# Linear Relativen-Widths for Linear Operators Preserving an Intersection of Cones

Authors: S. P. Sidorov
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

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## Abstract

We introduce the definition of linear relativen-width and find estimates of linear relative n-widths for linear operators preserving the intersection of cones of p-monotonicity functions.


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## Body

## 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].LetX be a normed linear space and let V be a cone in X (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 Xn be a n-dimensional subspace of X. 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 approximationg*∈Xn∩V of f∈V defined by
(1)∥f-g*∥X=inf⁡g∈Xn∩V∥f-g∥X;(2)
estimation of the deviation ofA∩V from Xn∩V, that is,
(2)E(A∩V;Xn∩V)X:=sup⁡f∈A∩Vinf⁡g∈Xn∩V∥f-g∥X;(3)
estimation of relativen-width of A in X with the constraint V(3)dn(A∩V,V)X:=inf⁡XnE(A∩V;Xn∩V)X,the leftmost infimum taken over all affine subsetsXn of dimension ≤n, such that Xn∩V≠⌀;(4)
estimation of linear relativen-widths with the constraint V in X.The notion of relativen-width (3) was first introduced in 1984 by Konovalov [2]. 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 dn(A∩V,V)X and determine optimal subspaces Xn (if they exist) for all A, V, X. Nevertheless, some estimates of relative shape-preserving n-widths have been obtained in papers [3–5]. Estimates of relative (not necessary shape-preserving) widths have been obtained in works [6–11].LetA be a subset of X and let L:X→X be a linear operator. The value
(4)e(A,L):=sup⁡f∈A∥f-Lf∥X=sup⁡f∈A∥(I-L)f∥X
is the error of approximation of the identity operator I by the operator L on the set A.LetV be a cone in X, V≠⌀. We will say that the operator L preserves the shape in the sense of V, if L(V)⊂V. 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 oflinear relative n-width.In this paper we introduce the definition of linear relativen-width and find estimates of linear relative n-widths for linear operators preserving an intersection of cones of p-monotonicity functions.


## 2. Notations and Definitions

LetB[0,1] denote the space of all real-valued bounded function, defined on [0,1], with the uniform norm on [0,1], and ∥f∥B[0,1]=sup⁡x∈[0,1]|f(x)|. Denote by Bk[0,1], k⩾0, the space of all real-valued functions, whose kth derivative is bounded on [0,1], endowed with the sup-norm
(5)∥f∥Bk[0,1]=∑0≤i≤k‍1i!sup⁡x∈[0,1]|Dif(x)|,
where Di denotes the ith differential operator, Dif(x)=dif(x)/dxi, and D0=I is the identity operator, and the derivatives are taken from the right at 0 and from the left at 1.Denote byCk[0,1], k≥0, the space of all real-valued and k-times continuously differentiable functions defined on [0,1] equipped with sup-norm (5).A continuous functionf:[0,1]→R is said to be k-monotone, k≥1, on [0,1] if and only if for all choices of k+1 distinct t0,…,tk in [0,1] the inequality [t0,…,tk]f≥0 holds, where [t0,…,tk]f denotes the kth divided difference of f at 0≤t0<t1<⋯<tk≤1. Note that 2-monotone functions are just convex functions.LetΔp denote the set of all p-monotone functions defined on [0,1] and Δ0:={f∈C[0,1]:f(t)≥0, t∈[0,1]} denotes the cone of all nonnegative functions. If f∈Cp[0,1], then f∈Δp if and only if f(p)(t)≥0, t∈[0,1]. It is said that a linear operator L of C[0,1] into C[0,1] preserves p-monotonicity, if L(Δp)⊂Δp.Letσ=(σi)i⩾0 be a sequence of real numbers, σi∈{-1,0,1}, and let h, k be two integers such that 0⩽h⩽k and σh·σk≠0. Denote
(6)Δh,k(σ):={f∈C[0,1]:σpf∈Δp,h≤p≤k}.
The cone Δh,k(σ) is the intersection of cones of p-monotonicity functions with h≤p≤k taken with signs σp; that is, Δh,k(σ)=∩p=hkσpΔp.Denoteei(x)=xi, i=0,1,…. Denote Πm:=span⁡{e0,e1,…,em} and Pm:={f∈Πm:∥f∥Bm[0,1]≤1}.Recall that a linear operator mappingC[0,1] into a linear space of finite dimension n is called anoperator of finite rankn.


