Fractional Fourier transforms on $L^p$ and applications

TL;DR

This paper develops the $L^p$ theory for the fractional Fourier transform (FRFT) on the real line, addressing convergence, inversion, and regularity issues, and establishing multiplier and Littlewood-Paley results for $1 \\le p < 2$.

Contribution

It introduces a new $L^p$ framework for FRFT, overcoming convergence challenges and establishing key properties like inversion, regularity, and multiplier theorems.

Findings

Established FRFT properties for $L^1$ functions using a chirp operator.

Proved convergence of FRFT means via fractional Gauss and Abel means.

Derived $L^p$ multiplier results and a Littlewood-Paley theorem for FRFT.

Abstract

This paper is devoted to the $L^p(\mathbb R)$ theory of the fractional Fourier transform (FRFT) for $1\le p < 2$. In view of the special structure of the FRFT, we study FRFT properties of $L^1$ functions, via the introduction of a suitable chirp operator. However, in the $L^1(\mathbb{R})$ setting, problems of convergence arise even when basic manipulations of functions are performed. We overcome such issues and study the FRFT inversion problem via approximation by suitable means, such as the fractional Gauss and Abel means. We also obtain the regularity of fractional convolution and results on pointwise convergence of FRFT means. Finally we discuss $L^p$ multiplier results and a Littlewood-Paley theorem associated with FRFT.