On $L^2$ -functions with bounded spectrum

TL;DR

This paper characterizes the class of continuous maps that preserve the Paley-Wiener space of functions with bounded spectrum, showing that only injective affine maps maintain this property when composed with functions in this space.

Contribution

The paper provides a complete description of maps that preserve the bounded spectrum property in $L^2$-functions, identifying injective affine maps as the only such transformations.

Findings

Only injective affine maps preserve the $PW$ space under composition.

Characterization of maps maintaining bounded spectral support.

Insight into structure of spectral-preserving transformations.

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 $\varphi : \mathbb R^m\rightarrow\mathbb R^n$ such that $f\circ\varphi\in PW(\mathbb R^m)$ for every function $f\in PW(\mathbb R^n)$. Only injective affine maps $\varphi$ have this property.