# Asymptotic Formulae via a Korovkin-Type Result

Authors: Daniel Cárdenas-Morales; Pedro Garrancho; Ioan Raşa
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We present a sort of Korovkin-type result that provides a tool to obtain asymptotic formulae for sequences of linear positive operators.


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## Body

## 1. Introduction

This paper deals with the approximation of continuous functions by sequences of positive linear operators. In this setting, on studying a sequence of operators, sayLn, one usually intends to prove firstly that the sequence defines an approximation process; that is, for each function f of some space, Lnf converges to f, in a certain sense as n tends to infinity; afterwards, one searches for quantitative results that estimate the degree of convergence, and finally one measures the goodness of the estimates, mainly through inverse and saturation results. An outstanding tool to achieve these saturation results is given by the asymptotic formulae that provide information about the so-called optimal degree of convergence. The most representative expression of this type was stated for the classical Bernstein operators by Voronovskaja in [1] and reads as follows: for f∈C[0,1] and x∈(0,1), if f′′(x) exists, then(1.1)lim⁡n→∞n(Bnf(x)-f(x))=x(1-x)2f′′(x).
This work dwells upon this type of expressions.A classical key ingredient to prove an asymptotic formula for a sequence of positive linear operatorsLn is Taylor's theorem. From it, the formula appears after some minor work if one is able to find easy-to-use expressions for the first moments of the operators, namely,(1.2)Lneix(x)=Ln((e1-xe0)i)(x),i=0,1,2,4,
(ei and eix denote the monomials ei(t)=ti and eix(t)=(t-x)i). On the contrary, if the calculation of the moments gets complicated, then a drawback appears.The main purpose of this paper is to present a tool that helps to overcome these situations. The basic ideas behind the main result (Theorem2.1 below) lies in [2, Chapter 5], the novelty here being that instead of using the Taylor formula of the function f to be approximated, we consider the Taylor formula of f∘φ-1 for a certain function φ.It is the intention of the authors that the paper offers a clear and quick procedure to obtain asymptotic expressions for a wide variety of sequences of linear positive operators.


