Maximally concentrated signals in the special affine fourier transformation domain

The problem of maximizing the energy of a signal bandlimited to E₁ = [-σ, σ] on an interval T₁ = [-τ, τ] in the time domain, which is called the energy concentration problem, was solved by a group of mathematicians, D. Slepian, H. Landau, and H. Pollak, at Bell Labs in the 1960s. The goal of this article is to solve the energy concentration problem for the n-dimensional special affine Fourier transformation which includes the Fourier transform, the fractional Fourier transform, the Lorentz transform, the Fresnel transform, and the linear canonical transform (LCT) as special cases. The solution in dimensions higher than one is more challenging because the solution depends on the geometry of the two sets E₁ and T₁. We outline the solution in the cases where E₁ and T₁ are n dimensional hyper-rectangles and discs.