Composition of Segal–Bargmann transforms

Abstract

We introduce and discuss some basic properties of some integral transforms in the framework of specific functional Hilbert spaces, the holomorphic Bargmann-Fock spaces on C and C 2 and the sliced hyperholomorphic Bargmann-Fock space on H. The first one is a natural integral transform mapping isometrically the standard Hilbert space on the real line into the two-dimensional Bargmann-Fock space. It is obtained as composition of the one and two dimensional Segal-Bargmann transforms and reduces further to an extremely integral operator that looks like a composition operator of the one-dimensional Segal-Bargmann transform with a specific symbol. We study its basic properties, including the identification of its image and the determination of a like-left inverse defined on the whole two-dimensional Bargmann-Fock space. We also examine their combination with the Fourier transform which lead to special integral transforms connecting the two-dimensional Bargmann-Fock space and its analogue on the complex plane. We also investigate the relationship between special subspaces of the twodimensional Bargmann-Fock space and the slice-hyperholomorphic one on the quaternions by introducing appropriate integral transforms. We identify their image and their action on the reproducing kernel.2010 Mathematics Subject Classification. Primary 44A15; Secondary 32A17, 32A10.

Bivariate poly-analytic Hermite polynomials

“…readily follows by rewriting H m,n,m ,n in the form , for fixed n, j, k, we claim that A 2 n,j,k (C 2 ) is a Hilbert subspace of H(C 2 z,w ). The special case A 2 0,0,0 (C 2 ) is realized as A 2 0,0,0 (C 2 ) = ker(A ξ ) ∩ ker(A ξ * ) ∩ ker(A ξ ) ∩ H(C 2 z,w ) and therefore is contained in the two-dimensional Bargmann-Fock space F 2 (C 2 ) of L 2 -holomorphic functions on C 2 and coincides with the phase space obtained as the composition of 1d-and 2d-Segal-Bargmann transforms (see [6]). The generalization of the considered transform, to the context of the phase spaces A 2 n,j,k (C 2 ), can be constructed as a coherent state transform from L 2 (R, C) onto A 2 n,j,k (C 2 ).…”