Existence of the best possible uniform approximation of a function of several variables by a sum of functions of fewer variables

Let $\varphi_i$ be some maps of a set $X$ onto sets $i=1,\dots,n$, $n\geqslant 2$. Approximations of real function $f$ on $X$ by sums $g_1\circ \varphi _1+\dots +g_n\circ \varphi _n$ are considered, where the $g_i$ are real function on $X_i$. Under certain constraints on the $\varphi_i$ the existence of the best possible approximation is proved in three cases. In the first case the function $f$ and the approximating sums are bounded, but the functions $\varphi_i$ can be unbounded. In the second case $f$ and the $g_i$ are bounded. In the third case $f$ and the $g_i$ are continuous, $X$ and the $X_i$ are compact sets with metrics, and the maps $\varphi_i$ are continuous.