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

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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 \((\mathcal {S}(\mathbb {R}^n))^*=\mathcal {S'}(\mathbb {R}^n)\).

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The first author thanks the University Grant Commission (UGC) for providing financial support. The second author thanks the Council of Scientific and Industrial Research (CSIR) for providing finacial support. We thank the referee for meticulously reading our manuscript and giving us several valuable suggestions which improved the clarity of the paper and the handling editor for the help during the editorial process.

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Correspondence to D. Venku Naidu.

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Bais, S.R., Mohan, P. & Venku Naidu, D. A characterization of translation and modulation invariant Hilbert space of tempered distributions. Arch. Math. 122, 429–436 (2024).

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