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

TL;DR

This paper provides a concrete estimate of the root mean square error when approximating a band-limited function in L^2(R) by a partial sum of its Hermite series, incorporating Fourier transform properties and explicit bounds.

Contribution

It introduces a new explicit error bound for Hermite series approximation of band-limited functions in L^2(R), considering local and tail behaviors.

Findings

Derived a concrete error estimate involving function and Fourier tail terms.

Provided an explicit upper bound for the error term S_a(K,T).

Applicable to functions with integrable derivatives on a finite interval.

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 $K=2n, \quad n \in Z_+,$ $$ \left[\frac{1}{2T}\int_{-T}^T[f(t)-(S_Kf)(t)]^2dt\right]^{1/2}\leq \left(1+\frac 1K\right)\left(\left[ \frac{1}{2T}\int_{|t|> T}f(t)^2dt\right]^{1/2} +\left[\frac{1}{2T} \int_{|\omega|>N}|\hat f(\omega)|^2d\omega\right]^{1/2} \right) +\frac{1}{K}\left[\frac{1}{2T}\int_{|t|\leq T}f_N(t)^2dt\right]^{1/2} +\frac{1}{\pi}\left(1+\frac{1}{2K}\right)S_a(K,T), $$ 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 $S_{a}(K,T)$.