Manzhos T. Existence, characterization and uniqueness of elements of the best approximation with restrictions.

In the thesis we investigate the problem of the best uniform restricted range approximation of vector-valued continuous functions by generalized polynomials, that is which values at each point t belong to a convex, nonempty set ?(t) with smooth boundary. In particular, existence, uniqueness, strict uniqueness, characterization of element of best approximation are proved. The estimation of error of the best approximation with restrictions through error of the best approximation without restrictions is received. Besides the concept of admissible pair sets is introduced and its properties are considered. Similar problems are considered for the case of best uniform restricted range approximation of vector-function by generalized polynomials that interpolate approximated function at the fixed points. In addition, theorems of many variables real-valued function best approximation element existence and characterization by generalized polynomials with restrictions on partial derivatives are proved.