On the Extension and Sampling Theorem for the Coupled Fractional Fourier Transform

The fractional Fourier transform, denoted by F
θ
, which is a generalization of the Fourier transform, depends on a parameter 0 ≤ θ ≤ π/2, so that when θ = 0, F
0
is the identity transformation and when θ = π/2, F
π/2
is the standard Fourier transform. The transform has been extended to higher dimensions by taking tensor products of one-dimensional transforms.In 2018 the author of this article introduced a novel generalization of the fractional Fourier transform to two dimensions, which is called the coupled fractional Fourier transform and is denoted by F
α,β
. This transform depends on two independent angles α and β, with 0 ≤ α, β ≤ π/2, so that F
0,0
is the identity transformation and F
π/2,π/2
, is the two-dimensional Fourier transform. For other values of α and β, we obtain other interesting configurations of the transform. One immediate application of this transform is in time-frequency representation because of its close relationship to the Wigner distribution function.The goal of this article is to extend the transform to a space of generalized functions and then introduce a sampling theorem for signals that are bandlimited in the domain of the transform.