  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 decayness 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. L.
Qian (Proc. Amer. Math. Soc., 2003) established the convergence rate of
$O(\sqrt{n}\exp(-\frac{\pi-\delta}2n))$ for this method, where $\delta<\pi$ is
the bandwidth and $n$ is the number of sample data. C. Micchelli {\it et al.}
(J. Complexity, 2009) proposed a different regularized method and obtained the
corresponding convergence rate of
$O(\frac1{\sqrt{n}}\exp(-\frac{\pi-\delta}2n))$. This latter rate is by far the
best among all regularized methods for the Shannon series. However, their
regularized method involves the solving of a linear system and is implicit and
more complicated. The main objective of this note is to show that the Gaussian
regularization of the Shannon series can also achieve the same best convergence
rate as that by C. Micchelli {\it et al}. We also show that the Gaussian
regularization method can improve the convergence rate for the useful average
sampling. Finally, the outstanding performance of numerical experiments
justifies our results.
