Vtyjkyuk

I will use the following formulation of the Kolmogorov-Arnold theorem:

(Theorem) A continuous multivariate function $f$ can be expressed as
beginalign
f(x_1, cdots, x_m) = sum_i=1^2m+1 Phi left( sum_j=1^m phi_i,j(x_j) right)
endalign
where $Phi, phi_i,j$ are all continuous

Define a "symmetric function" as a function of $m$ variable whose value given $m$ arguments is the same regardless of the order of the arguments.

For symmetric functions, can we prove something like:
beginalign
f(x_1, cdots, x_m) = sum_i=1^2m+1 Phi left( sum_j=1^m phi_i(x_j) right)
endalign

Intuitively, it seems like it would hold because the symmetry of $f$ should eliminate the need for $phi$ to peek at the position ($j$) of the input $x_j$.