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

Let \(\mathcal {S}(\mathbb {R}^n)\) $\mathcal{S}({\mathbb{R}}^n)$ be the Schwartz space and \(\mathcal {S'}(\mathbb {R}^n)\) ${\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ be the space of tempered distributions on \(\mathbb {R}^n\) ${\mathbb{R}}^n$ . In this article, we prove that if \(\mathcal {H} \subseteq \mathcal {S'}(\mathbb {R}^n)\) $\mathcal{H}\subseteq {\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ is a non-zero Hilbert space of tempered distributions which is translation and modulation invariant such that \(\begin{aligned} |(f,g)| \le C \Vert f\Vert _{\mathcal {H}} \end{aligned}\) $$\begin{array}{c}\hfill |(f,g)|\le {C\Vert f\Vert }_{\mathcal{H}}\end{array}$$ for some \(C>0\) $C>0$ and for all \(f\in \mathcal {H}\) $f\in \mathcal{H}$ , then \(\mathcal {H}=L^2(\mathbb {R}^n)\) $\mathcal{H}=L^2({\mathbb{R}}^n)$ , where \(g(x) = e^{-x^2}\) $g(x)=e^{-x^2}$ for all \(x\in \mathbb {R}^n\) $x\in {\mathbb{R}}^n$ and \((\cdot , \cdot )\) $(·,·)$ denotes the standard duality pairing between \(\mathcal {S'}(\mathbb {R}^n)\) ${\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ and \(\mathcal {S}(\mathbb {R}^n)\) $\mathcal{S}({\mathbb{R}}^n)$ with respect to which \((\mathcal {S}(\mathbb {R}^n))^*=\mathcal {S'}(\mathbb {R}^n)\) ${(\mathcal{S}({\mathbb{R}}^n))}^{\ast }={\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ .