Directional short-time Fourier transform of ultradistributions

TL;DR

This paper introduces and analyzes the k-directional short-time Fourier transform and its synthesis operator within Gelfand Shilov spaces, exploring their properties, duals, and applications to directional wave front analysis.

Contribution

It defines the k-directional short-time Fourier transform and its synthesis operator in Gelfand Shilov spaces, extending Fourier analysis tools to ultradistributions with directional considerations.

Findings

Established the properties of the k-directional short-time Fourier transform.

Analyzed the dual spaces and synthesis operators in Gelfand Shilov spaces.

Investigated directional wave front sets and their complements.

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 $\mathcal S^{\prime \alpha}_{\alpha}(\mathbb R^n)$.