## 2. The Main Result

Letφ be any ∞-times continuously differentiable function on [0,1], such that φ(0)=0,φ(1)=1, and φ′(x)>0 for x∈(0,1). We denote by eφ,ix the function(2.1)eφ,ix(t)=(φ(t)-φ(x))i,
and Di denotes the usual ith differential operator, though we keep on using the common notation f′ and f′′ for the first and second derivatives of a function f.Although we restrict our attention to the spaceC[0,1] of all continuous functions defined on [0,1], the following main result of the paper remains valid for any compact real interval with the obvious modifications.Theorem 2.1.
LetLn:C[0,1]→C[0,1] be a sequence of linear positive operators, and let x∈(0,1) be fixed. Let us assume that there exist a sequence of positive real numbers λn→+∞ (as n→+∞) and two functions p,q∈C[0,1], p being strictly positive on (0,1), such that for all i∈{0,1,2,4}(2.2)lim⁡n→+∞λn(Lneφ,ix(x)-eφ,ix(x))=p(x)D2eφ,ix(x)+q(x)D1eφ,ix(x).
Then for each f∈C[0,1], twice differentiable at the point  x,
(2.3)lim⁡n→+∞λn(Lnf(x)-f(x))=p(x)f′′(x)+q(x)f′(x).Remark 2.2.
Notice that fori=0, identity (2.2) becomes limn→+∞λn(Lne0(x)-1)=0, which is obviously fulfilled if the operators Ln preserve the constants.   This property is satisfied by most classical sequences of linear operators (see, e.g., [3]),   among them being the ones that we study in the present paper.
Moreover, fori=1,2,4, identity (2.2) becomes, respectively,
(2.4)lim⁡n→+∞λnLneφ,1x(x)=p(x)φ′′(x)+q(x)φ′(x),lim⁡n→+∞λnLneφ,2x(x)=2p(x)φ′(x)2,lim⁡n→+∞λnLneφ,4x(x)=0.
These are, actually, the identities that we will explicitly use throughout the paper. However, we have decided to write the hypotheses of the theorem as in (2.2) for the sake of brevity and to put across that we can think of the result as if it were of Korovkin type, since we can guarantee the convergence of λn(Lnf(x)-f(x)) and obtain its limit for any f∈C[0,1], whenever we have it for four test functions.Proof.
The classical Taylor theorem, applied to the functionf∘φ-1, yields for t∈[0,1] that
(2.5)f(t)=(f∘φ-1)(φ(t))=(f∘φ-1)(φ(x))+D1(f∘φ-1)(φ(x))(φ(t)-φ(x))+D2(f∘φ-1)(φ(x))2(φ(t)-φ(x))2+h(t-x)(φ(t)-φ(x))2,
where h is a continuous function which vanishes at 0. Equivalently we can write for t∈[0,1](2.6)f(t)=f(x)+D1(f∘φ-1)(φ(x))eφ,1x(t)+D2(f∘φ-1)(φ(x))2eφ,2x(t)+hx(t)eφ,2x(t),
where hx(t)∶=h(t-x). Applying the operator Ln and then evaluating at the fixed point x, we obtain the equality
(2.7)Lnf(x)=f(x)Lneφ,0x(x)+D1(f∘φ-1)(φ(x))Lneφ,1x(x)+D2(f∘φ-1)(φ(x))2Lneφ,2x(x)+Ln(hxeφ,2x)(x).
Now we subtract f(x) from both sides and multiply by λn to get
(2.8)λn(Lnf(x)-f(x))-λnLn(hxeφ,2x)(x)=f(x)λn(Lneφ,0x(x)-1)+D1(f∘φ-1)(φ(x))λnLneφ,1x(x)+D2(f∘φ-1)(φ(x))2λnLneφ,2x(x).
Taking into account the basic identities
(2.9)D1(f∘φ-1)(φ(x))=f′(x)φ′(x),D2(f∘φ-1)(φ(x))=1φ′(x)(f′′(x)φ′(x)-f′(x)φ′′(x)(φ′(x))2)
and using (2.4), we derive that
(2.10)lim⁡n→+∞(λn(Lnf(x)-f(x))-λnLn(hxeφ,2x)(x))=p(x)f′′(x)+q(x)f′(x).
The proof will be over once we prove that λnLn(hxeφ,2x)(x)→0 as n→+∞.
To this purpose letϵ>0 and let θx be an open set containing x such that for t∈θx, |hx(t)|<ϵ. Then if we define w(t)∶=max{0,|hx(t)|-ϵ}eφ,2x(t), we have that for all t∈[0,1](2.11)|hx(t)eφ,2x(t)|≤ϵeφ,2x(t)+max⁡{0,|hx(t)|-ϵ}eφ,2x(t)=ϵeφ,2x(t)+w(t).
On the other hand, w vanishes on θx, so there is a constant M such that for all t∈[0,1],
(2.12)|w(t)|≤Meφ,4x(t).
Finally, the linearity and positivity ofLn allow us to write, from (2.11) and (2.12),
(2.13)|λnLn(hxeφ,2x)(x)|≤ϵλnLneφ,2x(x)+λnMLneφ,4x(x),
from where taking limits and using again (2.4),
(2.14)lim⁡sup⁡n→+∞|λnLn(hxeφ,2x)(x)|≤2ϵp(x)φ′(x)2.
This ends the proof, as p(x) and φ′(x) are strictly positive and ϵ was arbitrary.


## 3. Applications

The first two applications correspond to sequences of operators recently introduced in [4, 5]. They represent respective modifications of the classical Bernstein operators and the well-known modified Meyer-König and Zeller operators (see [6]) which, instead of preserving the linear functions, hold fixed e0 and e2.   Thus they are inside this new line  of work which originated with the paper [7] and found further development in a long list of papers  (see, e.g., [8–14]).The aforementioned preserving property usually makes it quite difficult to compute the first moments and consequently to obtain asymptotic formulae. Here our result, applied withλn=n and φ=e2, enters the scene.Two further applications with different values ofφ are presented afterwards in less detail.


