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 H²( C⁺) H 2 ( C + ) onto L²( R⁺,( 2 π )⁻¹ t (π t) \, dt ) L 2 ( R + , ( 2 π ) - 1 t sinh ( π t ) d t ) . 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.