A convolution and product theorem for the fractional Fourier transform

The fractional Fourier transform (FRFT), which is a generalization of the Fourier transform, has many applications in several areas, including signal processing and optics, In two recent papers, Almeida and Mendlovic et al, derived 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, 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 FRFT that preserves the convolution theorem for the Fourier transform and is also easy to implement in the designing of filters