  We consider the reconstruction of a bandlimited function from its finite
localized sample data. Truncating the classical Shannon sampling series results
in an unsatisfactory convergence rate due to the slow decay of the sinc
function. To overcome this drawback, a simple and highly effective method,
called the Gaussian regularization of the Shannon series, was proposed in the
engineering and has received remarkable attention. It works by multiplying the
sinc function in the Shannon series with a regularized Gaussian function.
Recently, it was proved that the upper error bound of this method can achieve a
convergence rate of the order
$O(\frac{1}{\sqrt{n}}\exp(-\frac{\pi-\delta}{2}n))$, where $0<\delta<\pi$ is
the bandwidth and $n$ the number of sample data. The convergence rate is by far
the best convergence rate among all regularized methods for the Shannon
sampling series. The main objective of this article is to present the
theoretical justification and numerical verification that the convergence rate
is optimal when $0<\delta<\pi/2$ by estimating the lower error bound of the
truncated Gaussian regularized Shannon sampling series.
