Generalizations of the Bargmann Transform

We present a new way of obtaining the Bargmann transform between L 2 (R n) and the Fock space F = F(C n) via a simple restriction principle applied to holo-morphic functions. This same principle also recovers the transform between functions on a compact Lie group and holomorphic functions on its complexii-cation studied by Gross, Hall, Hijab et al., see 1] and 2], and it gives in a similar way canonical intertwining operators between real and complex symmetric domains 6]. This idea of restriction was rst applied to the Weyl transform in 7]. 1 The Bargmann Transform In this lecture we give several generalizations of the classical Bargmann transform between the Schrr odinger and the Fock model of the metaplectic representation, based on a restriction principle for reproducing kernel Hilbert spaces introduced in 7] in 1 lecture by Bent rsted