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Matrix valued discrete-continuous functions with the prolate spheroidal property and bispectrality

W. Riley Casper, F. Alberto Grunbaum, Milen Yakimov, Ignacio Zurrian

Abstract

Classical prolate spheroidal functions play an important role in the study of time-band limiting, scaling limits of random matrices, and the distribution of the zeros of the Riemann zeta function. We establish an intrinsic relationship between discrete-continuous bispectral functions and the prolate spheroidal phenomenon. The former functions form a vast class, parametrized by an infinite dimensional manifold, and are constructed by Darboux transformations from classical bispectral functions associated to orthogonal polynomials. Special cases include spherical functions. We prove that all such Darboux transformations which are self-adjoint in a certain sense give rise to integral operators possessing commuting differential operators and to discrete integral operators possessing commuting shift operators. One particularly striking implication of this is the correspondence between discrete and continuous pairs of commuting operators. Moreover, all results are proved in the setting of matrix valued functions, which provides further advantages for applications. Our methods rely on the use of noncommutative matrix valued Fourier algebras associated to discrete-continuous bispectral functions. We produce the commuting differential and shift operators in a constructive way with explicit upper bounds on their orders and bandwidths, which is illustrated with many concrete examples.

Matrix valued discrete-continuous functions with the prolate spheroidal property and bispectrality

Abstract

Classical prolate spheroidal functions play an important role in the study of time-band limiting, scaling limits of random matrices, and the distribution of the zeros of the Riemann zeta function. We establish an intrinsic relationship between discrete-continuous bispectral functions and the prolate spheroidal phenomenon. The former functions form a vast class, parametrized by an infinite dimensional manifold, and are constructed by Darboux transformations from classical bispectral functions associated to orthogonal polynomials. Special cases include spherical functions. We prove that all such Darboux transformations which are self-adjoint in a certain sense give rise to integral operators possessing commuting differential operators and to discrete integral operators possessing commuting shift operators. One particularly striking implication of this is the correspondence between discrete and continuous pairs of commuting operators. Moreover, all results are proved in the setting of matrix valued functions, which provides further advantages for applications. Our methods rely on the use of noncommutative matrix valued Fourier algebras associated to discrete-continuous bispectral functions. We produce the commuting differential and shift operators in a constructive way with explicit upper bounds on their orders and bandwidths, which is illustrated with many concrete examples.
Paper Structure (26 sections, 12 theorems, 168 equations, 3 figures)

This paper contains 26 sections, 12 theorems, 168 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Matrix valued discrete-continuous bispectrality
  3. The prolate spheroidal property
  4. Results
  5. Bispectrality and Fourier algebras
  6. Basic definitions
  7. Bispectral Darboux transformations
  8. Shift and differential operators
  9. Bifiltration
  10. Orthogonal polynomials
  11. Basic definitions
  12. Classical discrete-continuous bispectral functions
  13. Behaviour of Fourier algebras under bispectral Darboux transformations
  14. Estimates in the classical case
  15. Estimates for all bispectral Darboux transformations
  16. ...and 11 more sections

Key Result

Theorem 2.7

Let $\widetilde{\Psi}$ be a bispectral Darboux transformation of a bispectral function $\Psi$. Then:

Figures (3)

  • Figure 1: The classical orthogonal polynomials on ${\mathbb R}$.
  • Figure 2: Classical discrete-continuous bispectral functions.
  • Figure 3: Differential operators commuting with integral operators for the classical discrete-continuous bispectral functions.

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Theorem 2.7
  • proof
  • Remark 2.8
  • Definition 2.9
  • ...and 35 more