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 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 is mapped to the phase space of wave functions, or Fock space, . 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 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 . We also lay out that the Segal–Bargmann transform can be expanded to a convergent series with a dictionary of . In other words, we analyse the signal f in one basis and reconstruct it in a basis of the Segal–Bargmann module.