# Optimal Simultaneous Approximation via𝒜-Summability

Authors: Francisco Aguilera; Daniel Cárdenas-Morales; Pedro Garrancho
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We present optimal convergence results for themth derivative of a function by sequences of linear operators. The usual convergence is replaced by 𝒜-summability, with 𝒜 being a sequence of infinite matrices with nonnegative real entries, and the operators are assumed to be m-convex. Saturation results for nonconvergent but almost convergent sequences of operators are stated as corollaries.


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## Body

## 1. Introduction

The notion of almost convergence of a sequence introduced by Lorentz [1] in 1948 entered the Korovkin-type approximation theory (see [2]) through the papers of King and Swetits [3] and Mohapatra [4]. A step forward was given by Swetits [5] in 1979 who applied in the theory the more general notion of 𝒜-summability that Bell [6] had introduced a few years earlier.After Swetits, within a shape preserving approximation setting and using as well𝒜-summability, one finds in the literature two recent papers of the authors, [7, 8], where they studied, on one hand, qualitative and quantitative Korovkin-type results, and on the other, results on asymptotic formulae. In this paper we continue this line of work which naturally takes us to the topic of saturation. Indeed, after having established an asymptotic formula, a natural way to keep on is to study optimal results to control the goodness of the approximation errors. Here saturation enters the picture. Now, before detailing our specific aim, we present the general framework of the paper which includes the definition of 𝒜-summability.Let𝒜:={A(n)}={akj(n)} be a sequence of infinite matrices with nonnegative real entries; then a sequence of real numbers {xj} is said to be 𝒜-summable to ℓ if (whenever the series below converges for all k and n)
(1)limk→∞∑j=1∞‍akj(n)xj=ℓuniformlyforn∈ℕ={1,2,…}.
Notice that 𝒜-summability extends classical convergence, matrix summability, the  Cesaro summability, and almost convergence amongst others.Now, letm∈ℕ∪{0}, let Cm[a,b] denote the space of all m-times continuously differentiable functions on the real interval [a,b], let Dm denote the usual mth differential operator, and finally, let ℒ={Lj:Cm[a,b]→Cm[a,b]} be a sequence of linear operators fulfilling the following properties:(P0)
for eachf∈Cm[a,b] and x∈[a,b], DmLjf(x) is 𝒜-summable to Dmf(x), or equivalently
(2)𝒜Dm∘ℒk,nf(x):=∑j=1∞‍akj(n)DmLjf(x)
converges to Dmf as k tends to infinity, uniformly in n,(P1)
eachLj is m-convex; that is, it maps m-convex functions onto m-convex functions; recall that a function f∈Cm[a,b] is said to be m-convex whenever Dmf(t)≥0 for all t∈[a,b],(P2)
there exist a sequence of real positive numbersλk→+∞ and three strictly positive functions w0,w1, and w2 defined on (a,b) with wi∈C2-i(a,b) such that for f∈Cm[a,b], m+2-times differentiable in some neighborhood of a point x∈(a,b),
(3)limk→+∞λk(𝒜Dm∘ℒk,nf(x)-Dmf(x))=1w2D1(1w1D1(1w0Dmf))(x)
uniformly in n.The asymptotic formula (3) informs us that the order of convergence of 𝒜Dm∘ℒk,nf(x) towards Dmf(x) is not better than λk-1 if the right-hand side of (3) is different from 0. Thus, λk-1 is called the optimal order of convergence, and those functions that possess it form the saturation class. As for our specific aim with this paper, the results of Section 2 give us information about this saturation class, while Section 3 is devoted to state a sort of converse result of asymptotic formulae. We follow the line of two respective papers of two of the authors, namely [9, 10], which at the same time have their foundations on two outstanding papers of Lorentz and Schumaker [11] and Berens [12]. The last section of the paper contains some applications. Now we close this one with some remarks and notation that we will use throughout the paper.Firstly we point out that if (P1) fulfills andDmf≤Dmg on [a,b], then for all t∈[a,b], 𝒜Dm∘ℒk,nf(t)≤𝒜Dm∘ℒk,ng(t).Secondly, if we consider a bounded subintervalJ⊂[a,b] and fix a point c∈J, it is well known that the functions u0(t)=w0(t), u1(t)=w0(t)∫ct‍w1(s)ds, and u2(t)=w0(t)∫ct‍w1(t1)∫ct1‍w2(t2)dt2dt1 form in J an extended complete Tchebychev system T={u0,u1,u2} (see [13]). Moreover {u0,u1} is a fundamental system of solutions of the second-order differential equation in the unknown v (see the right-hand side of (3)) that follows:
(4)𝒟v:=w2-1D1(w1-1D1(w0-1v))≡0.
Besides 𝒟u2≡1.In this respect, we refer the reader to [11] to recall the class LipMT1, M≥0, formed by those functions f, differentiable on (a,b), fulfilling
(5)|ΔTf(t2)-ΔTf(t1)|≤M∫t1t2‍w2(s)ds,
where ΔTf=(1/w1)D1((1/w0)f). Notice that if w2≡1, then f∈LipMT1 amounts to the fact that ΔTf belongs to the classical class LipM1.Finally, ifαk(n) is a double sequence of real numbers such that limk→+∞αk(n)=0 uniformly in n∈ℕ and βk is another sequence of real numbers with limk→+∞βk=0, then we use the notation αk(n)=o(n)(βk) to indicate that
(6)limk→+∞αk(n)βk=0unifomlyinn∈ℕ.


