Asymptotic Formulae via a Korovkin-Type Result

Abstract

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

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, say 𝐿_𝑛, one usually intends to prove firstly that the sequence defines an approximation process; that is, for each function 𝑓 of some space, 𝐿_𝑛𝑓 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, thenlim_𝑛→∞𝑛𝐵_𝑛𝑓(𝑥)(𝑥)=𝑥()−𝑓1−𝑥2𝑓^′′(𝑥). (1.1) This work dwells upon this type of expressions.

A classical key ingredient to prove an asymptotic formula for a sequence of positive linear operators 𝐿_𝑛 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,𝐿_𝑛𝑒^𝑥_𝑖(𝑥)=𝐿_𝑛𝑒₁−𝑥𝑒₀^𝑖(𝑥),𝑖=0,1,2,4, (1.2) (𝑒_𝑖 and 𝑒^𝑥_𝑖 denote the monomials 𝑒_𝑖(𝑡)=𝑡^𝑖 and 𝑒^𝑥_𝑖(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 (Theorem 2.1 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

Let 𝜑 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 𝑒^𝑥_𝜑,𝑖 the function𝑒^𝑥_𝜑,𝑖(𝑡)=𝜑(𝑡)−𝜑(𝑥)^𝑖, (2.1) and 𝐷^𝑖 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 space 𝐶[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. Let 𝐿_𝑛∶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 𝜆_𝑛→+∞ (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}lim_𝑛→+∞𝜆_𝑛𝐿_𝑛𝑒^𝑥_𝜑,𝑖(𝑥)−𝑒^𝑥_𝜑,𝑖(𝑥)=𝑝(𝑥)𝐷²𝑒^𝑥_𝜑,𝑖(𝑥)+𝑞(𝑥)𝐷¹𝑒^𝑥_𝜑,𝑖(𝑥). (2.2) Then for each 𝑓∈𝐶[0,1], twice differentiable at the point 𝑥, lim_𝑛→+∞𝜆_𝑛𝐿_𝑛𝑓(𝑥)−𝑓(𝑥)=𝑝(𝑥)𝑓^′′(𝑥)+𝑞(𝑥)𝑓^′(𝑥). (2.3)

