The Segal-Bargmann Transform in Clifford Analysis

Abstract

The Segal-Bargmann transform plays an essential role in signal processing, quantum physics, infinite-dimensional analysis, function theory and further topics. The connection to signal processing is the short-time Fourier transform, which can be used to describe the Segal-Bargmann transform. The classical Segal-Bargmann transform $\mathcal{B}$ maps a square-integrable function to a holomorphic function square-integrable with respect to a Gaussian identity. In signal processing terms, a signal from the position space $L_2(\mathbb{R}^m,\mathbb{R})$ is mapped to the phase space of wave functions, or Fock space, $\mathcal{F}^2(\mathbb{C}^m,\mathbb{C})$. We extend the classical Segal-Bargmann transform to a space of Clifford algebra-valued functions. We show how the Segal-Bargmann transform is related to the short-time Fourier transform and use this connection to demonstrate that $\mathcal{B}$ is unitary up to a constant and maps Sommen's orthonormal Clifford Hermite functions $\left\{\phi_{l,k,j}\right\}$ to an orthonormal basis of the Segal-Bargmann module $\mathcal{F}^2(\mathbb{C}^m,\mathcal{C}\ell_m^{\mathbb{C}})$. We also lay out that the Segal-Bargmann transform can be expanded to a convergent series with a dictionary of $\mathcal{F}^2(\mathbb{C}^m,\mathcal{C}\ell_m^{\mathbb{C}})$. In other words, we analyse the signal $f$ on one basis and reconstruct it on a basis of the Segal-Bargmann module.