Can any smooth function be written in this form?

Question

Can any smooth function $F: \mathbb{R}^n \to \mathbb{R}$ be written in the form$$F(x) = F(a) + \sum_{\mu = 1}^n (x^\mu - a^\mu)H_\mu(x),$$where $a = (a^1, \dots, a^n) \in \mathbb{R}^n$ and the $H_\mu$ are $C^\infty$ functions?

Answer 1

This is an application of the fundamental theorem of calculus. By translation, let us assume $a=0$. Then $$ f (x) - f (0) = \int_0^1 \frac {d }{dt}\bigg|_{t=s} f (tx) \,ds =\int_0^1 \sum_{i=1}^n \frac {\partial f}{\partial x_i}(s x) x_i \, ds. $$ Now, interchange summation and integration and use differentiation under the integral (+ induction) to see that each of the functions $$ x \mapsto \int_0^1 \frac {\partial f}{\partial x_i} (sx) \, ds $$ is $C^\infty $.