Directional Stockwell transform of distributions

TL;DR

This paper introduces the directional Stockwell transform, combining features of the short-time Fourier and ridgelet transforms, with theoretical results on its properties and a distributional framework for distributions.

Contribution

It presents the first comprehensive study of the directional Stockwell transform, including identities, continuity results, and a distributional framework.

Findings

Proved an extended Parseval identity.

Derived a reconstruction formula.

Established continuity on test function spaces.

Abstract

We introduce and study the directional Stockwell transform as a hybrid of the directional short-time Fourier transform and the ridgelet transform. We prove an extended Parseval identity and a reconstruction formula for this transform, as well as results for the continuity of both the directional Stockwell transform and its synthesis transform on the appropriate space of test functions. Additionally, we develop a distributional framework for the directional Stockwell transform on the Lizorkin space of distributions $\mathcal{S}_{0}'(\mathbb{R}^n)$.