel
  16. ...and 9 more sections

Key Result

Theorem 1

Suppose that Jacobi parameters $(a_n : n \in \mathbb{N}_0)$ and $(b_n : n \in \mathbb{N}_0)$ are $N$-periodically modulated and such that $\mathfrak{X}_0(0)$ is not diagonalizable. Assume further that Then there is an explicit polynomial $\tau$ of degree $1$ (see eq:61b) such that the measure $\mu$ restricted to is purely discrete. More preciselyFor a set $X \subset \mathbb{R}$ by $X'$ we denote

Figures (1)

  • Figure 1: An example for $N=4$. If $0 = x_4$, then we are in the case \ref{['perMod:IIa']}, while $0 \in \{x_1, x_2, x_3, x_6, x_7, x_8 \}$ corresponds to the case \ref{['perMod:IIb']}.

Theorems & Definitions (67)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • ...and 57 more