Azerbaijan National Academy of Science

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2) A Chebyshev type theorem was proved for sums of ridge functions to be extremal to a given continuous function;

3) Explicit formulas for an exact computation of the approximation error and construction of best approximating function were obtained in problems of approximation of multivariate functions by sums of functions of fewer variables and ridge functions in both continuous and integral metrics;

4) It was shown that if each continuous function defined on a compact Hausdorff space is represented by linear superpositions, then all functions on this space possess such representation. In particular, from this result it follows that the famous Kolmogorov superposition theorem and several other superposition theorems, which were proven for continuous functions, are also valid for discontinuous functions.

5) The problem of approximation by neural networks with weights varying on a finite set of directions was considered. Necessary and sufficient conditions for density of such networks were found.

6) A family of new Morrey type spaces was constructed and integral representations for generalized mixed derivatives of functions from these spaces were obtained. By means of these integral representations, differential and differential-difference properties of functions from the constructed spaces were studied. The obtained theoretical results were applied to some higher order differential equations.