In this paper, we prove inversion theorems and Parseval identity for the multidimensional fractional Fourier transform. Analogous to the existing fractional convolutions on functions of single variable, we also introduce a generalized fractional convolution on functions of several variables and we derive their properties including convolution theorem and product theorem for the multidimensional fractional Fourier transform.

In this article, we extend the Stockwell transform to the space of all square integrable quaternion-valued functions on the real line, using the convolution of quaternion-valued functions. We prove that the extended Stockwel transform satisfies the Parseval's formula, inversion formula, and uncertainty principle. We also characterize the range of the Stockwell transform on $L^2(\mathbb{R},\mathbb{H})$ and prove a convolution theorem for the extended Stockwell transform. Applying the convolution theorem, we extend the transform to a suitable Boehmian space of quaternion-valued functions.

The coupled fractional Fourier transform is a two-dimensional fractional Fourier transform ${\mathcal{F}}_{\alpha ,\beta }$ that depends on two angles $\alpha ,\beta$ that are coupled in such a way that the transform parameters are $\gamma =(\alpha +\beta )/2$ and $\eta =(\alpha -\beta )/2.$ It generalizes the two-dimensional Fourier transform and it serves as useful tool in some applications in optics and signal processing. In this article we derive new properties of the transform, such as its additive property. We then extend some of them to ${\mathcal{L}}^2({\mathbb{R}}^2)$ and show that the transform is a unitary operator on ${\mathcal{L}}^2({\mathbb{R}}^2).$