Умови існування екстремального елемента для задачі відшукання відстані між двома множинами, єдиності екстремального елемента еквівалентної їй задачі, властивості функції відстані

Abstract

The theory of approximation of a function is a direction of mathematics which is intensively developing. The work of P. L. Chebyshev in 1857 is considered the beginning of the modern theory of approximations. It is devoted to polynomials that deviate the least from zero. In this work, the concept of the best approximation was introduced. Later, problems were investigated in which individual functions approached with polynomials, trigonometric polynomials, rational functions, etc. in different metrics. These tasks are a partial case of the problem of the best approximation of an element of linear normed space by convex set of this space. General theorems of existence, uniqueness of an extremal element, properties of the best approximation functional, duality theorems and criteria of an extremal element for this problem are established [1]. The more general problem are problem of finding the distance between two sets of linear normalized space is also considered [2, 3]. In [4, 5] the relations of duality, criteria of extremal element and sequence are proved for this problem. In this article established the conditions of the existence of an extremal element for the problem of finding the distance between two sets of linearly normalized space, the conditions of the unity of an extremal element for its equivalent problem, the properties of the function of the distance and formulas for finding an extremal element for the problem of finding the distance between two closed spheres of this space.