A result of Kolmogorov and Arnold says that continuous functions on $\mathbb{R}^n$ can be represented as sums of the form

$$ f(x_1,\dots,x_n)=\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^n\phi_{p,q}(x_p)\right),$$

where $\Phi_p$ and $\phi_{p,q}$ are unary continuous functions.

I'm curious about analogous results for functions on the Hilbert cube.

Suppose I have a continuous function $f:[0,1]^\omega \rightarrow [0,1]$. Is it always possible to write this in the form

$$ f(x_0,x_1,\dots) = \sum_{q<\omega} \Phi_q\left( \sum_{p<\omega}\phi_{p,q}(x_p) \right) ,$$

where $\Phi_p$ and $\phi_{p,q}$ are continuous functions and all of the sums converge uniformly? If this fails are there known analogous results?

EDIT: What we do get immediately is this.

For each $n>\omega$ let

$$f_n(x_0,\dots,x_{n-1})=\inf_{\overline{y}\in[0,1]^\omega} f(x_0,\dots,x_{n-1},\overline{y}),$$

i.e. $f_n$ is an approximation of $f$ from below by a continuous function on $[0,1]^n$. Let $f_{0}=0$ and $g_{n}(x_0,\dots,x_{n-1})=f_n(x_0,\dots,x_{n-1})-f_{n-1}(x_0,\dots,x_{n-2})$, so that in particular

$$f(x_0,x_1,\dots)=\sum_{1\leq n < \omega}g_{n}(x_0,\dots,x_{n-1}).$$

Now we can use Kolmogorov superposition on each $g_{n}$,

$$g_n(x_0,\dots,x_{n-1}) = \sum_{q=1}^{2n} \Phi_{n,q}\left( \sum_{p=0}^{n-1} \phi_{n,p,q}(x_p) \right).$$

So then we have

$$f(x_0,x_1,\dots)=\sum_{1\leq n < \omega} \sum_{q=1}^{2n} \Phi_{n,q}\left( \sum_{p=0}^{n-1} \phi_{n,p,q}(x_p) \right) $$.

By Dini's theorem the outermost sum is uniformly convergent and we can rearrange this sum to be one of the desired form, but it's unclear if the rearranged sum converges uniformly, in particular it's not clear to me that it is monotonic.

For a related posts see:

Kolmogorov superposition for smooth functions

Kolmogorov-Arnold theorem for (just-)functions

Continuous functions of three variables as superpositions of two variable functions