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Given an arbitrary compact set $K \subset \mathbb{R}^n$ and an open set $V \subset \mathbb{R}^n$, I want to construct a bump function $\zeta$ that is $1$ on $K$ and has its support in $V$, where I assume that $K \subset V$. Also it would be good to have $0 \le \zeta\le 1$ everywhere.

Could anybody give me a reference or explain to me how I can construct such a function?

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  • $\begingroup$ Evans PDEs and Lee introduction to smooth manifolds. $\endgroup$ – TKM Dec 12 '14 at 20:16
  • $\begingroup$ @TKM on which page in Evans book? -Cannot find it $\endgroup$ – user159356 Dec 12 '14 at 20:23
  • $\begingroup$ I think it's an exercise in Evans, Chapter 5. Though it may also be in the appendix, where he talks about convolutions. $\endgroup$ – Gyu Eun Lee Dec 12 '14 at 22:58
  • $\begingroup$ @TobiasHurth : I think you mean $K\subset V$ instead of the other way around? $\endgroup$ – DisintegratingByParts Dec 13 '14 at 21:26
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These bump functions, or test functions, are extremely important in distribution theory. They can be constructed using partitions of unity. I didn't find any good references online in this context, but if you can get hold of Hörmanders ``The Analysis of Linear Partial Differential Operators I'', then it's an excellent reference.

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  • $\begingroup$ well, the partition of unity is rather the existence proof, don't you think so? $\endgroup$ – user66906 Dec 12 '14 at 22:52
  • $\begingroup$ Doesn't the subsection "construction of bump functions" on the wiki page does exactly what the op asked for? $\endgroup$ – Viktor Glombik Apr 12 at 9:40

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