## 3. The Example of Linear Operator Preserving the ConeΔh,k(σ)

This section gives an example of linear operator of finite rankn preserving the cone Δh,k(σ) in the case σk=σk-2=1.Denotes(i):=min⁡{s>i:σs≠0} for i∈{h,…,k-1}. Denote tj=j/(n-1), j=0,1,…,n-1. Let (δp)p=0k-2 be the binary sequence defined by
(7)δp={0,ifp=k-2orifσpσs(p)≠-1,p≤k-31,ifσpσs(p)=-1,p≤k-3.Letk, n∈N, n≥k+2, and Λnh,k:Ck-2[0,1]→Ck-2[0,1] be the linear operator defined by
(8)Λnh,kf(x)={∑l=0k-2‍1l!el(x)×(∫t0t1Dl+1(Λnh,kf-f)(t)dtDlf(t0)-δl∫t0t1Dl+1(Λnh,kf-f)(t)dt)+(n-1)ek-1(x)(k-1)!×[Dk-2f(t1)-Dk-2f(t0)],ifx∈[t0,t1],∑l=0k-2‍1l!(x-tj)l×(∫tjtj+1Dl+1(Λnh,kf-f)(t)dtDlΛnh,kf(tj)-δl∫tjtj+1Dl+1(Λnh,kf-f)(t)dt)+(n-1)ek-1(x-tj)(k-1)!×[Dk-2f(tj+1)-Dk-2f(tj)],ifx∈(tj,tj+1],j=1,2,…,n-2.Theorem 1.
Consider thatΛnh,k:Ck-2[0,1]→Ck-2[0,1] is a continuous linear operator of finite rank n, such that Λnh,k(Δh,k(σ))⊂Δh,k(σ) and for all 0≤i≤k-2 there exists c>0 not depending on n such that
(9)sup⁡f∈Pk∥Di(Λnh,kf)-Dif∥B[0,1]≤cn-2.Proof.
SinceDk-2Λnh,kf is a piecewise linear function on [0,1] with the set of breakpoints {(tj,Dk-2f(tj))}j=0,…,n-1, then for every f such that Dk-2f≥0 the inequality Dk-2Λnh,kf≥0 holds. Moreover, Λnh,k(σpΔp)⊂σpΔp for p=k+1, k+2.
Note that for everyf∈Δk we have Dk-2(Λnh,kf-f)≥0 (since Dk-2f is convex and Dk-2Λnh,kf is a piecewise linear interpolation).
Letf∈Δh,k(σ) and suppose (by induction) that for i≥h+1 the inequality
(10)σlDl(Λnh,kf-f)≥0,l=i+1,…,k-2,
holds on [t0,t1]. For any x∈[t0,t1] we have
(11)Di(Λnh,kf-f)(x)=Di(Λnh,kf-f)(t0)+∫t0x‍Di+1(Λnh,kf-f)(z)dz.
Consider the following three cases.
(
1
) If σi=0, then δi=0 and it follows from DiΛnh,kf(0)=Dif(0) that
(12)sgn⁡Di(Λnh,kf-f)=sgn⁡Di+1(Λnh,kf-f).
(
2
) If σi≠0 and σi=σs(i), then it follows from (11) that
(13)sgn⁡Di(Λnh,kf-f)=sgn⁡Di+1(Λnh,kf-f)=⋯=sgn⁡Ds(i)(Λnh,kf-f)=σi.
(
3
) If σi≠0 and σi=-σs(i) then sgn⁡Di+1(Λnh,kf-f)=σs(i) and
(14)Di(Λnh,kf-f)(x)=Di(Λnh,kf-f)(t0)+∫t0x‍Di+1(Λnh,kf-f)(z)dz=-∫t0t1Di+1(Λnh,kf-f)(z)dz+∫t0x‍Di+1(Λnh,kf-f)(z)dz=-∫xt1‍Di+1(Λnh,kf-f)(z)dz=σi∫xt1‍σs(i)Di+1(Λnh,kf-f)(z)dz.
Using (10) we get σiDi(Λnh,kf-f)≥0 on [t0,t1], or σiDiΛnh,kf≥σiDif on [t0,t1]; that is, Λnh,kf∈σiΔi[t0,t1] if f∈σiΔi[t0,t1].
It can be shown analogously (by induction withj as induction variable) that Λnh,kf∈σiΔi[tj,tj+1] if f∈σiΔi[tj,tj+1] for any j=1,…,n-2. Therefore Λnh,k(σiΔi[0,1])⊂σiΔi[0,1].
After completing induction steps withi=k-3, k-4,…,h we can conclude that Λnh,k(Δh,k(σ))⊂Δh,k(σ).
It can be easily verified that
(
1
)  Λnh,kep=ep for all p=0,1,…,h-1 (since ±ep∈Δh,k(σ) for all p=0,1,…,h-1);
(
2
) if x∈[tj,tj+1) for some 0≤j≤n-2, then
(15)Dk-2(Λnh,kek-ek)(x)=k!(tj+1-x)(x-tj)2.
It follows from (15) that the inequality
(16)0≤Dk-2(Λnh,kek-ek)≤k!8(n-1)-2,
holds on [0,1].
Suppose (by induction) that for a fixedi≥h+1 there exist cl>0, l=i+1,…,k-2, such that for every x∈[0,1](17)0≤σlDl(Λnh,kf-f)(x)≤cl(n-1)-2,hhhhhhhhhhhhhhhl=i+1,…,k-2.
It follows from
(18)Di(Λnh,kek-ek)(0)=Diek(0)-δi∫01‍Di+1(Λnh,kek-ek)(t)dt
that for x∈[0,1](19)|Di(Λnh,kek-ek)(x)|=|Di(Λnh,kek-ek)(0)+∫0x‍Di+1(Λnh,kek-ek)(t)dt|=2|∫01‍Di+1(Λnh,kek+2-ek+2)(t)dt|≤2ci+1(n-1)21(k-i)!.
We have used the fact that if f∈C[0,1], f≥0, and there exists a constant a∈R such that f≤a on [0,1], then for every p∈N(20)0≤∫0x‍∫0tp-1‍⋯∫0t1‍f(t1)dt1⋯dtp≤axpp!.
Then (19) implies with ci=2(ci+1/(k-i)!)(21)∥Di(Λnh,kek+2)-Diek+2∥B[0,1]≤ci(n-1)-2,∥Λnh,kek-ek∥Bk-2[0,1]:=∑i=0k-2‍1i!∥Di(Λnh,kek-ek)∥B[0,1]≤(n-1)-2∑i=0k-2‍cii!.
Direct verification shows thatΛnh,kep=ep for all p=h,…,k-1.
Thus, (9) is verified and theorem is proved.


