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 $\mathcal S^\alpha_{\beta}(\mathbb R^n)$ and $\mathcal S^\alpha_{\beta}(\mathbb R^{k+n})$ respectively, and their duals. Also, we investigate directional regular sets and their complements - directional wave fronts, for elements of $\mathcal S^{\prime \alpha}_{\alpha}(\mathbb R^n)$.