eu BASIC PROPERTIES OF LINEAR RELATIVE n-WIDTHS

The paper examines the basic properties of linear relative widths as well as the different ways of finding their estimates. AMS Subject Classification: 41A35, 41A29

main question of this section how different the properties of Konovalov linear relative widths from linear width are.Section 4 describes basic properties of Korovkin linear relative width.Both in Section 3 and Section 4 we investigate different ways of finding the lower estimations of linear relative widths.

Definitions of Linear Relative Width
A nonempty subset V of a linear space is said to be a cone if it satisfies the following properties: 1. V t is closed with respect to taking sums of its elements: if f, g ∈ V , then f + g ∈ V ; 2. V is a conic set: if f ∈ V , λ 0, then λf ∈ V .
First we remind the definition of relative width given by Konovalov in [6].
Definition 1.Let X be a normed linear space, V be a cone of X, A ∩ V be a nonempty subset of X. Relative n-width of A ∩ V in X with the constraint V is defined by the left-most infimum is taken over all affine subsets X n of dimension ≤ n, such that X n ∩ V = ∅.
The definition of relative n-width was first introduced in 1984 by Konovalov [6].Though he considered a problem not connected with preserving shapes, the concept of relative n-width arises in the theory of shape-preserving approximation naturally.Of course, it is impossible to obtain d n (A∩V, V ) X and determine optimal subspaces X n (if they exist) for all A, V , X. Nevertheless, some estimations of relative shape-preserving n-widths have been obtained in papers [7], [8], [9].
Let L : X → X be a linear operator and V be a cone in X, V = ∅.We will say that the operator L is shape-preserving operator relative to the cone V , if Let A be a subset of X and L : X → X.The value is the error of approximation of identity operator I by the operator L on the set A.
One might consider the problem of finding (if exists) a linear operator of finite rank n, which gives the minimal error of approximation of identity operator on some set over all finite rank n linear operators L preserving the shape in the sense V .It leads us naturally to the notion of linear relative n-width.
The definition of linear relative n-width was introduced in the papers [3], [4], [5].The paper [3] finds estimates of linear relative n-widths for linear operators preserving an intersection of cones of p-monotonicity functions.Definition 2. A linear operator L : X → X is called the operator with finite rank n, if the dimension of linear subspace L(X) is equals to n, dim{L(X)} = n.
Two definitions of linear relative widths are presented below.The first one is based on Konovalov's ideas [6].Definition 3. Let X be a linear normed space.Let V be a cone in X. Konovalov linear n-width of a set A ∩ V ⊂ X in X with constraint V is defined by [5] where infimum is taken over all linear continuous operators L n such that L n : X → X is of finite rank n and L n (V ) ⊂ V .
To determine the negative impact of the property of shape-preserving on the order of linear approximation, the following definition based on ideas of Korovkin was introduced in [3].Definition 4. Let X be a linear normed space.Let V be a cone in X. Korovkin linear n-width of a set A ⊂ X in the space X with constraint V is defined by where infimum is taken over all linear continuous operators L n such that L n : X → X is of finite rank n and L n (V ) ⊂ V .
Estimation of linear relative n-widths is of interest in the theory of shapepreserving approximation as, knowing the value of relative linear n-width, we can judge how good or bad (in terms of optimality) this or that finitedimensional method preserving the shape in the sense V is.

Basic Properties of Konovalov Linear Relative n-Widths
Let X be a linear normed space, A ⊂ X, A = ∅.Let δ n (A) X denote the linear n-width of the set A in X, where infimum is taken over all linear continuous operators L n : X → X with finite rank n.
The well-known basic properties of linear widths are listed below (see, for example [10]).
Proposition 1.Let X be a linear normed space, A ⊂ X, A = ∅.Then 1. δ n (A) X = δ n (A) X , where A denotes the closure of A.

Foe every
As it will be shown in this section, some of the properties of linear widths mentioned in Proposition 1 do not hold for Konovalov relative linear widths.