## 2. Saturation Results

In this section we obtain local saturation results in the approximation process of𝒜Dm∘ℒk,nf(x) towards Dmf(x). Firstly we state without proof three lemmas; Lemma 1 coincides with [10, Lemma 1], Lemma 2 follows the same pattern as [10, Lemma 2], and finally Lemma 3 is a very direct consequence of (P1).Lemma 1.
LetJ be a bounded open subinterval of [a,b]. Let g,h∈C(J) and t0,t1,t2∈J such that t0∈(t1,t2), g(t1)=g(t2)=0 and g(t0)>0. Then there exist a real number ϵ<0, a solution of the differential equation (4) on J, say z, and a point x∈(t1,t2) such that ϵh(x)+z(x)=g(x) and, for all t∈(t1,t2), ϵh(t)+z(t)≥g(t).Lemma 2.
Letf,g∈Cm[a,b] and let x∈(a,b). Assume that there exists a neighborhood Nx of x where Dmf≤Dmg. Then
(7)𝒜Dm∘ℒk,nf(x)≤𝒜Dm∘ℒk,ng(x)+o(n)(λk-1).Lemma 3.
f
∈
C
m
[
a
,
b
] is a solution of the differential equation (4) in some neighborhood of x∈(a,b) if and only if
(8)𝒜Dm∘ℒk,nf(x)-Dmf(x)=o(n)(λk-1).The following two propositions, of interest by themselves, prepare the way to prove the announced results. An important role is played by the notion of convexity with respect to the extended complete Tchebychev system{u0,u1} that here we relate to the monotonic convergence of the process and allows us to compare the degree of approximation for two different functions.Proposition 4.
Letf∈Cm[a,b]; then (a)
Dmf is convex with respect to {u0,u1} on (a,b) if and only if for each x∈(a,b)(9)𝒜Dm∘ℒk,nf(x)≥Dmf(x)+o(n)(λk-1),(b)
if𝒜Dm∘ℒk,nf(x)≥𝒜Dm∘ℒk+1,nf(x)+o(n)(λk-1) for all x∈(a,b), then Dmf is convex with respect to {u0,u1} on (a,b).Proof.
(a) Letx∈(a,b). Assume that Dmf is convex with respect to {u0,u1} on (a,b) and let z∈〈u0,u1〉 such that
(10)z(x)=Dmf(x),D1z(x)=D+1(Dmf)(x)
(here D+1 denotes the right first derivative operator). Then, from [11, Lemma 2.2], we have that z(t)≤Dmf(t) for all t∈(a,b), and directly from Lemma 2, if we take Z∈Cm[a,b] such that DmZ(t)=z(t) for all t∈(a,b), we derive that
(11)𝒜Dm∘ℒk,nZ(x)≤𝒜Dm∘ℒk,nf(x)+o(n)(λk-1),
or equivalently
(12)𝒜Dm∘ℒk,nZ(x)-DmZ(x)≤𝒜Dm∘ℒk,nf(x)-Dmf(x)+o(n)(λk-1).
Finally we apply Lemma 3 to the fuction Z and obtain the required inequality as follows:
(13)𝒜Dm∘ℒk,nf(x)≥Dmf(x)+o(n)(λk-1).
To prove the converse we assume the contrary; that is, thatDmf is not convex with respect to {u0,u1} on (a,b); then there exist three points x1,x2, and s such that
