A Paley–Wiener Theorem for the Mehler–Fock Transform

In this note, we prove a Paley–Wiener Theorem for the Mehler–Fock transform. In particular, we show that it induces an isometric isomorphism from the Hardy space \(\mathcal H^2(\mathbb C^+)\) ${\mathcal{H}}^2({\mathbb{C}}^{+})$ onto \(L^2(\mathbb R^+,( 2 \pi )^{-1} t \sinh (\pi t) \, dt ) \) $L^2({\mathbb{R}}^{+},{(2\pi )}^{-1}tsinh(\pi t)dt)$ . The proof we provide here is very simple and is based on an old idea that seems to be due to G. R. Hardy. As a consequence of this Paley–Wiener theorem we also prove a Parseval’s theorem. In the course of the proof, we find a formula for the Mehler–Fock transform of some particular functions.