Linear superpositions with mappings which lower dimension

Abstract

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