Linear Relative -Widths for Linear Operators Preserving an Intersection of Cones

We introduce the definition of linear relative -width and find estimates of linear relative -widths for linear operators preserving the intersection of cones of -monotonicity functions. 1. Introduction In various applications of CAGD (computer-aided geometric design) it is necessary to approximate functions preserving its properties such as monotonicity, convexity, and concavity. The survey of the theory of shape-preserving approximation can be found in [1]. Let be a normed linear space and let be a cone in (a convex set, closed under nonnegative scalar multiplication). It is said that has the shape in the sense of whenever . Let be a -dimensional subspace of . Classical problems of approximation theory are of interest in the theory of shape-preserving approximation as well:(1)problems of existence, uniqueness, and characterization of the best shape-preserving approximation of defined by (2)estimation of the deviation of from , that is, (3)estimation of relative -width of in with the constraint ？the leftmost infimum taken over all affine subsets of dimension , such that ;(4)estimation of linear relative -widths with the constraint in . The notion of relative -width (3) was first introduced in 1984 by Konovalov [2]. Though he considered a problem not connected with preserving shapes, the concept of relative -width arises in the theory of shape-preserving approximation naturally. Of course, it is impossible to obtain and determine optimal subspaces (if they exist) for all , , . Nevertheless, some estimates of relative shape-preserving -widths have been obtained in papers [3–5]. Estimates of relative (not necessary shape-preserving) widths have been obtained in works [6–11]. Let be a subset of and let be a linear operator. The value is the error of approximation of the identity operator by the operator on the set . Let be a cone in , . We will say that the operator preserves the shape in the sense of , if . One might consider the problem of finding (if exists) a linear operator of finite rank , which gives the minimal error of approximation of identity operator on some set over all finite rank linear operators preserving the shape in the sense . It leads us naturally to the notion of linear relative -width. In this paper we introduce the definition of linear relative -width and find estimates of linear relative -widths for linear operators preserving an intersection of cones of -monotonicity functions. 2. Notations and Definitions Let denote the space of all real-valued bounded function, defined on , with the uniform norm on , and . Denote by , , the space