A class of impulsive eigenfunctions of multidimensional Fourier transform

Abstract

The coordinate axes of ${\mathbb {R}}^{N}$ are arbitrarily partitioned into two sets; each set defines a hyperplane passing through the origin and these two hyperplanes are orthogonal. After a review of impulse functions over such hyperplanes and their Fourier transforms, it is shown that an impulse function over the union of these two hyperplanes is an eigenfunction of the $N$-dimensional Fourier transform. Furthermore, based on the simple rotation property of the Fourier transform, it is also shown that impulse functions over unions of finite number of arbitrarily rotated versions of those two hyperplane sets are also eigenfunctions of the $N$-dimensional Fourier transform.