# Test functions as a taylor series with integral remainder

• May 11th 2011, 10:52 AM
measureman
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?
• May 11th 2011, 11:35 AM
girdav
You can show the derivative properties by the Leibniz's rule (notice that $\displaystyle \phi_0^{(k)}(0)=0$ if $\displaystyle k\geq 1$.
• May 11th 2011, 11:43 AM
measureman
Quote:

Originally Posted by girdav
You can show the derivative properties by the Leibniz's rule (notice that $\displaystyle \phi_0^{(k)}(0)=0$ if $\displaystyle k\geq 1$.

Ah cheers, I tried this and thought I ended up with \chi'(x) = \phi'(x) - now I notice the term that cancels.
• May 13th 2011, 08:53 AM
measureman
I now have another problem.

I have shown that if \phi is a test function on R, with the first m - 1 derivatives at the origin being zero, then \phi = x^m\psi for some test function \psi. (*)

So it follows from my first post that $\displaystyle \phi(x) = \phi_0(x)\sum_{j = 0}^{m - 1} \frac{x^j}{j!}\phi^{(j)}(0) + x^m\mu(\phi(x))$ (**)
for some mapping \mu from test functions to test functions.

If v is a distribution, then apparently I can define $\displaystyle u(\phi) = v(\mu(\phi))$, and u is a distribution. Does anyone know how to prove this?

The hint in the book is to use (*) and (**), but I just can't see it.