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 $S_{ a}(K,T)$.
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