Fourier-like multipliers and applications for integral operators

TL;DR

This paper explores the properties of signals that are nearly time and bandlimited, demonstrating their inclusion within uncertainty principles and analyzing associated integral operators and wavelet multipliers.

Contribution

It introduces wavelet multipliers for integral operators with bounded kernels, studying their boundedness, Schatten class properties, and their relation to phase space restriction operators.

Findings

Almost time and bandlimited signals can be approximated by eigenfunctions of phase space operators.

Wavelet multipliers are unitarily equivalent to phase space restriction operators.

The paper extends uncertainty principles to nearly time and bandlimited signals.

Abstract

Timelimited functions and bandlimited functions play a fundamental role in signal and image processing. But by the uncertainty principles, a signal cannot be simultaneously time and bandlimited. A natural assumption is thus that a signal is almost time and almost bandlimited. The aim of this paper is to prove that the set of almost time and almost bandlimited signals is not excluded from the uncertainty principles. The transforms under consideration are integral operators with bounded kernels for which there is a Parseval Theorem. Then we define the wavelet multipliers for this class of operators, and study their boundedness and Schatten class properties. We show that the wavelet multiplier is unitary equivalent to a scalar multiple of the phase space restriction operator. Moreover we prove that a signal which is almost time and almost bandlimited can be approximated by its projection on the span of the first eigenfunctions of the phase space restriction operator, corresponding to the largest eigenvalues which are close to one.