Generalizations of the fractional Fourier transform and their analytic properties

TL;DR

This paper introduces generalized quadratic-phase integral transforms extending the fractional Fourier transform, characterizes their group structure, and analyzes their dependence on parameters in various function spaces.

Contribution

It provides a comprehensive characterization of these generalized transforms and establishes conditions for their continuous dependence on parameters.

Findings

Characterization of one-parameter groups of quadratic-phase integral transforms.

Necessary and sufficient conditions for continuous dependence on parameters.

Analysis of these transforms in L2, pointwise, and almost-everywhere senses.

Abstract

We consider one-parameter families of quadratic-phase integral transforms which generalize the fractional Fourier transform. Under suitable regularity assumptions, we characterize the one-parameter groups formed by such transforms. Necessary and sufficient conditions for continuous dependence on the parameter are obtained in L2, pointwise, and almost-everywhere senses.