Exponentially sparse representations of Fourier integral operators

TL;DR

This paper studies the sparsity of Fourier integral operators' Gabor-matrix representations, showing how regularity of phase and symbol affects decay rates, with implications for efficient computation.

Contribution

It demonstrates that regularity conditions of phase and symbol lead to sub-exponential or exponential decay in the matrix representation, extending understanding beyond pseudodifferential operators.

Findings

Gevrey regularity yields sub-exponential decay.

Analytic regularity results in exponential decay.

Ultra-analytic regularity does not produce super-exponential decay.

Abstract

We investigate the sparsity of the Gabor-matrix representation of Fourier integral operators with a phase having quadratic growth. It is known that such an infinite matrix is sparse and well organized, being in fact concentrated along the graph of the corresponding canonical transformation. Here we show that, if the phase and symbol have a regularity of Gevrey type of order $s>1$ or analytic ($s=1$), the above decay is in fact sub-exponential or exponential, respectively. We also show by a counterexample that ultra-analytic regularity ($s<1$) does not give super-exponential decay. This is in sharp contrast to the more favorable case of pseudodifferential operators, or even (generalized) metaplectic operators, which are treated as well.