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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Linear superpositions with mappings which lower dimension


Author: Y. Sternfeld
Journal: Trans. Amer. Math. Soc. 277 (1983), 529-543
MSC: Primary 26B40; Secondary 54F45
MathSciNet review: 694374
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0694374-9
PII: S 0002-9947(1983)0694374-9
Article copyright: © Copyright 1983 American Mathematical Society