### 3.1. Modified Bernstein Operators Which Preservex2

This section deals with the following sequence of operators presented in [4] (we use the same notation) as a byproduct of some interesting results, defined for f∈C[0,1] and n>1 as(3.1)Bn,0,2f(x)=∑k=0nf(k(k-1)n(n-1))(nk)xk(1-x)n-k.
As pointed out in [13], this provides an example of a sequence of positive linear polynomial operators that preserve e0 and e2 and represents an approximation process for functions f∈C[0,1].The presence of the square root in the definition makes it difficult to obtain easy-to-handle expressions for the momentsBn,0,2eix(x),i=0,1,2,4, and consequently to obtain an asymptotic formula. We will apply our theorem to get it, though we first prove a quantitative result missed in [4].Proposition 3.1.
Letf∈C[0,1],  x∈[0,1], and let δ>0. Then
(3.2)|Bn,0,2f(x)-f(x)|≤ω(f,δ)(1+1δ2x(1-x)1-(1-x)n-1n-1).Proof.
From the usual quantitative estimate in terms ofω(f,·) stated in [15], we can write
(3.3)|Bn,0,2f(x)-f(x)|≤ω(f,δ)(1+1δ2x(x-Bn,0,2e1(x))).
Now, for n>1, (k-1)/(n-1)≤k/n, so k(k-1)/n(n-1)≤k/n and Bn,0,2e1≤Bne1=e1. Thus
(3.4)x-Bn,0,2e1(x)≥0.
On the other hand, for n>1, (1/2)(k/n+(k-1)/(n-1))≥k(k-1)/n(n-1)≥(k-1)/(n-1), so
(3.5)12∑k=1n(kn+k-1n-1)(nk)xk(1-x)n-k≥Bn,0,2e1(x)≥∑k=1nk-1n-1(nk)xk(1-x)n-k,
and using the equalities
(3.6)∑k=1nkn(nk)xk(1-x)n-k=x,∑k=1nk-1n-1(nk)xk(1-x)n-k=nn-1∑k=1n(kn-1n)(nk)xk(1-x)n-k=nx-1n-1+(1-x)nn-1,
we get
(3.7)1-x2(n-1)(1-(1-x)n-1)≤x-Bn,0,2e1(x)≤1-xn-1(1-(1-x)n-1),
from which the result follows.Corollary 3.2.
For allf∈C[0,1] and x∈(0,1), whenever f′′(x) exists,
(3.8)lim⁡n→∞n(Bn,0,2f(x)-f(x))=-1-x2f′(x)+(1-x)x2f′′(x).Proof.
We will apply the theorem withφ=e2,λn=n,q(x)=-(1-x)/2, and p(x)=(1-x)x/2. As the operators Bn,0,2 preserve the constants, it suffices to check that (2.4) holds true.
First computations yield(3.9)Bn,0,2e0(x)=1,Bn,0,2e2(x)=x2,Bn,0,2e4(x)=2n(n-1)x2+4(n-2)n(n-1)x3+(n-3)(n-2)n(n-1)x4,Bn,0,2e6(x)=4n2(n-1)2x2+32(n-2)n2(n-1)2x3+38(n-3)(n-2)n2(n-1)2x4+12(n-4)(n-3)(n-2)n2(n-1)2x5+(n-5)(n-4)(n-3)(n-2)n2(n-1)2x6,Bn,0,2e8(x)=8n3(n-1)3x2+208(n-2)n3(n-1)3x3+652(n-2)(n-3)n3(n-1)3x4+576(n-4)(n-3)(n-2)n3(n-1)3x5+188(n-5)(n-4)(n-3)(n-2)n3(n-1)3x6+24(n-6)(n-5)(n-4)(n-3)(n-2)n3(n-1)3x7+(n-7)(n-6)(n-5)(n-4)(n-3)(n-2)n3(n-1)3x8.
Thus for i=1,2,4 the quantities Bn,0,2eφ,ix appear after some calculations using the following identities:
(3.10)eφ,1x=ee2,1x=e2-x2e0,eφ,2x=ee2,2x=e4-2x2e2+x4e0,eφ,4x=ee2,4x=e8-4x2e6+6x4e4-4x6e2+x8e0,Bn,0,2eφ,1x(x)=0,Bn,0,2eφ,2x(x)=2(1-x)x2(n-1)n(1-3x+2nx),Bn,0,2eφ,4x(x)=4(1-x)x2(n-1)3n3(n4(12-12x)x4+n3(112-324x+216x2)x3+n2(159-1041x+1881x2-1011x3)x2+n(52-759x+2921x2-4089x3+1887x4)x+(2-102x+876x2-2580x3+3060x4-1260x5)).
Finally we are in a position to apply Theorem2.1 and then prove the corollary, since we show below that assumptions in (2.4) are fulfilled:
(3.11)lim⁡n→∞nBn,0,2eφ,1x(x)=0=2xq(x)+2p(x)=q(x)φ′(x)+p(x)φ′′(x),lim⁡n→∞nBn,0,2eφ,2x(x)=4(1-x)x3=8x2p(x)=2p(x)φ′(x)2,lim⁡n→∞nBn,0,2eφ,4x(x)=0.


