TITLE: Kolmogorov superposition for smooth functions
QUESTION [16 upvotes]: Kolmogorov superposition theorem states that a continuous function $f(x_1,\ldots,x_n)$ can be written as 
$$f(x_1,\ldots,x_n)=\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^{n}\phi_{q,p}(x_p)\right)$$
for certain continuous functions $\Phi_q$ and $\phi_{q,p}$.
For smooth, $f\in C^{\infty}$, can we obtain such a representation with the $\Phi_q$ and $\phi_{q,p}$ smooth too? Moreover, I am particularly interested in the case in which more than smooth, $f$ belongs to some Denjoy-Carleman quasianalytic class, but the question about smoothness would be a point to start. 
I don't assume independence of $\Phi_q$ and $\phi_{q,p}$ of their parameters or of $f$.

REPLY [6 votes]: The inner functions $\phi_{qp}$ are independent in Kolmogorov's superposition theorem and they are very "bad" functions like cantor function. The smoother $f$ is, the more singular $\Phi_q$ has to be to cancel out the singularities of $\phi_{pq}$ except in some special cases, such as  $f$ is a constant. There is no linear relation between the smoothness of $f$ and it representing function $\Phi_q$. The smoothness of $f$ mainly depends on the behaviour of $\Phi_q$ on the image of singular points of $\sum_p \phi_{qp}$ .