Test functions as a taylor series with integral remainder

Hi All,

Given a test function $\displaystyle \phi$ (C^infinity function with compact support), I have shown that we can write $\displaystyle \phi = \sum_{j = 0}^{m - 1} \frac{x^j}{j!}\phi^{(j)}(0) + \frac{1}{(m - 1)!}\int_0^x (x - t)^{m - 1}\phi^{(m)}(t) \, \mathrm{d}t.$

Choose a test function $\displaystyle \phi_0$ on $\displaystyle \R$ such that $\displaystyle \phi_0(x) = 1$ on a neighbourhood of $\displaystyle x = 0$. I have to show the following:

On this neighbourhood, any test function $\displaystyle \phi$ on $\displaystyle \R$ can be written as $\displaystyle \phi = \phi_0(x)\sum_{j = 0}^{m - 1} \frac{x^j}{j!}\phi^{(j)}(0) + \chi(x)$

where $\displaystyle \chi(x)$ is a test function on $\displaystyle \R$ such that $\displaystyle \chi^{(j)}(0) = 0$ for $\displaystyle j = 1, \ldots m - 1$.

I cannot see where this is coming from.

I have considered the function defined by $\displaystyle \chi(x) = \phi(x) - \phi_0(x)\sum_{j = 0}^{m - 1} \frac{x^j}{j!}\phi^{(j)}(0)$, but I do not think this has the desired derivative properties.

Can anyone help?