A characterization of translation and modulation invariant Hilbert space of tempered distributions

Abstract

Let $\mathcal {S}(\mathbb {R}^n)$ be the Schwartz space and $\mathcal {S'}(\mathbb {R}^n)$ be the space of tempered distributions on $\mathbb {R}^n$. In this article, we prove that if $\mathcal {H} \subseteq \mathcal {S'}(\mathbb {R}^n)$ is a non-zero Hilbert space of tempered distributions which is translation and modulation invariant such that

$$\begin{aligned} |(f,g)| \le C \Vert f\Vert _{\mathcal {H}} \end{aligned}$$

for some $C>0$ and for all $f\in \mathcal {H}$, then $\mathcal {H}=L^2(\mathbb {R}^n)$, where $g(x) = e^{-x^2}$ for all $x\in \mathbb {R}^n$ and $(\cdot , \cdot )$ denotes the standard duality pairing between $\mathcal {S'}(\mathbb {R}^n)$ and $\mathcal {S}(\mathbb {R}^n)$ with respect to which $(\mathcal {S}(\mathbb {R}^n))^*=\mathcal {S'}(\mathbb {R}^n)$.