### 3.2. The Modified Meyer-König and Zeller Operators

Forf∈C[0,1], x∈[0,1], and n>1, we consider the operators defined as(3.12)Rnf(x)=∑k=0∞f(k(k-1)(n+k)(n+k-1))(n+kk)xk(1-x)n+1,
for x<1 and Rnf(1)=f(1).They were introduced in [5] as a modification of the well-known modified Meyer-König and Zeller operators (see [6]).It turns to be another example of a sequence of positive linear operators that preservee0 and e2 and represents an approximation process for functions f∈C[0,1].Here again, the presence of the square root in the definition makes it difficult to obtain easy-to-handle expressions forRneix(x), i=0,1,2,4, and consequently to obtain by usual means an asymptotic formula. With this aim we shall apply our theorem.Corollary 3.3.
For allf∈C[0,1] and  x∈(0,1), whenever  f′′(x) exists,
(3.13)lim⁡n→∞n(Rnf(x)-f(x))=-(1-x)22f′(x)+x(1-x)22f′′(x).Proof.
We will apply the theorem withφ=e2,λn=n,q(x)=-(1-x)2/2 and p(x)=x(1-x)2/2. As the operators Rn preserve the constants, it suffices to check that (2.4) holds true.
Direct computations with the use of mathematical software (Mathematica) give(3.14)Rne0(x)=1,Rne2(x)=x2,Rne4(x)=4nx3(2+n)(3+n)+n2(1+n)x4(2+n)(3+n)(4+n)+4n3x5(2+n)(3+n)(4+n)(5+n)+O(n-2),Rne6(x)=12n8x5(1+n)(2+n)2(3+n)2(4+n)2(5+n)2+n10(17+n)x6(1+n)(2+n)2(3+n)2(4+n)2(5+n)2(6+n)2+12n12x7(1+n)(2+n)2(3+n)2(4+n)2(5+n)2(6+n)2(7+n)2+O(n-2),Rne8(x)=24n19x7(1+n)2(2+n)3(3+n)3(4+n)3(5+n)3(6+n)3(7+n)3+n22(59+n)x8(1+n)2(2+n)3(3+n)3(4+n)3(5+n)3(6+n)3(7+n)3(8+n)3+24n25x9(1+n)2(2+n)3(3+n)3(4+n)3(5+n)3(6+n)3(7+n)3(8+n)3(9+n)3+O(n-2).
Now, using again (3.10) we can write
(3.15)Rneφ,1x(x)=0,Rneφ,2x(x)=4x3n3(1-x)2(2+n)(3+n)(4+n)(5+n)+O(n-2),Rneφ,4x(x)=n25x7(24+59x+24x2)(1+n)2(2+n)3(3+n)3(4+n)3(5+n)3(6+n)3(7+n)3(8+n)3(9+n)3+3n22x8(1+n)2(2+n)3(3+n)3(4+n)3(5+n)3(6+n)3(7+n)3(8+n)3-4n12x7(12+17x+12x2)(1+n)(2+n)2(3+n)2(4+n)2(5+n)2(6+n)2(7+n)2+6n3x7(4+x+4x2)(2+n)(3+n)(4+n)(5+n)+O(n-2)
and then the following identities which prove the corollary:
(3.16)lim⁡n→∞nRneφ,1x(x)=0=2xq(x)+2p(x)=q(x)φ′(x)+p(x)φ′′(x),lim⁡n→∞nRneφ,2x(x)=4x3(1-x)2=8x2p(x)=2p(x)φ′(x)2,lim⁡n→∞nMn,4eφ,4x(x)=0.


### 3.3. The Modified Bernstein Operators Which Preserve a General Functionτ

Letτ be any function fulfilling the same properties as the general function φ considered in the paper. For f∈C[0,1], x∈[0,1], and n>0, we consider the following operators defined from the classical Bernstein operators Bn:(3.17)Bnτf(x)=∑k=0n(nk)τ(x)k(1-τ(x))n-k(f∘τ-1)(kn).
This sequence of linear operators was studied by the authors in [13]. Here we show a nice way to obtain its asymptotic formula.Corollary 3.4.
For allf∈C[0,1] and  x∈(0,1), whenever  f′′(x) exists,
(3.18)lim⁡n→+∞n(Bnτf(x)-f(x))=-τ(x)(1-τ(x))τ′′(x)2τ′(x)3f′(x)+τ(x)(1-τ(x))2τ′(x)2f′′(x).Proof.
It follows the same pattern as the proof of the previous corollaries. It suffices to make use of the following identities which one can obtain directly from the corresponding ones for the Bernstein operators (i.e.,Bneix(x),i=1,2,4; see, e.g., [2]):
(3.19)Bnτeτ,1x(x)=0,Bnτeτ,2x(x)=τ(x)(1-τ(x))n,Bnτeτ,4x(x)=3(1-τ(x))2τ(x)2n2+τ(x)(1-τ(x))(1-6τ(x)+6τ(x)2)n3.