(14)a<x1<s<x2<b,S(Dmf,x1,x2)(s)<Dmf(s),
where S(Dmf,x1,x2) is the unique function of the space 〈u0,u1〉 which interpolates Dmf at x1 and x2.
Now we apply Lemma1 with h=u2 and g=Dmf-S(Dmf,x1,x2) and derive the existence of ϵ<0, a solution z^ of 𝒟z≡0 and s1∈[x1,x2] satisfying
(15)ϵu2(t)+z^(t)≥Dmf(t)-S(Dmf,x1,x2)(t),t∈(x1,x2),(16)ϵu2(s1)+z^(s1)=Dmf(s1)-S(Dmf,x1,x2)(s1).
Let us take U2,Z^,S^∈Cm[a,b] such that DmU2=u2, DmZ^=z^ and DmS^=S(Dmf,x1,x2) on (a,b) and apply then Lemma 2 taking into account (15). This yields that
(17)ϵ𝒜Dm∘ℒk,nU2(s1)+𝒜Dm∘ℒk,nZ^(s1)≥𝒜Dm∘ℒk,nf(s1)-𝒜Dm∘ℒk,nS^(s1)+o(n)(λk-1).
After introducing equality (16) we get
(18)ϵ(𝒜Dm∘ℒk,nU2(s1)-DmU2(s1))+𝒜Dm∘ℒk,nZ^(s1)-DmZ^(s1)≥𝒜Dm∘ℒk,nf(s1)-Dmf(s1)-(𝒜Dm∘ℒk,nS^(s1)-DmS^(s1))+o(n)(λk-1).
Finally, multiplying by λk and applying (P2) we obtain the following inequality which contradicts our assumption:
(19)ϵ≥𝒜Dm∘ℒk,nf(s1)-Dmf(s1)+o(n)(λk-1).
(b) If 𝒜Dm∘ℒk,nf(x)≥𝒜Dm∘ℒk+1,nf(x)+o(n)(λk-1) for x∈(a,b), then directly from (P0) we have that 𝒜Dm∘ℒk,nf(x)≥Dmf(x)+o(n)(λk-1) and it suffices to use (a) to complete the proof.Proposition 5.
LetM≥0 and let f,w∈Cm[a,b]. Then the following items are equivalent (i)
MDmw±Dmf are convex with respect to {u0,u1} on (a,b),(ii)
for eachx∈(a,b)(20)|𝒜Dm∘ℒk,nf(x)-Dmf(x)|≤M(𝒜Dm∘ℒk,nw(x)-Dmw(x))+o(n)(λk-1).Proof.
It suffices to apply Proposition4 replacing f by Mw±f.With appropriate choices of the functionw and applying the results of [11], we give two saturation results; the first one is stated in terms of classic Lipschitz spaces, while the second one puts across the relationship with the asymptotic formula.Theorem 6.
Letf∈Cm[a,b]. Then
(21)λk|𝒜Dm∘ℒk,nf(x)-Dmf(x)|≤Mw2(x)+o(n)(1),ppppppppx∈(a,b)
if only if, on (a,b),
(22)ΔTDmf=w1-1D1(w0-1(Dmf))∈LipM1.Proof.
Takew∈Cm[a,b] such that Dmw(t)=w0(t)∫ct‍w1(t1)∫ct1‍dt2dt1 and then apply Proposition 5. Thus the result follows directly after using (P2) and [11, Theorem 3.2] taking into account that 𝒟(Dmw)=w2-1.Theorem 7.
Letf∈Cm[a,b]. Then
(23)λk|𝒜Dm∘ℒk,nf(x)-Dmf(x)|≤M+o(n)(1),x∈(a,b)
if and only if, almost everywhere on (a,b),
(24)1w2D1(1w1D1(1w0Dmf))≤M.Proof.
Takew∈Cm[a,b] such that Dmw(t)=w0(t)∫ct‍w1(t1)∫ct1‍w2(t2)dt2dt1 and then apply Proposition 5. Thus the result follows directly after using (P2) and [11, Theorem 3.2] taking into account that 𝒟(Dmw)≡1.


