An orthogonal polynomial analogue of the Landau-Pollak-Slepian time-frequency analysis

TL;DR

This paper develops a novel time-frequency analysis framework for orthogonal polynomials on [-1,1], paralleling classical bandlimited function analysis, and establishes localization, approximation, and uncertainty principles within this new setting.

Contribution

It introduces a time-frequency theory for orthogonal polynomials, linking spectral decomposition of a specific operator to polynomial theory, and proves an uncertainty principle in this context.

Findings

Spectral decomposition related to orthogonal polynomials is established.

Localization and approximation properties of eigenfunctions are demonstrated.

An uncertainty principle for orthogonal polynomial time-frequency analysis is proven.

Abstract

The aim of this article is to present a time-frequency theory for orthogonal polynomials on the interval [-1,1] that runs parallel to the time-frequency analysis of bandlimited functions developed by Landau, Pollak and Slepian. For this purpose, the spectral decomposition of a particular compact time-frequency-operator is studied. This decomposition and its eigenvalues are closely related to the theory of orthogonal polynomials. Results from both theories, the theory of orthogonal polynomials and the Landau-Pollak-Slepian theory, can be used to prove localization and approximation properties of the corresponding eigenfunctions. Finally, an uncertainty principle is proven that reflects the limitation of coupled time and frequency locatability.