Directional short-time Fourier transform and directional regularity

Abstract

We give some new results related to the directional short-time Fourier transform (DSTFT) and extend them on the spaces ${\mathcal{K}}_1({\mathbb{R}}^n)$ and ${\mathcal{K}}_1(\mathbb{R})\hat{\otimes }\mathcal{U}({\mathbb{C}}^n)$ and their duals. Then, we define multi-directional STFT and, for tempered distributions, directional regular sets and their complements, directional wave fronts. Different windows with mild conditions on their support show the invariance of these notions related to window functions. Smoothness of $f$ follows from the assumptions of the directional regularity in any direction.