Does anyone have a reference for the following result? I am pretty sure that it is true, and should not be hard to prove, but i would surprise me if it is not already proven in many places:



> 
> Let $G$ be a locally compact Abelian group and $U$ an open precompact set in $G$. Then for all $f \in C\_C(G)$ we can find $n$ and $f\_1,..,f\_n \in C\_C(G)$ so that 
>  $$f=f\_1+..+f\_n (\*)$$
>  and for all $i$ we have ${\rm supp}(f\_i) \subset t\_i+U$ for some $t\_i \in G$.
> 
> 
> 


Here $C\_C(G)$ denotes the space of compactly supported continuous functions on $G$.



Can any compactly supported continuous function be written as a linear combination of functions with small support
Regarding partitions of unity: Every locally compact group is paracompact, thus it is normal, thus there exist partitions of unity. But you do not need this argument: For general locally compact groups and any compact subset $A$ and any finite precompact open cover of $A$ there exists a continuous function with the value $1$ on $A$ and vanishing outside the union of the cover which can be written as the sum of continuous functions supported on the elements of the cover (Folland, *Real Analysis*, page 134). You use the cover described in the comment by Lee and then you take these functions given by the theorem and multiply them with the function $f$ (the proof uses the Alexandrov compactification which is compact, thus it is normal).


First I thought you were only interested in an approximation (because you had mentioned Stone Weierstraß), for an approximation you can use this proof:
If $\mathcal{U}$ is a local basis at identity consisting of compact sets and $(f\_U)\_{U\in \mathcal{U}}$ is any family of continuous, non-negative, symmetric functions such that $\int\_G f\_U=1$ and $\mathrm{supp}(f\_U)\subset U$ for all $U\in \mathcal{U}$, then $(f\_U)\_{U\in \mathcal{U}}$ is an *approximate identity* for the convolution algebra of compactly supported continuous functions (reference: Gerald B. Folland, *A Course in Abstract Harmonic Analysis*, proposition 2.42). Since convolutions can be approximated by finite linear combinations of left-translates of the functions, we get your result.


