Metaplectic operators with quasi-diagonal kernels
Gianluca Giacchi, Luigi Rodino
TL;DR
The paper addresses how metaplectic operators, whose kernels are not diagonal, can nonetheless exhibit quasi-diagonal behavior after Gaussian smoothing. By computing the kernels and their smoothed versions, the authors identify a localization manifold $oldsymbol{\Gamma_S}=igligl\{(x,D^{T}x): x otin R(C)^{ot}igriglig\}$ on which the smoothed kernels concentrate, and prove off-diagonal decay with respect to this manifold under several structural conditions on the symplectic blocks. The work establishes two practical sufficient conditions (invertible $C$ or $D=I$) and then treats the general case via a decomposition and Gaussian integration, showing quasi-diagonality holds in broad settings, though higher-dimensional counterexamples reveal the limits of necessity. The results connect to time-frequency analysis by linking smoothed metaplectic kernels to Wigner kernels and Gabor matrices, with implications for kernel-based representations and transforms in signal processing and quantum harmonic analysis.
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.