You can show the derivative properties by the Leibniz's rule (notice that if .
Given a test function (C^infinity function with compact support), I have shown that we can write
Choose a test function on such that on a neighbourhood of . I have to show the following:
On this neighbourhood, any test function on can be written as
where is a test function on such that for .
I cannot see where this is coming from.
I have considered the function defined by , but I do not think this has the desired derivative properties.
Can anyone help?
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 (**)
for some mapping \mu from test functions to test functions.
If v is a distribution, then apparently I can define , 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.