Likewise theta functions of rank $r$ on $\mathbb{R}^d$: analytic properties and associated Segal-Bargmann transform

TL;DR

This paper introduces a new Hilbert space of theta functions on , characterizes its structure, and explores its image under the Segal-Bargmann transform, resulting in the theta-Bargmann Fock space.

Contribution

It defines and analyzes a novel space of theta functions on , providing explicit descriptions, orthonormal bases, and connections to the Segal-Bargmann transform.

Findings

Constructed an explicit orthonormal basis for the theta function space.

Characterized the image of the space under the Segal-Bargmann transform.

Established the structure of the theta-Bargmann Fock space.

Abstract

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.