Directional short-time Fourier transform of ultradistributions

Abstract

We define and analyse the $k$-directional short-time Fourier transform and its synthesis operator over Gelfand Shilov spaces ${\mathcal{S}}_{\beta }^{\alpha }({\mathbb{R}}^n)$ and ${\mathcal{S}}_{\beta }^{\alpha }({\mathbb{R}}^{k+n})$ respectively, and their duals. Also, we investigate directional regular sets and their complements - directional wave fronts, for elements of ${\mathcal{S}}_{\alpha }^{\prime \alpha }({\mathbb{R}}^n)$.