Abstract

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 of all real-valued functions, whose th derivative is bounded on , endowed with the sup-norm where denotes the th differential operator, , and is the identity operator, and the derivatives are taken from the right at 0 and from the left at 1.

Denote by , , the space of all real-valued and -times continuously differentiable functions defined on equipped with sup-norm (5).

A continuous function is said to be -monotone, , on if and only if for all choices of distinct in the inequality holds, where denotes the th divided difference of at . Note that 2-monotone functions are just convex functions.

Let denote the set of all -monotone functions defined on and , denotes the cone of all nonnegative functions. If , then if and only if , . It is said that a linear operator of into preserves -monotonicity, if .

Let be a sequence of real numbers, , and let , be two integers such that and . Denote The cone is the intersection of cones of -monotonicity functions with taken with signs ; that is, .

Denote , . Denote and .

Recall that a linear operator mapping into a linear space of finite dimension is called an operator of finite rank .

3. The Example of Linear Operator Preserving the Cone

This section gives an example of linear operator of finite rank preserving the cone in the case .

Denote for . Denote , . Let be the binary sequence defined by

Let , , , and be the linear operator defined by

Theorem 1. Consider that is a continuous linear operator of finite rank , such that and for all there exists not depending on such that

Proof. Since is a piecewise linear function on with the set of breakpoints , then for every such that the inequality holds. Moreover, for , .
Note that for every we have (since is convex and is a piecewise linear interpolation).
Let and suppose (by induction) that for the inequality holds on . For any we have
Consider the following three cases.
If , then and it follows from that
If and , then it follows from (11) that
If and then and Using (10) we get on , or on ; that is, if .
It can be shown analogously (by induction with as induction variable) that if for any . Therefore .
After completing induction steps with , we can conclude that .
It can be easily verified that
   for all (since for all );
if for some , then It follows from (15) that the inequality holds on .
Suppose (by induction) that for a fixed there exist , , such that for every It follows from that for We have used the fact that if , , and there exists a constant such that on , then for every
Then (19) implies with
Direct verification shows that for all .
Thus, (9) is verified and theorem is proved.

4. The Main Result

Let be a linear normed space. Recall that linear -width of a set in is defined by [12] where infimum is taken over all linear continuous operators of finite rank .

Dealing with the problem of approximation of smooth functions by some class of linear operators, we may find that operators of this class have some property which limits the degree of approximation of smooth functions by operators of this class. Let us cite the well-known instances. By definition, every positive linear operator is shape-preserving with respect to the cone of all nonnegative functions . It was shown by Korovkin [13] that if linear polynomial operator preserves positiveness, the degree of approximation of continuous functions by this operator is low. He proved that the order of approximation by positive linear polynomial operators of degree cannot be better than in even for the functions 1, , and . Moreover, Videnskiĭ [14] has shown that the result of [13] does not depend on the properties of the polynomials but rather on the limitation of dimension.

To determine the negative impact of the property of shape-preserving on the order of linear approximation we introduce the following definition based on ideas of Korovkin.

Let be a linear normed space. Let be a cone in and let be a set and .

Definition 2. Let one define Korovkin linear relative -width of set in with the constraint by where infimum is taken over all linear continuous operators of finite rank satisfying .

Note that if then .

If we compare the value of Korovkin linear relative -width of set in with the constraint to the value of linear -width of the set in we can evaluate the negative impact of the shape-preserving constraint on the intrinsic error of approximation by means of the shape-preserving linear operators of finite rank compared to the error of unconstrained linear finite-rank approximation on the same set.

This section examines approximation properties of linear finite dimensional operators preserving the cone , that is, such that In this section we will find estimates of Korovkin linear relative -widths for linear operators preserving the cone in the space , that is, estimates of .

First we will prove a preliminary result.

Lemma 3. Let be a linear operator of finite rank , , satisfying (24). Then there exists such that

Proof. It is sufficient to show that (25) holds for any linear operator of finite rank , such that . Denote , where , . Let be the system of functions generating the linear space ; that is, .
Consider the matrix . Rank of matrix is not equal to 0, rank . Indeed, if rank , then , , for every , which is impossible.
Take a nontrivial vector , such that
Let denote the set of functions , such that , . Let a function be such that where where points , , and are arranged in ascending order.
Denote by the set of all vectors with .
It follows from Theorem 2.1 in [15] that where
It follows from that . Then
Given , we have since .
Denote . It follows from [16] that there exist functions , , such that and such that
  , where with for and ;
   for all ;
  .
Let , , be such that , . It follows from the proof of Theorem 2 in [16] that there exist positive real numbers , , such that
Then , , and, consequently, we have It follows from that where , , do not depend on .
It follows from (36) and (29) that
It follows from (31), (32), (35), (36), and (37) that Thus, as it follows from (30) there exists a constant such that