## 3. Converse Result of the Asymptotic Formula

This section is devoted to give a converse result of the asymptotic formula stated in (3). It turns to be an extension of the results of [12]. A rough statement of the problem would read as follows: under the general framework of the paper, assume the existence of a function g such that for f∈Cm[a,b],
(25)limk→+∞λk(𝒜Dm∘ℒk,nf(x)-Dmf(x))=g(x)uniformlyinn.
Is fm+2-times differentiable at x? is it true that 𝒟(Dmf)=g?The answer, affirmative in certain sense, represents the content of this section. We will make use of two lemmas. We state them without proof as they resemble closely [10, Lemmas 3, 4].Lemma 8.
Letf∈Cm[a,b]. If
(26)limsupk→+∞λk(𝒜Dm∘ℒk,nf(x)-Dmf(x))≥0,x∈(a,b),
then Dmf is convex with respect to {u0,u1} on (a,b).Lemma 9.
Letx∈(a,b) and let H∈Cm[a,b] such that for all t∈(a,b)DmH(t)=w0(t)∫at‍h(s)w1(s)ds; then
(27)liminft→xh(t)-h(x)W2(t)-W2(x)≤liminfk→+∞λk(𝒜Dm∘ℒk,nH(x)-DmH(x)),limsupk→+∞λk(𝒜Dm∘ℒk,nH(x)-DmH(x))≤limsupt→xh(t)-h(x)W2(t)-W2(x).Theorem 10.
Letf∈Ck[a,b] and let ψ a finitely valued function in L1(a,b) such that for x∈(a,b)(28)liminfk→+∞λk(𝒜Dm∘ℒk,nf(x)-Dmf(x))≤ψ(x)≤limsupk→+∞λk(𝒜Dm∘ℒk,nf(x)-Dmf(x));
then almost everywhere on (a,b), ψ=𝒟(Dmf).Proof.
It follows the same pattern as [10, Theorem 1]. We detail it however for the sake of completeness. Let Ψ(t)=w0(t)∫at‍w1(s)∫as‍ψ(ν)w2(v)dνds, and let G∈Cm[a,b] such that for all t∈(a,b)(29)DmG(t)=Dmf(t)-Ψ(t).
For q∈ℕ, let mq and Mq be, respectively, the minor and major functions of ψ with respect to w2 such that
(30)|mq(t)-∫at‍ψ(s)w2(s)ds|<1q,t∈(a,b),|Mq(t)-∫at‍ψ(s)w2(s)ds|<1q,t∈(a,b),
whose existence is guaranteed from the theory of Lebesgue integration (see e.g., [14]). In particular it follows that
(31)limsupt→xmq(t)-mq(x)W2(t)-W2(x)≤ψ(x)≤liminft→xMq(t)-Mq(x)W2(t)-W2(x).
From the assumptions and Lemma 9, if we consider m¯q∈Cm[a,b] such that for all t∈(a,b)Dmm¯q(t)=w0(t)∫at‍mq(s)w1(s)ds, we have that
(32)limsupk→∞λk(𝒜Dm∘ℒk,nm¯q-Dmm¯q)(x)≤limsupt→xmq(t)-mq(x)W2(t)-W2(x)≤ψ(x)≤limsupk→∞λk(𝒜Dm∘ℒk,nLnf-f)(x),
hence
(33)limsupk→∞λk(𝒜Dm∘ℒk,n(f-m¯q)-Dm(f-m¯q))(x)≥0.
Now Lemma 8 yields that for each q∈ℕ, Dm(f-m¯q) is convex with respect to {u0,u1} on (a,b). Letting q tend to infinity we derive that DmG is convex respect to {u0,u1} on (a,b). If we proceed this way with Mq, we conclude that -DmG is convex respect to {u0,u1} on (a,b) as well. Hence, in this interval 𝒟(DmG)=0 and consequently, almost everywhere on (a,b)(34)𝒟(Dmf)=𝒟(Ψ),
from where the proof follows recalling the definition of Ψ at the top of the proof, the one of 𝒟 in (4), and finally using (P2).


