Directional short-time Fourier transform and directional regularity

TL;DR

This paper extends the theoretical framework of the directional short-time Fourier transform (DSTFT) to new function spaces, introduces multi-directional STFT, and explores directional regularity, wave fronts, and invariance properties.

Contribution

It provides new results on DSTFT in advanced function spaces, defines multi-directional STFT, and studies directional regularity and wave fronts for tempered distributions.

Findings

Extended DSTFT to $ ext{K}_1$ and tensor product spaces.

Defined multi-directional STFT and directional wave fronts.

Established invariance of regularity notions under mild window support conditions.

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 $\mathcal K_{1}({\mathbb R})\widehat{\otimes}\mathcal U(\mathbb C^n)$ 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.