# real analysis – Kolmogorov overlay on the Hilbert cube

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?