The Segal-Bargmann Transform in Clifford Analysis

2021/06/18 by Bernstein, Swanhild, Schufmann, Sandra · 1 citation
#30G35 #42B10 #42C05 #44A15 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.09956

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 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 L₂(ℝ^m,ℝ) is mapped to the phase space of wave functions, or Fock space, F²(ℂ^m,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 B is unitary up to a constant and maps Sommen's orthonormal Clifford Hermite functions \ϕ_l,k,j\ to an orthonormal basis of the Segal-Bargmann module F²(ℂ^m,Cℓ_m^ℂ). We also lay out that the Segal-Bargmann transform can be expanded to a convergent series with a dictionary of F²(ℂ^m,Cℓ_m^ℂ). In other words, we analyse the signal f on one basis and reconstruct it on a basis of the Segal-Bargmann module.

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