## 4. The Main Result

LetX be a linear normed space. Recall that linear n-width of a set A⊂X in X is defined by [12]
(22)δn(A)X:=inf⁡Lne(A,Ln),
where infimum is taken over all linear continuous operators Ln:X→X of finite rank n.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 functionsΔ0. 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 n cannot be better than n-2 in C[0,1] even for the functions 1, x, and x2. 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.LetX be a linear normed space. Let V be a cone in X and let A⊂X be a set and A∩V≠⌀.Definition 2.
Let one define Korovkin linear relativen-width of set A in X with the constraint V by
(23)δn(A,V)X:=inf⁡Ln(V)⊂Ve(A,Ln),
where infimum is taken over all linear continuous operators Ln:X→X of finite rank n satisfying Ln(V)⊂V.Note that ifV=X then δn(A,V)X=δn(A)X.If we compare the value of Korovkin linear relativen-width δn(A,V)X of set A in X with the constraint V to the value of linear n-width δn(A)X of the set A in X we can evaluate the negative impact of the shape-preserving constraint Ln(V)⊂V on the intrinsic error of approximation by means of theshape-preserving linear operators of finite rank n compared to the error ofunconstrained linear finite-rank approximation on the same set.This section examines approximation properties of linear finite dimensional operatorsLn preserving the cone Δh,k(σ), that is, such that
(24)Ln(Δh,k(σ))⊂Δh,k(σ).
In this section we will find estimates of Korovkin linear relative n-widths for linear operators preserving the cone Δh,k(σ) in the space Ck-2[0,1], that is, estimates of δn(A,Δh,k(σ))Ck-2[0,1].First we will prove a preliminary result.Lemma 3.
LetLn:Ck-2[0,1]→Bk-2[0,1] be a linear operator of finite rank n, n>k+2, satisfying (24). Then there exists c>0 such that
(25)sup⁡x∈[0,1]|Dk-2Lnep(x)-Dk-2ep(x)|≥cn-2∀h≤p≤k.Proof.
(
1
) It is sufficient to show that (25) holds for any linear operator Ln:Ck-2[0,1]→Bk-2[0,1] of finite rank n, such that Ln(Δh,k(σ))⊂σk-2Δk-2. Denote Z={zi}, where zi=i/n, i=0,…,n. Let {v1,…,vn} be the system of functions generating the linear space {Dk-2Lnf:f∈C[0,1]}; that is, span⁡{v1,…,vn}={Dk-2Lnf:f∈C[0,1]}.
Consider the matrixA=(vj(zi))j=1,…,ni=0,…,n. Rank of matrix A is not equal to 0, rank A≠0. Indeed, if rank A=0, then Dk-2Lnf(zi)=∑j=1naj(f)vi(zi)=0, i=0,…,n, for every f∈Ck-2[0,1], which is impossible.
Take a nontrivial vectorδ=(δ0,…,δn)∈Rn+1, such that
(26)∑i=0n‍|δi|=1,∑i=0n‍δivj(zi)=0,j=1,…,n.
