# Prove there exists a infinitely differentiable function whose value of partial derivatives of all orders at $0$ is a given function

Let $n \geq 1$ be an integer, and $C: \mathbb{N}^n \to \mathbb{R}$ be a function. Prove that there exits a infinitely differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ whose value and partial derivatives of all orders at $0 \in R^n$ satisfy $$\left( {\partial^{|\beta|}f \over{\partial x_{1}^{\beta_{1}}\partial x_{2}^{{\beta }_{2}}}...\partial x_{n}^{{\beta}_{n}}} \right) (0)=C(\beta),$$for all $\beta\in \mathbb{N}^n.$

I know a borel theorem in one-variable case,but I am not sure whether they have some correlations. Can someone help me prove thia question?

• Surely this function $C$ should be symmetric at least? – Pedro Apr 8 '15 at 2:10
• I think so,even though the problem didn't say this point – python3 Apr 8 '15 at 2:12
• C does not have to be symmetrical – python3 Apr 8 '15 at 3:37
• math.stackexchange.com/a/63062/3217 – Georges Elencwajg Apr 8 '15 at 20:24