Conditions of Extremality of an Admissible Element for the Problem of Best Simultaneous Approximation in the Sense of the Squared Norm Of Several Elements of a Linear Normed Space by a Subset of this Space

• Дата: 2026
• Автор: Гудима, Уляна
• Формат: Стаття
• Мова: Українська
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2026
• Онлайн доступ: https://mcm-math.kpnu.edu.ua/article/view/354647
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

An important class in approximation theory is formed by problems of best simultaneous approximation of several elements of a linear normed space by a subset of this space. The problem of best simultaneous approximation of several elements of a linear normed space by a subset of this space consists in finding, within a given subset of the space, a point for which the maximum distance to each of several fixed points of the space is minimal, that is, does not exceed the maximal distance from each of these points to any other point of the given subset. There arise problems, in particular in approximation theory, in which the deviations between fixed elements of a linear normed space and elements of the approximating set are defined by so-called distorted metrics (weighted norms, seminorms, sublinear functionals, convex functionals, etc.). In particular, in [1] a criterion of extremality of an admissible element was established for the problem of best simultaneous approximation, in the sense of weighted distances, of several elements of a linear normed space by a convex subset of this space; in [2] conditions of extremality of an admissible element were obtained for the problem of best simultaneous approximation of several elements of a certain polynormed space by a subset of this space; in [3] a criterion of extremality of an admissible element was proved for the problem of best approximation, in the sense of a seminorm, of an element of a linear normed space by a convex subset of this space; in [4] conditions for the existence of an extremal element were established for a generalized Steiner problem in a polynormed space, where deviations between elements are defined by sublinear functionals; in [5] conditions of extremality of an admissible element were established for the problem of finding a generalized Chebyshev center of several closed balls in a certain polynormed space relative to a subset of this space. From a unified standpoint, problems of best simultaneous approximation of several elements of a linear normed space by convex subsets of this space are considered in [6, 7]. If, in the problem of best simultaneous approximation of several elements of a linear normed space where distances between points are defined by the norm, the norm is replaced by the square of the norm, one obtains the problem of best simultaneous approximation in the sense of the squared norm, which is studied in this paper.