LetFk-2(Z,δ) denote the set of functions f∈Ck-2[0,1], such that Dk-2f(zi)=sgn⁡δi, i=0,…,n. Let a function g∈F(Z,δ) be such that
(27)sup⁡0≤j≤n-1sup⁡x∈(zj,zj+1)[zj,x,zj+1](Dk-2g)=λ(Z,δ),
where
(28)λ(Z,δ):=inf⁡f∈Fk-2(Z,δ)max⁡0≤j≤n-1sup⁡x∈[zj,zj+1][zj,x,zj+1](Dk-2f),
where points zj, x, and zj+1 are arranged in ascending order.
Denote byDn the set of all vectors δ=(δi)i=0n with δi∈{-1,0,1}.
It follows from Theorem 2.1 in [15] that
(29)max⁡0≤j≤n-1sup⁡x∈[zj,zj+1][zj,x,zj+1](Dk-2g)≤λ(Z),
where
(30)λ(Z):=sup⁡δ∈Dnλ(δ,Z)≤(3n2)2.
It follows fromDk-2Lng∈span⁡{v1,…,vn} that ∑i=1n‍δiDk-2Lng(zi)=0. Then
(31)1=∑i=0n‍|δi|=∑i=0n‍δiDk-2g(zi)=∑i=0n‍δi(Dk-2g(zi)-Dk-2Lng(zi))⩽∑i=0n‍|δi||Dk-2Lng(zi)-Dk-2g(zi)|⩽∥Dk-2Lng-Dk-2g∥B[0,1].
(
2
) Given z∈[0,1], we have
(32)|Dk-2Lng(z)-Dk-2g(z)|=|Dk-2Ln(g-ek-2(k-2)!Dk-2g(z))(z)+1(k-2)!Dk-2g(z)(Dk-2Lnek-2-Dk-2ek-2)(z)|=|Dk-2Ln(g-ek-2(k-2)!Dk-2g(z))(z)|+1(k-2)!|Dk-2g(z)||(Dk-2Lnek-2-Dk-2ek-2)(z)|≤|Dk-2Ln(g-ek-2(k-2)!Dk-2g(z))(z)|+1(k-2)!|(Dk-2Lnek-2-Dk-2ek-2)(z)|,
since ∥Dk-2g∥B[0,1]=1.
Denotegz=g-(ek-2/(k-2)!)Dk-2g(z). It follows from [16] that there exist functions φz,j∈span⁡{eh,…,ek}, j=1,2, such that
(33)φz,j+(-1)jgz∈Δh,k(σ),j=1,2,
and such that
(
1
)  φz,j∈Δh,k(σ(r)), where σ(j):=(σi(j))i≥0 with σi(j)=σi for i≠j and σj(j)=0;
(
2
)  Dk-2φz,j(z)=0<Dk-2φz,j(x) for all x∈[0,1]∖{z};
(
3
)  Dkφz,j=2σk·max⁡0≤j≤n-1sup⁡x∈[zj,zj+1][zj,x,zj+1](Dk-2g).
Letbi,j, i=h,…,k, be such that φz,j=∑i=hk‍bi,jei, j=1,2. It follows from the proof of Theorem 2 in [16] that there exist positive real numbers ai, i=h,…,k, such that
(34)|bi,j|≤ai|Dkφz,j|,j=1,2.
ThenLn(φz,j+(-1)jgz)∈σk-2Δk, j=1,2, and, consequently, we have
(35)|Dk-2Ln(g-ek-2(k-2)!Dk-2g(z))(z)|≤max⁡{|Dk-2Lnφz,1(z)|,|Dk-2Lnφz,2(z)|}.
It follows from Dk-2φz,j(z)=0 that
(36)|Dk-2Lnφz,j(z)|=|Dk-2Lnφz,j(z)-Dk-2φz,j(z)|≤∥Dk-2Lnφz,j-Dk-2φz,j∥B[0,1]≤∑i=hk‍|bi,j|∥Dk-2Lnei-Dk-2ei∥B[0,1]≤|Dkφz,j|∑i=hk‍|ai|∥Dk-2Lnei-Dk-2ei∥B[0,1]≤2max⁡0≤j≤n-1sup⁡x∈[zj,zj+1][zj,x,zj+1](Dk-2g)×∑i=hk‍|ai|∥Dk-2Lnei-Dk-2ei∥B[0,1],
where ai, i=h,…,k, do not depend on n.
It follows from (36) and (29) that
(37)|Dk-2Lnφz,j(z)|≤2λ(Z)∑i=hk‍|ai|∥Dk-2Lnei-Dk-2ei∥B[0,1].
It follows from (31), (32), (35), (36), and (37) that
(38)12λ(Z)≤∑i=hk‍|ai|∥Dk-2Lnei-Dk-2ei∥B[0,1].
Thus, as it follows from (30) there exists a constant C>0 such that
(39)Cn-2≤∑i=hk‍∥Dk-2Lnei-Dk-2ei∥B[0,1].Theorem 4.
Letn≥k+2. If σkσk-2=1 then
(
1
) there exist c1,c2>0 not depending on n such that
(40)c1n-2<δ(Pk,Δh,k(σ))Ck-2[0,1]<c2n-2;
(
2
) for all 0≤m≤k-1(41)δn(Pm,Δh,k(σ))Ck[0,1]=0.
If σkσk-2≠1 then
