Fourier characterizations of Pilipovic spaces

TL;DR

This paper characterizes Pilipovic spaces using Fourier transforms, establishing conditions involving fractional Fourier transforms that determine membership in these function spaces.

Contribution

It provides a new Fourier-analytic characterization of Pilipovic spaces through estimates involving fractional Fourier transforms.

Findings

Characterization of Pilipovic spaces via Fourier transforms.

Equivalence between space membership and fractional Fourier transform estimates.

Framework for analyzing Pilipovic spaces using harmonic analysis techniques.

Abstract

Let $f$ be a function or distribution on $\rr d$. We show that $f$ belongs to a certain Pilipovi{\'c} space, if and only if $f$ and suitable partial fractional Fourier transforms of $f$ satisfy certain types of estimates.