A convolution and product theorems for the N-dimensional fractional Fourier transform

Abstract

The $n$-dimensional fractional Fourier transform, which is a generalization of the one dimensional fractional Fourier transform, has many applications in several areas, including signal processing and optics. In a recent paper, derived n-dimensional fractional Fourier transforms of a product and of a convolution of two functions. Unfortunately, their convolution formulas do not generalize very nicely the classical result for the Fourier transform and Laplace transform, which states that the Fourier transform of the convolution of two functions is the product of their Fourier transforms. The purpose of this note is to introduce a new convolution structure for the n-dimensional fractional Fourier transform that preserves the convolution theorem for the one dimensional fractional Fourier transform.