Non-isometric translation and modulation invariant Hilbert spaces

TL;DR

This paper proves that certain Hilbert spaces of distributions that are invariant under non-isometric translation and modulation must be equivalent to the standard L^2 space, highlighting a rigidity property of such invariance.

Contribution

It establishes a uniqueness result showing that invariance under non-isometric translation and modulation forces the space to be L^2, revealing a fundamental limitation of such invariance properties.

Findings

Invariance implies the space is L^2.

The space's norm is equivalent to the L^2 norm.

Non-isometric invariance characterizes L^2 uniquely.

Abstract

Let $\mathcal H$ be a Hilbert space of distributions on $\mathbf R^d$ which contains at least one non-zero element in $\mathscr D '(\mathbf R^d)$. If there is a constant $C_0>0$ such that $$ \nm {e^{i\scal \cdo \xi}f(\cdo -x)}{\mathcal H}\le C_0\nm f{\mathcal H}, \qquad f\in \mathcal H ,\ x,\xi \in \mathbf R^d, $$ then we prove that $\maclH = L^2(\mathbf R^d)$, with equivalent norms.