The fractional Fourier transform (FrFT), which is a generalization of the Fourier transform, has become the focus of many research papers in recent years because of its applications in electrical engineering and optics. It is known that the Hermite functions of order n are eigenfunctions of the Fourier transform with eigenvalues $(i{)}^n$ and likewise the Hermite functions of order n are eigenfunctions of the FrFT of order α, where $0\le \alpha \le 1$ but with different eigenvalues ${\lambda }_n.$ Those eigenvalues ${\lambda }_n$ approach $(i{)}^n$ as α approaches 1. The FrFT has been extended to n dimensions using tensor product of n copies of the one-dimensional transform. In this article we introduce a new two-dimensional FrFT that is not a tensor product of two one-dimensional transforms. The definition of the new transform is based on using the Hermite functions of two complex variables as eigenfunctions of the transform. We then derive some of its properties, such as its inversion formula, convolution structure and theorem, and its analogue of Poisson summation formula.

Many systems of orthogonal polynomials and functions are bases of a variety of function spaces, such as the Hermite and Laguerre functions which are orthogonal bases of $L^2(-\infty , \infty )$ and $L^2(0,\infty ),$ and the Jacobi polynomials which are an orthogonal basis of a weighted $L^2(-1,1).$ The associated Legendre functions, and more generally, the spheroidal wave functions are also an orthogonal basis of $L^2(-1,1).$ The prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property. They are an orthogonal basis of both $L^2(-1,1)$ and a subspace of $L^2(-\infty , \infty ),$ known as the Paley-Wiener space of bandlimited functions. They also satisfy a discrete orthogonality relation. No other system of classical orthogonal functions is known to possess this strange property. This raises the question of whether there are other systems possessing this property. The aim of the article is to answer this question in the affirmative by providing an algorithm to generate such systems and then demonstrating the algorithm by a new example.

We derive sampling formula for signals that are bandlimited to a disc of radius $R$ in the linear canonical transform (LCT) domain. By bandlimitedness in a disc $D$ in the LCT domain, we mean that the LCT $F(\omega)$ of a signal $f(t)$ vanishes outside the disc $D.$ We first express the signal in polar coordinates and then obtain the sampling formula. The samples of the angle $\theta$ are taken at $2N+1$ uniformly distributed points on the unit circle while the samples of the radial distance $r$ are taken at the zeros of the Bessel function. As a special case, we obtain sampling formula for signals that are bandlimited to a disc in the fractional Fourier transform domain.

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 ${\mathcal{L}}^2({\mathbb{R}}^2).$