Some unexpected behaviors of operators with small time-frequency dispersion

TL;DR

This paper investigates pseudo-differential operators with small time-frequency dispersion and reveals unexpected behaviors that challenge common assumptions about their output being close to a scalar multiple of the input.

Contribution

It demonstrates that operators with small time-frequency dispersion can exhibit surprising behaviors, contradicting the usual heuristic assumptions.

Findings

Operators with small dispersion can behave unexpectedly

The output of such operators may significantly differ from scalar multiples

Challenges existing assumptions in time-frequency analysis

Abstract

We study the approximation properties of pseudo-differential operators with small time-frequency dispersion, meaning that their spreading functions are supported in a small neighborhood of the origin. It is commonly assumed that for such operators $H$, the output $Hf$ can differ only a little from a scalar multiple of the input $f$. However, we disprove this heuristic statement, hence revealing some unexpected behaviors of such operators.