Directional Stockwell transform of distributions

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 $\mathcal{S}_{0}'(\mathbb{R}^n)$.