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Suppose I have two functions that are Schwartz class, say $f,g \in S(\mathbb{R})$, and suppose I have another function $\psi(x)$ such that \begin{equation} g(x) = \psi(x)f(x) \end{equation} I would like to find a way to understand why this means that $\psi$ must be differentiable but I struggle to find a start as I am very new to Schwartz class functions.

I thought maybe smoothness is enough to begin with. In any case, away from points where $f(x) \neq 0$ I have no problems, but somehow I need to find a way to say something about the differentiability fo $\psi$ in general, provided $f,g$ are Schwartz (and not both identically zero).

Any help would be great !

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    ψ(x) must not only be differantiable but also mustn't grow faster than polynoms. This question might help you a little. http://math.stackexchange.com/questions/104565/if-fx-hxgx-is-h-differentiable-if-f-and-g-are2012-02-01

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Even if $f$ and $g$ are not identically $0$, $\psi$ doesn't need to be differentiable. Indeed, take $f(x)=g(x)=$ a test function, whose support is contained in $[0,1]$ for example. Theses functions are in the Schwartz class, and the equality $g=\psi f$ is satisfied for each $\psi$ such that $\psi(x)=1$ for $x\in [0,1]$. But of course, out of this interval, $\psi$ is allowed to be anything, in particular not even continuous.

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    @superM Maybe I misunderstand something but in this case $\psi$ would be the quotient of two smooth non-vanishing functions, so necessary differentiable.2012-02-01