### 3.4. The Modified Bernstein Operators Which Preservexj

This section deals with the family of sequences of positive linear polynomial operatorsBn,0,j,j=1,2,…, presented in [4] and defined for f∈C[0,1] and n≥j as(3.20)Bn,0,jf(x)=∑k=0nf((k(k-1)⋯(k-j+1)n(n-1)⋯(n-j+1))1/j)(nk)xk(1-x)n-k.The first two elementsBn,0,1 and Bn,0,2 are, respectively, the Bernstein operators and those ones studied in Section 3.1. The operator Bn,0,j holds fixed the functions e0 and ej.The next corollary deals withBn,0,3 and shows an application of Theorem 2.1 with φ=e3.Corollary 3.5.
For allf∈C[0,1] and x∈(0,1),  whenever f′′(x) exists,
(3.21)lim⁡n→+∞n(Bn,0,3f(x)-f(x))=x(1-x)2f′′(x)-(1-x)f′(x).Proof.
It follows the same pattern as Corollary3.2 although some more cumbersome calculations, which we have carried out with the use of Mathematica, are required. We detail below the identities that allow us to end the proof. We make use of the notation
(3.22)(n-3)(i)=(n-3)(n-4)⋯(n-i-2),Bn,0,3e0(x)=1,Bn,0,3e3(x)=x3,Bn,0,3e6(x)=(n-3)(3)n(n-1)(n-2)x6+9(n-3)(2)n(n-1)(n-2)x5+18(n-3)n(n-1)(n-2)x4+6n(n-1)(n-2)x3,Bn,0,3e9(x)=(n-3)(6)n2(n-1)2(n-2)2x9+27(n-3)(5)n2(n-1)2(n-2)2x8+243(n-3)(4)n2(n-1)2(n-2)2x7+862(n-3)(3)n2(n-1)2(n-2)2x6+1242(n-3)(2)n2(n-1)2(n-2)2x5+540(n-3)n2(n-1)2(n-2)2x4+36n2(n-1)2(n-2)2x3,Bn,0,3e12(x)=(n-3)(9)n3(n-1)3(n-2)3x12+54(n-3)(8)n3(n-1)3(n-2)3x11+1107(n-3)(7)n3(n-1)3(n-2)3x10+11025(n-3)(6)n3(n-1)3(n-2)3x9+56808(n-3)(5)n3(n-1)3(n-2)3x8+149580(n-3)(4)n3(n-1)3(n-2)3x7+186876(n-3)(3)n3(n-1)3(n-2)3x6+94284(n-3)(2)n3(n-1)3(n-2)3x5+13608(n-3)n3(n-1)3(n-2)3x4+216n3(n-1)3(n-2)3x3.
For i=1,2,4 the quantities Bn,0,3eφ,ix=Bn,0,3ee3,ix, required to apply Theorem 2.1, appear after some calculations using the following identities:
(3.23)eφ,1x=ee3,1x=e3-x3e0,eφ,2x=ee3,2x=e6-2x3e3+x6e0,eφ,4x=ee3,4x=e12-4x3e9+6x6e6-4x9e3+x12e0.Finally, motivated by the well-known Voronovskaja formula for the classical Bernstein operators and by the results in Corollaries3.2 and 3.5, we close this section and the paper stating the following conjecture.Conjecture 3.6.
For allf∈C[0,1],x∈(0,1), and j∈{4,5,…},  whenever f′′(x) exists,
(3.24)lim⁡n→+∞n(Bn,0,jf(x)-f(x))=x(1-x)2f′′(x)-(j-1)(1-x)2f′(x).


