Geodesic
Existence of the best possible uniform approximation of a function of several variables by a sum of functions of fewer variables
Sbornik. Mathematics, Tome 187 (1996) no. 5, pp. 623-634 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     title = {Existence of the~best possible uniform approximation of a~function of several variables by a~sum of functions of fewer variables},
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A. L. Garkavi; V. A. Medvedev; S. Ya. Havinson. Existence of the best possible uniform approximation of a function of several variables by a sum of functions of fewer variables. Sbornik. Mathematics, Tome 187 (1996) no. 5, pp. 623-634. http://geodesic.mathdoc.fr/item/SM_1996_187_5_a0/

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