Exact and approximate expansions with pure Gaussian wavepackets

TL;DR

This paper develops frames of Gaussian wavepackets using parabolic dilation, rotation, and translation, providing asymptotic analysis and demonstrating rapid decay of coefficients away from the data's wavefront set.

Contribution

It introduces a new construction of Gaussian wavepacket frames with asymptotic properties and analyzes coefficient decay relative to the wavefront set.

Findings

Frames constructed with Gaussian wavepackets satisfy Daubechies-like frame criteria asymptotically.

Coefficients decay rapidly away from the wavefront set of the data.

The method provides a precise asymptotic understanding of the frame coefficients.

Abstract

We construct frames of wavepackets produced by parabolic dilation, rotation and translation of (a finite sum of) Gaussians and give asymptotics on the analogue of Daubechies frame criterion. We show that the coefficients in the corresponding approximate expansion decay fast away from the wavefront set of the original data.