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

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

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

I am curious to know the similar results obtained for the functions of the Hilbert cube.

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

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

or $ Phi_p $ and $ phi_ {p, q} $ the continuous functions and all the sums converge uniformly? If this fails, are there any known similar results?