Summation of Gaussian shifts as Jacobi's third Theta function

Abstract: A proper choice of parameters of the Jacobi modular identity (Jacobi Imaginary transformation) implies that the summation of Gaussian shifts on infinity periodic grids can be represented as the Jacobi's third Theta function. As such, connection between summation of Gaussian shifts and the solution to a Schr\"{o}dinger equation is explicitly shown. A concise and controllable upper bound of the saturation error for approximating constant functions with summation of Gaussian shifts can be immediately obtained in terms of the underlying shape parameter of the Gaussian. This shed light on how to choose a shape parameter and provides further understanding on using Gaussians with increasingly flatness.