We introduce the definition of linear relative

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 [

Let

problems of existence, uniqueness, and characterization of the best shape-preserving approximation

estimation of the deviation of

estimation of relative

the leftmost infimum taken over all affine subsets

estimation of linear relative

The notion of relative

Let

Let

In this paper we introduce the definition of linear relative

Let

Denote by

A continuous function

Let

Let

Denote

Recall that a linear operator mapping

This section gives an example of linear operator of finite rank

Denote

Let

Consider that

Since

Note that for every

Let

Consider the following three cases.

It can be shown analogously (by induction with

After completing induction steps with

It can be easily verified that

Suppose (by induction) that for a fixed

Then (

Direct verification shows that

Thus, (

Let

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

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

Let one define Korovkin linear relative

Note that if

If we compare the value of Korovkin linear relative

This section examines approximation properties of linear finite dimensional operators

First we will prove a preliminary result.

Let

Consider the matrix

Take a nontrivial vector

Let

Denote by

It follows from Theorem 2.1 in [

It follows from

Denote

Let

Then

It follows from (

It follows from (

Let

To prove (

As it was shown in the proof of Theorem

Finally, (

Denote

Let

Note that

On the other hand, if

Estimation of linear relative

The paper shows that if a linear operator with finite rank

The author declares that there is no conflict of interests regarding the publication of this paper.

This work is supported by RFBR (Grants 14-01-00140 and 13-01-00238).