Which functions are in $C_0^\infty$, except for the Bump function

Asked Nov 11 '15 at 07:42

Active Nov 11 '15 at 07:42

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We all know the standard example for a function in $C_0^\infty(\Omega)$, namely \begin{align} \Phi(x) = \begin{cases} e^{ -\frac{1}{1 - x^2}} & \mbox{ for } |x| < 1\\ 0 & \mbox{ otherwise.} \end{cases} \end{align} However, I have never heard of a different function in $C_0^\infty$. Of course, you can do a rescaling of the given function or introduce other minor changes (I assume without a proof, that you could replace the $x^2$ with $x^4$). But are there any types of functions in $C_0^\infty$. I would be very happy to hear some examples or references.

Thanks!

(Also, if you can come up with better tags, please feel free to edit.)

asked Nov 11 '15 at 07:42

Thomas

4,363

1

You can take convolution of any $f$ support in $B\subset\subset \Omega$ with a smooth bump function to get another bump function too. – Nov 11 '15 at 08:00
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Any smooth function multiplied by $\Phi$. – copper.hat Nov 11 '15 at 08:09
See the links I give in my answer at How many smooth functions are non-analytic? for a fairly thorough discussion and list of references. – Dave L. Renfro Nov 11 '15 at 15:11
Thanks for all the comments, I feel stupid for not realizing, that I could just combine $\phi$ with another function. If any of you would take the time to write your comment as an answer I will gladly accept it. Thanks! – Thomas Nov 12 '15 at 09:52

Which functions are in $C_0^\infty$, except for the Bump function

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