Test functions as a taylor series with integral remainder
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?