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

Abstract

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 $K=2n,n\in Z_{+},$ $${\left[\frac{1}{2T}{\int }_{-T}^T[f(t)-(S_{K}f)(t){]}^{2}dt\right]}^{1/2}\le \left(1+\frac{1}{K}\right)\left({\left[\frac{1}{2T}{\int }_{|t|>T}f(t{)}^{2}dt\right]}^{1/2}+{\left[\frac{1}{2T}{\int }_{|\omega |>N}|\hat{f}(\omega ){|}^{2}d\omega \right]}^{1/2}\right)+\frac{1}{K}{\left[\frac{1}{2T}{\int }_{|t|\le T}{f}_N(t{)}^{2}dt\right]}^{1/2}+\frac{1}{\pi }\left(1+\frac{1}{2K}\right)S_a(K,T),$$ in which $S_{K}f$ 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 $S_a(K,T)$.