Remark 2.2. Notice that for 𝑖=0, identity (2.2) becomes lim_𝑛→+∞𝜆_𝑛(𝐿_𝑛𝑒₀(𝑥)−1)=0, which is obviously fulfilled if the operators 𝐿_𝑛 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, for 𝑖=1,2,4, identity (2.2) becomes, respectively, lim_𝑛→+∞𝜆_𝑛𝐿_𝑛𝑒^𝑥_𝜑,1(𝑥)=𝑝(𝑥)𝜑^′′(𝑥)+𝑞(𝑥)𝜑^′(𝑥),lim_𝑛→+∞𝜆_𝑛𝐿_𝑛𝑒^𝑥_𝜑,2(𝑥)=2𝑝(𝑥)𝜑′(𝑥)²,lim_𝑛→+∞𝜆_𝑛𝐿_𝑛𝑒^𝑥_𝜑,4(𝑥)=0. (2.4) 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 𝜆_𝑛(𝐿_𝑛𝑓(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 function 𝑓∘𝜑⁻¹, yields for 𝑡∈[0,1] that 𝑓(𝑡)=𝑓∘𝜑⁻¹𝜑(𝑡)=𝑓∘𝜑⁻¹𝜑(𝑥)+𝐷¹𝑓∘𝜑⁻¹𝜑(𝑥𝜑(𝑡)(𝑥+𝐷)−𝜑)₂𝑓∘𝜑⁻¹𝜑(𝑥)2𝜑(𝑡)−𝜑(𝑥)²+ℎ(𝑡−𝑥)𝜑(𝑡)−𝜑(𝑥)², (2.5) where ℎ is a continuous function which vanishes at 0. Equivalently we can write for 𝑡∈[0,1]𝑓(𝑡)=𝑓(𝑥)+𝐷¹𝑓∘𝜑⁻¹𝜑(𝑥)𝑒^𝑥_𝜑,1(𝑡)+𝐷²𝑓∘𝜑⁻¹𝜑(𝑥)2𝑒^𝑥_𝜑,2(𝑡)+ℎ_𝑥(𝑡)𝑒^𝑥_𝜑,2(𝑡), (2.6) where ℎ_𝑥(𝑡)∶=ℎ(𝑡−𝑥). Applying the operator 𝐿_𝑛 and then evaluating at the fixed point 𝑥, we obtain the equality 𝐿_𝑛𝑓(𝑥)=𝑓(𝑥)𝐿_𝑛𝑒^𝑥_𝜑,0(𝑥)+𝐷¹𝑓∘𝜑⁻¹𝜑(𝑥)𝐿_𝑛𝑒^𝑥_𝜑,1(𝑥)+𝐷²𝑓∘𝜑⁻¹𝜑(𝑥)2𝐿_𝑛𝑒^𝑥_𝜑,2(𝑥)+𝐿_𝑛ℎ_𝑥𝑒^𝑥_𝜑,2(𝑥). (2.7) Now we subtract 𝑓(𝑥) from both sides and multiply by 𝜆_𝑛 to get 𝜆_𝑛𝐿_𝑛𝑓(𝑥)−𝑓(𝑥)−𝜆_𝑛𝐿_𝑛ℎ_𝑥𝑒^𝑥_𝜑,2(𝑥)=𝑓(𝑥)𝜆_𝑛𝐿_𝑛𝑒^𝑥_𝜑,0(𝑥)−1+𝐷¹𝑓∘𝜑⁻¹𝜑(𝑥)𝜆_𝑛𝐿_𝑛𝑒^𝑥_𝜑,1(𝑥)+𝐷²𝑓∘𝜑⁻¹𝜑(𝑥)2𝜆_𝑛𝐿_𝑛𝑒^𝑥_𝜑,2(𝑥). (2.8) Taking into account the basic identities 𝐷¹𝑓∘𝜑⁻¹𝜑(𝑥)=𝑓^′(𝑥)𝜑^′(𝑥),𝐷²𝑓∘𝜑⁻¹𝜑(𝑥)=1𝜑^′(𝑥)𝑓^′′(𝑥)𝜑^′(𝑥)−𝑓^′(𝑥)𝜑^′′(𝑥)𝜑^′(𝑥)² (2.9) and using (2.4), we derive that lim_𝑛→+∞𝜆_𝑛𝐿_𝑛𝑓(𝑥)−𝑓(𝑥)−𝜆_𝑛𝐿_𝑛ℎ_𝑥𝑒^𝑥_𝜑,2(𝑥)=𝑝(𝑥)𝑓^′′(𝑥)+𝑞(𝑥)𝑓^′(𝑥). (2.10) The proof will be over once we prove that 𝜆_𝑛𝐿_𝑛(ℎ_𝑥𝑒^𝑥_𝜑,2)(𝑥)→0 as 𝑛→+∞.
To this purpose let 𝜖>0 and let 𝜃_𝑥 be an open set containing 𝑥 such that for 𝑡∈𝜃_𝑥, |ℎ_𝑥(𝑡)|<𝜖. Then if we define 𝑤(𝑡)∶=max{0,|ℎ_𝑥(𝑡)|−𝜖}𝑒^𝑥_𝜑,2(𝑡), we have that for all 𝑡∈[0,1]ℎ_𝑥(𝑡)𝑒^𝑥_𝜑,2(𝑡)≤𝜖𝑒^𝑥_𝜑,2(𝑡)+max0,ℎ_𝑥(𝑡)−𝜖𝑒^𝑥_𝜑,2(𝑡)=𝜖𝑒^𝑥_𝜑,2(𝑡)+𝑤(𝑡). (2.11) On the other hand, 𝑤 vanishes on 𝜃_𝑥, so there is a constant 𝑀 such that for all 𝑡∈[0,1], |𝑤(𝑡)|≤𝑀𝑒^𝑥_𝜑,4(𝑡). (2.12)
Finally, the linearity and positivity of 𝐿_𝑛 allow us to write, from (2.11) and (2.12), 𝜆_𝑛𝐿_𝑛ℎ_𝑥𝑒^𝑥_𝜑,2(𝑥)≤𝜖𝜆_𝑛𝐿_𝑛𝑒^𝑥_𝜑,2(𝑥)+𝜆_𝑛𝑀𝐿_𝑛𝑒^𝑥_𝜑,4(𝑥), (2.13) from where taking limits and using again (2.4), limsup_𝑛→+∞𝜆_𝑛𝐿_𝑛ℎ_𝑥𝑒^𝑥_𝜑,2(𝑥)≤2𝜖𝑝(𝑥)𝜑′(𝑥)². (2.14) 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 with 𝜆_𝑛=𝑛 and 𝜑=𝑒₂, enters the scene.

