The directional short-time fractional Fourier transform of distributions

TL;DR

This paper introduces the directional short-time fractional Fourier transform (DSTFRFT), establishes its fundamental properties, and develops a distributional framework for it, extending classical Fourier analysis tools to a new directional and fractional setting.

Contribution

It presents the DSTFRFT, proves key identities and continuity properties, and constructs a distributional framework, advancing the mathematical understanding of fractional Fourier analysis in multiple directions.

Findings

Proved an extended Parseval's identity for DSTFRFT.

Established a reconstruction formula for DSTFRFT.

Developed a distributional framework for DSTFRFT on tempered distributions.

Abstract

We introduce the directional short-time fractional Fourier transform (DSTFRFT) and prove an extended Parseval's identity and a reconstruction formula for it. We also investigate the continuity of both the directional short-time fractional Fourier transform and its synthesis operator on the appropriate space of test functions. Using the obtained continuity results, we develop a distributional framework for the DSTFRFT on the space of tempered distributions $\mathcal{S}'(\mathbb{R}^n)$. We end the article with a desingularization formula.