Metaplectic operators with quasi-diagonal kernels

TL;DR

This paper investigates the quasi-diagonal properties of metaplectic operators' kernels, revealing conditions under which these operators exhibit quasi-diagonality, with implications for time-frequency analysis.

Contribution

It introduces a new framework for understanding the quasi-diagonality of metaplectic operators' kernels using Gaussian smoothing, expanding their analytical characterization.

Findings

Metaplectic kernels are not diagonal but are quasi-diagonal under certain conditions.

Quasi-diagonality is characterized via Gaussian convolution smoothing.

Applications to time-frequency analysis are discussed.

Abstract

Metaplectic operators form a relevant class of operators appearing in different applications, in the present work we study their Schwartz kernels. Namely, diagonality of a kernel is defined by imposing rapid off-diagonal decay conditions, and quasi-diagonality by imposing the same conditions on the smoothing of the kernel through convolution with the Gaussian. Kernels of metaplectic operators are not diagonal. Nevertheless, as we shall prove, they are quasi-diagonal under suitable conditions. Motivation for our study comes from problems in time-frequency analysis, that we discuss in the last section.