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Let $f$ be a band-limited function in $L^2({\mathbb{R}})$. Fix $T >0$ andsuppose $f^{\prime}$ exists and is integrable on $[-T, T]$. This paper gives aconcrete estimate of the error incurred when approximating $f$ in the root meansquare by a partial sum of its Hermite series.  Specifically, we show, for $K=2n, \quad n \in Z_+,$  $$  \left[\frac{1}{2T}\int_{-T}^T[f(t)-(S_Kf)(t)]^2dt\right]^{1/2}\leq\left(1+\frac 1K\right)\left(\left[ \frac{1}{2T}\int_{|t|>T}f(t)^2dt\right]^{1/2} +\left[\frac{1}{2T} \int_{|\omega|>N}|\hatf(\omega)|^2d\omega\right]^{1/2} \right)+\frac{1}{K}\left[\frac{1}{2T}\int_{|t|\leq T}f_N(t)^2dt\right]^{1/2}+\frac{1}{\pi}\left(1+\frac{1}{2K}\right)S_a(K,T), $$ in which $S_Kf$ is the$K$-th partial sum of the Hermite series of $f, \hat f $ is the Fouriertransform 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 $S_{a}(K,T)$.

An estimate of the root mean square error incurred when approximating an $f \in L^2({\mathbb{R}})$ by a partial sum of its Hermite series