Linear superpositions with mappings which lower dimension

Abstract:

It is shown that for every $n$-dimensional compact metric space $X$, there exist $2n + 1$ functions $\{ {\varphi _j}\}_{j = 1}^{2n + 1}$ in $C(X)$ and $n$ mappings $\{ {\psi _i}\}_{i = 1}^n$ on $X$ with $1$-dimensional range each, with the following property: for every $0 \leqslant k \leqslant n$, every $k$ tuple $\{ {\psi _{i_l}}\}_{l = 1}^k$ of the ${\psi _i}$’s and every $2(n - k) + 1$ tuple $\{ {\varphi _{{j_m}}}\}_{m = 1}^{2(n - k) + 1}$ of the ${\varphi _j}$’s, each $f \in C(X)$ can be represented as $f(x) = \Sigma _{l = 1}^k{g_l}({\psi _{{i_l}}}(x)) + \Sigma _{m = 1}^{2(n - k) + 1}{h_m}({\varphi _{{j_m}}}(x))$, with ${g_l} \in C({\psi _{{i_l}}}(X))$ and ${h_m} \in C(R)$. It is also shown that in many cases the number $2(n - k) + 1$ is the smallest possible.