On eigenfunctions of continuous and discrete (anti)symmetric multivariate Fourier transforms

Abstract

We derive various full sets of eigenfunctions of Fourier transforms (continuous and discrete sine and cosine Fourier transforms in one variable, discrete symmetric and antisymmetric multivariate exponentials, sine and cosine Fourier transforms, and continuous symmetric and antisymmetric multivariate sine and cosine Fourier transforms), associated with Hermite polynomials in one variable. Symmetric and antisymmetric multivariate exponential Fourier transforms F satisfy the relation F4=1, whereas symmetric and antisymmetric multivariate sine and cosine Fourier transforms F satisfy the relation F2=1.