(42)δ(Pk,Δh,k(σ))Ck-2[0,1]=0.Proof.
To prove (40) it is sufficient to show that there exist c1,c2>0 not depending on n such that
(43)c1n-2<inf⁡Ln(Δh,k(σ))⊂Δh,k(σ)sup⁡f∈Pk∥f-Lnf∥Bk-2[0,1]<c2n-2,
where infimum is taken over all linear continuous operators Ln of Ck-2[0,1] into Bk-2[0,1] of finite rank n satisfying Ln(Δh,k(σ))⊂Δh,k(σ). Without loss of generality we will assume σk=σk-2=1. Then the upper estimate in (40) follows from Theorem 1. The lower estimate follows from Lemma 3.
As it was shown in the proof of Theorem1 there exists a continuous linear operator Λnh,k:Ck-2[0,1]→Ck-2[0,1] of finite rank n, such that Λnh,k(Δh,k(σ))⊂Δh,k(σ), and for all 0≤i≤k-2(44)sup⁡f∈Pm∥Di(Λnh,kf)-Dif∥B[0,1]=0,0≤m≤k-1.
It follows from (44) that
(45)inf⁡Ln(Δh,k(σ))⊂Δh,k(σ)sup⁡f∈Pm∥f-Lnf∥Bk-2[0,1]=0,hhhhhhhhhhhhhhhhhhhh0≤m≤k-1,
where infimum is taken over all linear continuous operators Ln of Ck-2[0,1] into Bk-2[0,1] of finite rank n satisfying Ln(Δh,k(σ))⊂Δh,k(σ), and consequently (41) holds.
Finally, (42) follows from [16], where it is shown (remark after Proposition 1) that there exists a linear operator that maps the cone of positive and concave functions onto the same cone and holds the space P3.DenoteB(k):={f∈Bk[0,1]:∥f∥Bk[0,1]≤1}.Theorem 5.
Letn≥k+2 and σkσk-2=1. Then there exist c1, c2>0 not depending on n such that
(46)c1n-2<δn(B(k),Δh,k(σ))Ck-2[0,1]<c2n-2.Proof.
Note thatPk⊂B(k). Then the lower estimate in (46) follows from (40).
On the other hand, iff∈B(k) and x,t∈[0,1] then
(47)f(t)=∑r=0k-1‍Drf(x)r!xr+1k!∫01‍(x-t)+kDkf(t)dt
and ∥Dkf∥B[0,1]≤1. The properties of linear operator Λnh,k defined in (8) imply that there exists c2>0 such that
(48)∥f-Λnh,kf∥Bk-2[0,1]≤c2(n-1)2.


## 5. Conclusion

Estimation of linear relativen-widths is of interest in the theory of shape-preserving approximation as, knowing the value of relative linear n-width, we can judge how good or bad (in terms of optimality) this or that finite dimensional method preserving the shape in the sense V is.The paper shows that if a linear operator with finite rankn preserves the shape in the sense of cone Δh,k(σ), the degree of simultaneous approximation of derivatives of order 0≤i≤k-2 of continuous functions by derivatives of this operator cannot be better than n-2 on both the set Pk and the ball B(k). 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 Δh,k(σ). It is worth noting that nonlinear approximation preserving k-monotonicity does not have this shortcoming [17].



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