A quaternionic analogue of the Segal-Bargmann transform

TL;DR

This paper introduces a quaternionic analogue of the Segal-Bargmann transform within the context of slice hyperholomorphic functions, providing explicit formulas, properties, and connections to quaternionic Fourier transforms.

Contribution

It presents a new quaternionic Segal-Bargmann transform, proves its independence from the slice, and derives its inverse and relation to quaternionic Fourier transforms.

Findings

Explicit expression of the inverse transform

Proof of independence from the slice

Connection to quaternionic Fourier transform

Abstract

The Bargmann-Fock space of slice hyperholomorphic functions is recently introduced by Alpay, Colombo, Sabadini and Salomon. In this paper, we reconsider this space and present a direct proof of its independence of the slice. We also introduce a quaternionic analogue of the classical Segal-Bargmann transform and discuss some of its basic properties. The explicit expression of its inverse is obtained and the connection to the left one-dimensional quaternionic Fourier transform is given.