## 3.1. Modified Bernstein Operators Which Preservex2

This section deals with the following sequence of operators presented in [4] (we use the same notation) as a byproduct of some interesting results, defined for f∈C[0,1] and n>1 as(3.1)Bn,0,2f(x)=∑k=0nf(k(k-1)n(n-1))(nk)xk(1-x)n-k.
As pointed out in [13], this provides an example of a sequence of positive linear polynomial operators that preserve e0 and e2 and represents an approximation process for functions f∈C[0,1].The presence of the square root in the definition makes it difficult to obtain easy-to-handle expressions for the momentsBn,0,2eix(x),i=0,1,2,4, and consequently to obtain an asymptotic formula. We will apply our theorem to get it, though we first prove a quantitative result missed in [4].Proposition 3.1.
Letf∈C[0,1],  x∈[0,1], and let δ>0. Then
(3.2)|Bn,0,2f(x)-f(x)|≤ω(f,δ)(1+1δ2x(1-x)1-(1-x)n-1n-1).Proof.
From the usual quantitative estimate in terms ofω(f,·) stated in [15], we can write
(3.3)|Bn,0,2f(x)-f(x)|≤ω(f,δ)(1+1δ2x(x-Bn,0,2e1(x))).
Now, for n>1, (k-1)/(n-1)≤k/n, so k(k-1)/n(n-1)≤k/n and Bn,0,2e1≤Bne1=e1. Thus
(3.4)x-Bn,0,2e1(x)≥0.
On the other hand, for n>1, (1/2)(k/n+(k-1)/(n-1))≥k(k-1)/n(n-1)≥(k-1)/(n-1), so
(3.5)12∑k=1n(kn+k-1n-1)(nk)xk(1-x)n-k≥Bn,0,2e1(x)≥∑k=1nk-1n-1(nk)xk(1-x)n-k,
and using the equalities
(3.6)∑k=1nkn(nk)xk(1-x)n-k=x,∑k=1nk-1n-1(nk)xk(1-x)n-k=nn-1∑k=1n(kn-1n)(nk)xk(1-x)n-k=nx-1n-1+(1-x)nn-1,
we get
(3.7)1-x2(n-1)(1-(1-x)n-1)≤x-Bn,0,2e1(x)≤1-xn-1(1-(1-x)n-1),
from which the result follows.Corollary 3.2.
For allf∈C[0,1] and x∈(0,1), whenever f′′(x) exists,
(3.8)lim⁡n→∞n(Bn,0,2f(x)-f(x))=-1-x2f′(x)+(1-x)x2f′′(x).Proof.
We will apply the theorem withφ=e2,λn=n,q(x)=-(1-x)/2, and p(x)=(1-x)x/2. As the operators Bn,0,2 preserve the constants, it suffices to check that (2.4) holds true.
First computations yield(3.9)Bn,0,2e0(x)=1,Bn,0,2e2(x)=x2,Bn,0,2e4(x)=2n(n-1)x2+4(n-2)n(n-1)x3+(n-3)(n-2)n(n-1)x4,Bn,0,2e6(x)=4n2(n-1)2x2+32(n-2)n2(n-1)2x3+38(n-3)(n-2)n2(n-1)2x4+12(n-4)(n-3)(n-2)n2(n-1)2x5+(n-5)(n-4)(n-3)(n-2)n2(n-1)2x6,Bn,0,2e8(x)=8n3(n-1)3x2+208(n-2)n3(n-1)3x3+652(n-2)(n-3)n3(n-1)3x4+576(n-4)(n-3)(n-2)n3(n-1)3x5+188(n-5)(n-4)(n-3)(n-2)n3(n-1)3x6+24(n-6)(n-5)(n-4)(n-3)(n-2)n3(n-1)3x7+(n-7)(n-6)(n-5)(n-4)(n-3)(n-2)n3(n-1)3x8.
Thus for i=1,2,4 the quantities Bn,0,2eφ,ix appear after some calculations using the following identities:
(3.10)eφ,1x=ee2,1x=e2-x2e0,eφ,2x=ee2,2x=e4-2x2e2+x4e0,eφ,4x=ee2,4x=e8-4x2e6+6x4e4-4x6e2+x8e0,Bn,0,2eφ,1x(x)=0,Bn,0,2eφ,2x(x)=2(1-x)x2(n-1)n(1-3x+2nx),Bn,0,2eφ,4x(x)=4(1-x)x2(n-1)3n3(n4(12-12x)x4+n3(112-324x+216x2)x3+n2(159-1041x+1881x2-1011x3)x2+n(52-759x+2921x2-4089x3+1887x4)x+(2-102x+876x2-2580x3+3060x4-1260x5)).
Finally we are in a position to apply Theorem2.1 and then prove the corollary, since we show below that assumptions in (2.4) are fulfilled:
(3.11)lim⁡n→∞nBn,0,2eφ,1x(x)=0=2xq(x)+2p(x)=q(x)φ′(x)+p(x)φ′′(x),lim⁡n→∞nBn,0,2eφ,2x(x)=4(1-x)x3=8x2p(x)=2p(x)φ′(x)2,lim⁡n→∞nBn,0,2eφ,4x(x)=0.


