# Can any compactly supported continuous function be written as a linear combination of functions with small support

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 $G$.

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The key word you want is "partition of unity", which you can find explained in any differential topology textbook. The proof is a consequence of the standard partition of unity argument applied to a finite open cover of $supp(f)$ of the form $\{t_i+U\}$, which you get using compactness $supp(f)$. Also, the "abelian" hypothesis is unnecessary. – Lee Mosher Jun 20 '13 at 16:34
@Lee Mosher Thank you. – Nick S Jun 20 '13 at 16:40
What exactly do you need a reference for? There are many simple proofs of it (Stone-Weierstraß, partition of unity, approximate identity), but probably there is no place where it is stated exactly in this form. – The User Jun 20 '13 at 17:02
Sorry, I thought you were interested in an approximation (because you had mentioned Stone-Weierstraß). For precise results you will need partitions of unity. – The User Jun 20 '13 at 18:08
And you do not have to look into differential topology textbooks, it is a standrd result from general topology that a space admits partitions of unity if and only if Urysohn’s lemma holds if and only if the space is normal. Then you have to consider the Alexandrov compactification. – The User Jun 20 '13 at 18:18

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.