On $L^2$ -functions with bounded spectrum

Abstract

We consider the class $PW({\mathbb{R}}^n)$ of functions in $L^2({\mathbb{R}}^n)$, whose Fourier transform has bounded support. We obtain a description of continuous maps $\phi :{\mathbb{R}}^m\to {\mathbb{R}}^n$ such that $f\circ \phi \in PW({\mathbb{R}}^m)$ for every function $f\in PW({\mathbb{R}}^n)$. Only injective affine maps $\phi$ have this property.