## 4. Applications

In this section we illustrate the use of some of the results of the paper. We will make use of the asymptotic formulae obtained in [8, Section 3] to state some saturation results for the classical Bernstein operators and for a modification of them. Here we consider almost convergence, as a particular case of 𝒜-summability. We refer the reader to [8, Subsections 3.1, 3.2] for further details.


### 4.1. Saturation of Bernstein Operators and Almost Convexity

LetpB(x)=x(1-x).Corollary 11.
LetM>0 and f∈C3[0,1]; then
(35)klogk|∑j=nn+k-1‍1kD3Bjf(x)-D3f(x)|≤M12(1-x)2+o(n)(1),x∈(0,1)
if and only if
(36)pB3(1e2D4f+2e3D3f)∈LipM1on(0,1).Corollary 12.
LetM>0 and f∈C3[0,1]; then
(37)klogk|∑j=nn+k-1‍1kD3Bjf(x)-D3f(x)|≤M+o(n)(1),pppppx∈(a,b)
if and only if
(38)|D3(pBD2f)|≤Ma.e.on(0,1).


### 4.2. Saturation of Modified Bernstein Operators and Almost Convexity

Here we consider the sequence of linear operatorsLj given in [8, Subsection 3.2].Corollary 13.
LetM>0 and f∈C[0,1]; then
(39)klogk|∑j=nn+k-1‍1kLjf(x)-f(x)|≤Mx(1-x)2+o(n)(1),ppppppppppppx∈(0,1)
if and only if
(40)D1f∈LipM1on(0,1).Corollary 14.
LetM>0 and f∈C[0,1]; then
(41)klogk|∑j=nn+k-1‍1kLjf(x)-f(x)|≤M+o(n)(1),x∈(0,1)
if and only if
(42)|(pBD2f)|≤Ma.e.on(0,1).


## 4.1. Saturation of Bernstein Operators and Almost Convexity

LetpB(x)=x(1-x).Corollary 11.
LetM>0 and f∈C3[0,1]; then
(35)klogk|∑j=nn+k-1‍1kD3Bjf(x)-D3f(x)|≤M12(1-x)2+o(n)(1),x∈(0,1)
if and only if
(36)pB3(1e2D4f+2e3D3f)∈LipM1on(0,1).Corollary 12.
LetM>0 and f∈C3[0,1]; then
(37)klogk|∑j=nn+k-1‍1kD3Bjf(x)-D3f(x)|≤M+o(n)(1),pppppx∈(a,b)
if and only if
(38)|D3(pBD2f)|≤Ma.e.on(0,1).


## 4.2. Saturation of Modified Bernstein Operators and Almost Convexity

Here we consider the sequence of linear operatorsLj given in [8, Subsection 3.2].Corollary 13.
LetM>0 and f∈C[0,1]; then
(39)klogk|∑j=nn+k-1‍1kLjf(x)-f(x)|≤Mx(1-x)2+o(n)(1),ppppppppppppx∈(0,1)
if and only if
(40)D1f∈LipM1on(0,1).Corollary 14.
LetM>0 and f∈C[0,1]; then
(41)klogk|∑j=nn+k-1‍1kLjf(x)-f(x)|≤M+o(n)(1),x∈(0,1)
if and only if
(42)|(pBD2f)|≤Ma.e.on(0,1).



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