Do you have some reference to a proof of the so-called Borel theorem, i.e. every power series is the Taylor series of some C∞ function?

**Answer**

Borel’s theorem states that given a sequence of real numbers (an)n∈N there exists a C∞ function f∈C∞(R) such that

f(n)(0)n!=an , i.e. the Taylor series associated to f is ΣanXn.

The function f is never unique: you can always add to it a flat function, one all of whose derivatives at zero are zero, like the well-known Cauchy function e−1/x2 .

There is a huge *caveat* however: you can’t go from the series to the function f .

Firstly, the series might not be convergent at any x≠0∈R ! An example is ΣanXn=ΣnnXn whose radius of convergence is zero.

Secondly, even if it does converge it might converge to the wrong function! For example if you start with Cauchy’s function you get the zero Taylor series. It converges to zero, of course, but that is definitely not the Cauchy function you started with. So we should not read too much in Borel’s theorem: it cannot force a non-analytic function to become analytic!

Borel’s theorem is also valid in several variables. Given a sequence of k-tuples (aI)I∈Nk of real numbers aI∈R, there exists a function f∈C∞(Rk), again highly non-unique, whose derivatives satisfy

∂If(0)I!=aI. [I have used multiindex notation with I=(i1,…,ik), I!=i1!…ik!etc.]

There is a vast generalization due to Whitney of Borel’s theorem. You can consider a closed subset Z⊂Rk and continuous functions ϕI∈C(Z) . Whitney gives necessary and sufficient growth and compatibility conditions on the ϕI ‘s which will guarantee that there exists a C∞ function f∈C∞(U) defined on an open neighbourhood U⊃Z of Z such that ∂If(0)I!=ϕI. Borel’ s theorem is then the case Z={0} .

**Bibliography:** Borel’s theorem in several variables is proved in R.Narasimhan’s book *Analysis on Real and complex Manifolds*, which also contains the precise statement of Whitney’s theorem.

**Attribution***Source : Link , Question Author : Community , Answer Author : Georges Elencwajg*