Relaxed quaternionic Gabor expansions at critical density

TL;DR

This paper extends Gabor expansion theory into the quaternionic domain, demonstrating that signals can be uniquely represented at critical density with a relaxed lattice structure, using two exponential kernels.

Contribution

It introduces a quaternionic Gabor expansion framework at critical density, allowing unique signal representations with a relaxed lattice structure.

Findings

Unique quaternionic Gabor expansions at critical density.

Relaxed lattice structure with an additional point per cell.

Extension of classical Gabor theory to quaternionic signals.

Abstract

Shifted and modulated Gaussian functions play a vital role in the representation of signals. We extend the theory into a quaternionic setting, using two exponential kernels with two complex numbers. As a final result, we show that every continuous and quaternion-valued signal $f$ in the Wiener space can be expanded into a unique $\ell^2$ series on a lattice at critical density $1$, provided one more point is added in the middle of a cell. We call that a relaxed Gabor expansion.