The fractional Fourier transform (FRFT), which is a generalization of the Fourier transform, has ubiquitous applications in optics and signal processing. The FRFT depends on an angle $, 0\leq \alpha \leq 2\pi,$ and reduces to the standard Fourier transform when $\alpha =\pi/2.$ The coupled fractional Fourier transform is a two-dimensional version of the fractional Fourier transform that depends on two angles $\alpha$ and $\beta$ and is not a tensor product of two one-dimensional fractional Fourier transforms. The two angles are coupled in such a way that the transform depends on the sum $\alpha + \beta$ and the difference $\alpha-\beta$ of the two angles. This transform has interesting applications, such as explaining the rotations of the Wigner distribution in four dimensions.

In this talk we extend the coupled fractional Fourier transform to Schwartz-like spaces and to the space of tempered distributions. We then investigate properties of pseudo-differential operators associated with the coupled fractional Fourier transform on a Schwartz-like space. We conclude the talk by applying the results to obtain solution of a generalized heat equation.