  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.
