The characterization of the image for the Weinstein heat transform

Based on reproducing kernel Hilbert space theory, this paper describes a result characterizing the image time-t heat transform associated with the Weinstein-Laplacian operator on half-space $R_{+}^{d+1}:=R^d\times R_{+}^{⁎}$. The guiding principle behind characterization is this: the functions in the image of the heat transform should be those functions having an analytic continuation to the complex space $C^{d+1}$, that are even in the last variable and that possess an appropriate $L^2$-finite norm. Furthermore, we prove that the associated Bargmann transform with this heat transform is an isometric isomorphism (we referred to here as the Weinstein-Bargmann transform) for which we compute the inverse in an explicit form.