## 3.2. The Modified Meyer-König and Zeller Operators

Forf∈C[0,1], x∈[0,1], and n>1, we consider the operators defined as(3.12)Rnf(x)=∑k=0∞f(k(k-1)(n+k)(n+k-1))(n+kk)xk(1-x)n+1,
for x<1 and Rnf(1)=f(1).They were introduced in [5] as a modification of the well-known modified Meyer-König and Zeller operators (see [6]).It turns to be another example of a sequence of positive linear operators that preservee0 and e2 and represents an approximation process for functions f∈C[0,1].Here again, the presence of the square root in the definition makes it difficult to obtain easy-to-handle expressions forRneix(x), i=0,1,2,4, and consequently to obtain by usual means an asymptotic formula. With this aim we shall apply our theorem.Corollary 3.3.
For allf∈C[0,1] and  x∈(0,1), whenever  f′′(x) exists,
(3.13)lim⁡n→∞n(Rnf(x)-f(x))=-(1-x)22f′(x)+x(1-x)22f′′(x).Proof.
We will apply the theorem withφ=e2,λn=n,q(x)=-(1-x)2/2 and p(x)=x(1-x)2/2. As the operators Rn preserve the constants, it suffices to check that (2.4) holds true.
Direct computations with the use of mathematical software (Mathematica) give(3.14)Rne0(x)=1,Rne2(x)=x2,Rne4(x)=4nx3(2+n)(3+n)+n2(1+n)x4(2+n)(3+n)(4+n)+4n3x5(2+n)(3+n)(4+n)(5+n)+O(n-2),Rne6(x)=12n8x5(1+n)(2+n)2(3+n)2(4+n)2(5+n)2+n10(17+n)x6(1+n)(2+n)2(3+n)2(4+n)2(5+n)2(6+n)2+12n12x7(1+n)(2+n)2(3+n)2(4+n)2(5+n)2(6+n)2(7+n)2+O(n-2),Rne8(x)=24n19x7(1+n)2(2+n)3(3+n)3(4+n)3(5+n)3(6+n)3(7+n)3+n22(59+n)x8(1+n)2(2+n)3(3+n)3(4+n)3(5+n)3(6+n)3(7+n)3(8+n)3+24n25x9(1+n)2(2+n)3(3+n)3(4+n)3(5+n)3(6+n)3(7+n)3(8+n)3(9+n)3+O(n-2).
Now, using again (3.10) we can write
(3.15)Rneφ,1x(x)=0,Rneφ,2x(x)=4x3n3(1-x)2(2+n)(3+n)(4+n)(5+n)+O(n-2),Rneφ,4x(x)=n25x7(24+59x+24x2)(1+n)2(2+n)3(3+n)3(4+n)3(5+n)3(6+n)3(7+n)3(8+n)3(9+n)3+3n22x8(1+n)2(2+n)3(3+n)3(4+n)3(5+n)3(6+n)3(7+n)3(8+n)3-4n12x7(12+17x+12x2)(1+n)(2+n)2(3+n)2(4+n)2(5+n)2(6+n)2(7+n)2+6n3x7(4+x+4x2)(2+n)(3+n)(4+n)(5+n)+O(n-2)
and then the following identities which prove the corollary:
(3.16)lim⁡n→∞nRneφ,1x(x)=0=2xq(x)+2p(x)=q(x)φ′(x)+p(x)φ′′(x),lim⁡n→∞nRneφ,2x(x)=4x3(1-x)2=8x2p(x)=2p(x)φ′(x)2,lim⁡n→∞nMn,4eφ,4x(x)=0.


