By S. Ya. Khavinson

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

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**Sample text**

Call the points x and x' in x equivalent with respect to Rt, i = 1, 2, ...

To shorten the arguments, we can assume that nl > 0. ) We can move this positive exponent in the word F to the extreme right position by circular permutations. 4) function w(x) for which the measureµ E 'P. 9) fl_ fl (Xo w (xo) W (xo) w 2 (xo) It is non-empty (it contains w). Therefore, the set of values taken at wni-l (x 0 ) by functions v from that set is a non-degenerate interval. Take a value from that interval satisfying the following requirements: is different from the numbers Xo, W (xo), ...

The identity §8. A COUNTEREXAMPLE 41 = map e: e(x) x plays the role of the identity. Right away, let us note an important (for us) example of functions in G. 4) is an entire analytic function and belongs to G. Choose in G certain functions v1 ( x), ... , Vn ( x), and define a homomorphism cp of a free group Go into the group G by setting cp (eo) = e. 1) and is indeed a homomorphism of a free group Go into G. 1) and cpF its image, we shall often write F(x) instead of (cpF)(x), since the presence of the variable x already distinguishes a function (an element of G) from a word.