Directional Short-Time Fourier Transform of Ultradistributions

Abstract

We define and analyze 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 \(\mathcal {S}^{\prime \alpha }_{\alpha }(\mathbb {R}^n)\).