  Let $f$ be a band-limited function in $L^2({\mathbb{R}})$. Fix $T >0$ and
suppose $f^{\prime}$ exists and is integrable on $[-T, T]$. This paper gives a
concrete estimate of the error incurred when approximating $f$ in the root mean
square by a partial sum of its Hermite series.
  Specifically, we show, for $K=2n, \quad n \in Z_+,$
  $$
  \left[\frac{1}{2T}\int_{-T}^T[f(t)-(S_Kf)(t)]^2dt\right]^{1/2}\leq
\left(1+\frac 1K\right)\left(\left[ \frac{1}{2T}\int_{|t|>
T}f(t)^2dt\right]^{1/2} +\left[\frac{1}{2T} \int_{|\omega|>N}|\hat
f(\omega)|^2d\omega\right]^{1/2} \right)
+\frac{1}{K}\left[\frac{1}{2T}\int_{|t|\leq T}f_N(t)^2dt\right]^{1/2}
+\frac{1}{\pi}\left(1+\frac{1}{2K}\right)S_a(K,T), $$ in which $S_Kf$ is the
$K$-th partial sum of the Hermite series of $f, \hat f $ is the Fourier
transform of $f$, $\displaystyle{N=\frac{\sqrt{2K+1}+% \sqrt{2K+3}}{2}}$ and
$f_N=(\hat f
\chi_{(-N,N)})^\vee(t)=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{\sin
(N(t-s))}{t-s}f(s)ds$. An explicit upper bound is obtained for $S_{a}(K,T)$.
