We introduce and study the Hilbert space of $(L^2,\Gamma,\chi)$-likewise theta functions on $\mathbb{ R}^d$ with respect to a given discrete subgroup $\Gamma$ of arbitrary rank and a character $\chi$ of $\Gamma$. A concrete description is given and an orthonormal basis is then constructed. Its range by the classical Segal-Bargmann transform is also characterized and leads to the so-called theta-Bargmann Fock space.
Categories: math.CV