There exist
First we will show that the analogue of property 2 of Proposition 1 does not hold in the case of Konovalov linear widths.
1}, and We can establish the following properties of δ n (A ∩ V, V ) X .
Theorem 1.Let X be a linear normed space, A ⊂ X, A = ∅, V ⊂ X is a non-empty cone.Then
Let us prove 3).Since b + (A) ⊃ A, we have On the other hand, for every f ∈ b + (A), there exist f * ∈ A and α ∈ [0, 1], such that f = αf * .We can write b , where b α (A) := {αf : f ∈ A}.We have Properties 1 and 4 easily follow from the definition of Konovalov linear relative width 3 .
The following example shows that in the proposition 1) of Theorem 1 we can not write A ∩ V (instead of A ∩ V as it is in Proposition 1).
Example 4. Let us consider X = R 3 with Euclidean norm Note that the property 2) of Theorem 1 is not true if V is an arbitrary set (not necessary a cone).The next example show that 2) for all α 1.
The next theorem points out several ways of finding lower estimations of Konovalov linear relative widths.
Proof.Every continuous linear operator L n of finite rank n may be written in the form The analogous property holds for continuous linear operators of finite rank n, defined in Y .Since X is a subspace of Y , it follows from HahnBanach theorem that there exists a continuous operator Ln on Y , for which Ln = L n for all f ∈ X and for all f ∈ A.
Using on example [10, p. 10] it is possible to achieve a strong inequality in 2 for particular choices of X, A, V .
Since linear cone-preserving approximation is not better than best conepreserving approximation, we get the following proposition.Theorem 3. Let X be a linear normed space and A ⊂ X, V ⊂ X.Then Proof.Denote X n = range L n .Then It follows from 3 follows that one of the methods for estimation of Konovalov linear relative width δ n (A∩V, V ) X from below is estimation of the corresponding relative n-width d n (A ∩ V, V ) X .
The following example 6 shows that , where L 2 2π (R) denotes the Banach space of all (equivalence classes of) functions f : R → R that are Lebesgue integrable to the 2-th power over [−π, π] and that satisfy f (x + 2π) = f (x) for a.e.x ∈ R. The space L 2 2π (R) is endowed with the norm Then every linear operator L n of finite rank n, such that L n (V ) ⊂ V , may be written in the form where a i , g i ∈ L 2 2π (R), i = 1, . . ., n, are a.e.non-negative functions.It is obvious that d n (A ∩ V, V ) L 2 2π (R) = 0. Our aim is to show that the set of all linear operators satisfying Thus It follows from h t (t) = 0 follows that n i=1 a i , h t g i (t) = 0.It is imposable for all t ∈ [−π, π] because of a i , g i 0 a.e.Theorem 4. Let X be a linear normed space and V be a cone in X.Then where δ n (A ∩ V ) X denotes the linear n-width of A ∩ V in X.
Proof.We have where infimum is taken over all continuous linear operators L n : X → X with finite rank n.The last quantity is the linear n-width of A ∩ V in X.
Note that if V = X then relative n-width of A in X is equal to n-width of A in X for every A, i.e. δ n (A ∩ X, X) X = δ n (A ∩ X) X = δ n (A) X .
The example 7 shows that there is a choice of X, A, V for which strong inequality holds in (3).Note that range rangeL = span{Ie 0 , Ie 1 , Ie 2 } and dimension of range rangeL are equal to 3.

Let b(A) denotes the balanced hull of
Analogues of theorems 2, 3, 4 are holds and can be find below.
Theorem 7. Let X be a linear normed space and A ⊂ X, V ⊂ X.Then Theorem 8. Let X be a linear normed space and V be a cone in X.Then where δ n (A) X is the linear n-width of A in X.
Of course, the value of Konovalov linear relative n-width is not greater than the value of Korovkin linear relative n-width: If we compare the value of Korovkin linear relative n-width δ n (A, V ) X of set A in X with the constraint V to the value of linear n-width δ n (A) X of the set A in X we can evaluate the negative impact of the shape-preserving constraint L n (V ) ⊂ V on the intrinsic error of approximation by means of the shape-preserving linear operators of finite rank n compare to the error of unconstrained linear finite-rank approximation on the same set.
It seems that the problem of estimation of Konovalov linear relative n-width δ n (A ∩ V, V ) X is much harder that the problem of estimation of Korovkin linear relative n-width δ n (A, V ) X , so it will be difficult to use (7) for getting lower estimations for the value of δ n (A, V ) X .On the other hand, (7) can potentially help for getting upper estimations for the value of Konovalov linear relative n-width δ n (A ∩ V, V ) X .