A Paley-Wiener theorem for the Mehler-Fock transform

Montes-Rodríguez, A. and Virtanen, J. (2025) A Paley-Wiener theorem for the Mehler-Fock transform. Computational Methods and Function Theory, 25 (1). pp. 119-127. ISSN 1617-9447 doi: 10.1007/s40315-024-00537-4

Abstract/Summary

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+) onto L2(R+, (2π )−1t sinh(πt) 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.