Action on Powers and Polynomials

Let us determine the action of the filter on a function which is a simple power on the real line. If $f(x)=x^{n}$ then we have

$$\begin{eqnarray*} f_{\varepsilon}(x) & = & \frac{1}{2\varepsilon} \int_{x-\v... ...^{j_{M}} \frac{(n+1)!}{k!(n+1-k)!}\, x^{n+1-k}\varepsilon^{k}, \end{eqnarray*}$$

where $k=2j+1$ and $j_{M}=n/2$ if $n$ is even, while $j_{M}=(n-1)/2$ if $n$ is odd. We have therefore

$$\begin{eqnarray*} f_{\varepsilon}(x) & = & \sum_{j=0}^{j_{M}} \frac{n!\varep... ...ac{n(n-1)(n-2)(n-3)\varepsilon^{4}}{5!}\, x^{n-4} + \ldots\;. \end{eqnarray*}$$

We see therefore that the filter preserves the original power, and that all other terms generated are of lower order and are damped by factors of $\varepsilon^{2}$. It follows that the filter will reproduce any order-$n$ polynomial, adding to it a lower-order polynomial, of order $n-2$, with all coefficients damped by powers of $\varepsilon^{2}$. Therefore, in the $\varepsilon\to 0$ limit the filter reduces to the identity, in so far as polynomials are concerned.