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

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]. LetX be a normed linear space and letV be a cone inX (a convex set, closed under nonnegative scalar multiplication). It is said that f ∈ X has the shape in the sense of V whenever f ∈ V. Let X n be a n-dimensional subspace of X. Classical problems of approximation theory are of interest in the theory of shape-preserving approximation as well:


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: 2

International Journal of Mathematics and Mathematical Sciences
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.

The Example of Linear Operator
Preserving the Cone Δ ℎ, () This section gives an example of linear operator of finite rank  preserving the cone Δ ℎ, () in the case   =  −2 = 1.
Thus, ( 9) is verified and theorem is proved.
International Journal of Mathematics and Mathematical Sciences

The Main Result
Let  be a linear normed space.Recall that linear -width of a set  ⊂  in  is defined by [12]   ()  := inf 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 Δ 0 .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  −2 in [0, 1] even for the functions 1, , and  2 .Moreover, Videnskiȋ [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  ∩  ̸ = ⌀.
where infimum is taken over all linear continuous operators   :  →  of finite rank  satisfying   () ⊂ .
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 finiterank approximation on the same set.

Conclusion
Estimation of linear relative -widths is of interest in the theory of shape-preserving approximation as, knowing the value of relative linear -width, we can judge how good or bad (in terms of optimality) this or that finite dimensional method preserving the shape in the sense  is.
The paper shows that if a linear operator with finite rank  preserves the shape in the sense of cone Δ ℎ, (), the degree of simultaneous approximation of derivatives of order 0 ≤  ≤  − 2 of continuous functions by derivatives of this operator cannot be better than  −2 on both the set   and the ball  () .Results show that the shape-preserving property of operators is negative in the sense that the error of approximation by means of such operators does not decrease with the increase of smoothness of approximated functions.In other words, there is saturation effect for linear finite-rank operators preserving the shape in the sense of cone Δ ℎ, ().It is worth noting that nonlinear approximation preserving -monotonicity does not have this shortcoming [17].