By S. Ya. Khavinson

This publication bargains with difficulties of approximation of constant or bounded services of a number of variables by way of linear superposition of features which are from a similar category and feature fewer variables. the most subject is the gap of linear superpositions $D$ regarded as a subspace of the distance of constant features $C(X)$ on a compact house $X$. Such homes as density of $D$ in $C(X)$, its closedness, proximality, and so forth. are studied in nice element. The method of those and different difficulties in keeping with duality and the Hahn-Banach theorem is emphasised. additionally, significant awareness is given to the dialogue of the Diliberto-Straus set of rules for locating the easiest approximation of a given functionality by means of linear superpositions.

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**Example text**

For example, In can be embedded into Rn by the identity map. 7, uniformly separate Borel measures. As it turns out, for such systems the number 2n + 1 clearly becomes a rigid characteristic of the dimension n. 1. IfdimX = n (n ~ 1) and a system of functions 'Pi(x) E C(X), ~ 2n+l, and there exists a system of (2n + 1) functions that does so. THEOREM i = 1, ... 1) to hold for all f(x) E C(X) when dimX = n, it is necessary that N ~ 2n + 1. 1 (its first part-the second part is the theorem of Ostrand and Tikhomirov) was proved by Sternfeld.

2. IfdimX = n ~ 2, then any system of functions 'Pi(x) E C(X), i = 1, ... , N, uniformly separating points in X contains at least 2n + 1 functions: N~2n+l. 1), or in a similar representation for an arbitrary function f(x) E B(X) and with Yi E B(JR), there are at least 2n + 1 terms. This is a complete solution of the questions about the possibility of decreasing the number of terms in the Kolmogorov-OstrandTikhomirov representation (and, moreover, obtained without appealing to any special structure of inner functions in Kolmogorov's theorem for X =In).

3) f(x) = 91 o