The Kolmogorov–Arnold representation theorem says that any multivariable function $f(\mathbf{x})$ where $\mathbf{x}\in\mathbb{R}^n$ can be written as
$$\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^n\phi_{q,p}(x_p)\right)$$
Here $\Phi_q$ and $\phi_{q,p}$ are just single variable functions, and $x_p$ is the $p$th component of $\mathbf{x}$.
If I restrict this to only functions that can be written as
$$\Phi\left(\sum_{p=1}^n\phi_{p}(x_p)\right)$$
What functions do I have left? Is there a theorem telling me which ones I have left, or perhaps a result telling me that I can create a sufficiently good approximation to any smooth function?
