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Are there functions in $C_0^\infty(\mathbb R)$ which are not related to $f(x)=\begin{cases}e^{-1/x}&x>0\newline 0&x\leq 0\end{cases}$ by translation, dilation, or composition?

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replace $e^x$ with something else that grows very quickly –  yoyo Mar 31 '11 at 20:21

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You can convolute a characteristic function with a mollifier (however, a mollifier can ultimately be some form of an exponential function).

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I was thinking along the same lines: you can multiply any function you like by a $\psi_i$ from a partition of unity w.r.t. an open cover of your choice, but then again, the existence of the latter is often proved using the much-loved exponential function. –  Gerben Mar 31 '11 at 22:23

For example, $f(x)=\dfrac{1}{2\pi}\int\limits_{-\infty}^{\infty}e^{-itx}\prod\limits_{k=1}^{\infty}\dfrac{\sin{(t\cdot2^{-k})}}{t\cdot2^{-k}}\,dt.$ It is co-called $\textbf{atomic function}$ which was introduced both by V.L. Rvachev, V.A. Rvachev, and W. Hilberg (independently) in 1971.

$\textbf{Link:}$ http://demonstrations.wolfram.com/ApproximateSolutionsOfAFunctionalDifferentialEquation/

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