## 3.3. The Modified Bernstein Operators Which Preserve a General Functionτ

Letτ be any function fulfilling the same properties as the general function φ considered in the paper. For f∈C[0,1], x∈[0,1], and n>0, we consider the following operators defined from the classical Bernstein operators Bn:(3.17)Bnτf(x)=∑k=0n(nk)τ(x)k(1-τ(x))n-k(f∘τ-1)(kn).
This sequence of linear operators was studied by the authors in [13]. Here we show a nice way to obtain its asymptotic formula.Corollary 3.4.
For allf∈C[0,1] and  x∈(0,1), whenever  f′′(x) exists,
(3.18)lim⁡n→+∞n(Bnτf(x)-f(x))=-τ(x)(1-τ(x))τ′′(x)2τ′(x)3f′(x)+τ(x)(1-τ(x))2τ′(x)2f′′(x).Proof.
It follows the same pattern as the proof of the previous corollaries. It suffices to make use of the following identities which one can obtain directly from the corresponding ones for the Bernstein operators (i.e.,Bneix(x),i=1,2,4; see, e.g., [2]):
(3.19)Bnτeτ,1x(x)=0,Bnτeτ,2x(x)=τ(x)(1-τ(x))n,Bnτeτ,4x(x)=3(1-τ(x))2τ(x)2n2+τ(x)(1-τ(x))(1-6τ(x)+6τ(x)2)n3.


## 3.4. The Modified Bernstein Operators Which Preservexj

This section deals with the family of sequences of positive linear polynomial operatorsBn,0,j,j=1,2,…, presented in [4] and defined for f∈C[0,1] and n≥j as(3.20)Bn,0,jf(x)=∑k=0nf((k(k-1)⋯(k-j+1)n(n-1)⋯(n-j+1))1/j)(nk)xk(1-x)n-k.The first two elementsBn,0,1 and Bn,0,2 are, respectively, the Bernstein operators and those ones studied in Section 3.1. The operator Bn,0,j holds fixed the functions e0 and ej.The next corollary deals withBn,0,3 and shows an application of Theorem 2.1 with φ=e3.Corollary 3.5.
For allf∈C[0,1] and x∈(0,1),  whenever f′′(x) exists,
(3.21)lim⁡n→+∞n(Bn,0,3f(x)-f(x))=x(1-x)2f′′(x)-(1-x)f′(x).Proof.
It follows the same pattern as Corollary3.2 although some more cumbersome calculations, which we have carried out with the use of Mathematica, are required. We detail below the identities that allow us to end the proof. We make use of the notation
(3.22)(n-3)(i)=(n-3)(n-4)⋯(n-i-2),Bn,0,3e0(x)=1,Bn,0,3e3(x)=x3,Bn,0,3e6(x)=(n-3)(3)n(n-1)(n-2)x6+9(n-3)(2)n(n-1)(n-2)x5+18(n-3)n(n-1)(n-2)x4+6n(n-1)(n-2)x3,Bn,0,3e9(x)=(n-3)(6)n2(n-1)2(n-2)2x9+27(n-3)(5)n2(n-1)2(n-2)2x8+243(n-3)(4)n2(n-1)2(n-2)2x7+862(n-3)(3)n2(n-1)2(n-2)2x6+1242(n-3)(2)n2(n-1)2(n-2)2x5+540(n-3)n2(n-1)2(n-2)2x4+36n2(n-1)2(n-2)2x3,Bn,0,3e12(x)=(n-3)(9)n3(n-1)3(n-2)3x12+54(n-3)(8)n3(n-1)3(n-2)3x11+1107(n-3)(7)n3(n-1)3(n-2)3x10+11025(n-3)(6)n3(n-1)3(n-2)3x9+56808(n-3)(5)n3(n-1)3(n-2)3x8+149580(n-3)(4)n3(n-1)3(n-2)3x7+186876(n-3)(3)n3(n-1)3(n-2)3x6+94284(n-3)(2)n3(n-1)3(n-2)3x5+13608(n-3)n3(n-1)3(n-2)3x4+216n3(n-1)3(n-2)3x3.
For i=1,2,4 the quantities Bn,0,3eφ,ix=Bn,0,3ee3,ix, required to apply Theorem 2.1, appear after some calculations using the following identities:
(3.23)eφ,1x=ee3,1x=e3-x3e0,eφ,2x=ee3,2x=e6-2x3e3+x6e0,eφ,4x=ee3,4x=e12-4x3e9+6x6e6-4x9e3+x12e0.Finally, motivated by the well-known Voronovskaja formula for the classical Bernstein operators and by the results in Corollaries3.2 and 3.5, we close this section and the paper stating the following conjecture.Conjecture 3.6.
For allf∈C[0,1],x∈(0,1), and j∈{4,5,…},  whenever f′′(x) exists,
(3.24)lim⁡n→+∞n(Bn,0,jf(x)-f(x))=x(1-x)2f′′(x)-(j-1)(1-x)2f′(x).



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*Source: 217464-2012-05-07.xml*