Two further applications with different values of 𝜑 are presented afterwards in less detail.

3.1. Modified Bernstein Operators Which Preserve 𝑥²

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 as𝐵_𝑛,0,2𝑓(𝑥)=^𝑛_𝑘=0𝑓𝑘(𝑘−1)𝑛()𝑛𝑘𝑥𝑛−1^𝑘(1−𝑥)^𝑛−𝑘. (3.1) 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 moments 𝐵_𝑛,0,2𝑒^𝑥_𝑖(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. Let 𝑓∈𝐶[0,1], 𝑥∈[0,1], and let 𝛿>0. Then 𝐵_𝑛,0,2𝑓(𝑥)(𝑥1−𝑓)≤𝜔𝑓,𝛿1+𝛿()()2𝑥1−𝑥1−1−𝑥^𝑛−1𝑛−1. (3.2)

Proof. From the usual quantitative estimate in terms of 𝜔(𝑓,⋅) stated in [15], we can write 𝐵_𝑛,0,2𝑓(𝑥)(𝑥1−𝑓)≤𝜔𝑓,𝛿1+𝛿2𝑥𝑥−𝐵_𝑛,0,2𝑒₁(𝑥). (3.3) Now, for 𝑛>1, (𝑘−1)/(𝑛−1)≤𝑘/𝑛, so 𝑘(𝑘−1)/𝑛(𝑛−1)≤𝑘/𝑛 and 𝐵_𝑛,0,2𝑒₁≤𝐵_𝑛𝑒₁=𝑒₁. Thus 𝑥−𝐵_𝑛,0,2𝑒₁(𝑥)≥0. (3.4) 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 12^𝑛_𝑘=1𝑘𝑛+𝑘−1𝑛𝑘𝑥𝑛−1^𝑘(1−𝑥)^𝑛−𝑘≥𝐵_𝑛,0,2𝑒₁(𝑥)≥^𝑛_𝑘=1𝑘−1𝑛𝑘𝑥𝑛−1^𝑘(1−𝑥)^𝑛−𝑘, (3.5) and using the equalities ^𝑛_𝑘=1𝑘𝑛𝑛𝑘𝑥^𝑘(1−𝑥)^𝑛−𝑘=𝑥,_𝑛𝑘=1𝑘−1𝑛𝑘𝑥𝑛−1^𝑘(1−𝑥)^𝑛−𝑘=𝑛𝑛−1^𝑛_𝑘=1𝑘𝑛−1𝑛𝑛𝑘𝑥^𝑘(1−𝑥)^𝑛−𝑘=𝑛𝑥−1+()𝑛−11−𝑥^𝑛,𝑛−1 (3.6) we get 1−𝑥2()()𝑛−11−1−𝑥^𝑛−1≤𝑥−𝐵_𝑛,0,2𝑒₁(𝑥)≤1−𝑥()𝑛−11−1−𝑥^𝑛−1, (3.7) from which the result follows.

Corollary 3.2. For all 𝑓∈𝐶[0,1] and 𝑥∈(0,1), whenever 𝑓^′′(𝑥) exists, lim_𝑛→∞𝑛𝐵_𝑛,0,2𝑓(𝑥)−𝑓(𝑥)=−1−𝑥2𝑓^′(𝑥)+(1−𝑥)𝑥2𝑓^′′(𝑥). (3.8)

Proof. We will apply the theorem with 𝜑=𝑒₂,𝜆_𝑛=𝑛,𝑞(𝑥)=−(1−𝑥)/2, and 𝑝(𝑥)=(1−𝑥)𝑥/2. As the operators 𝐵_𝑛,0,2 preserve the constants, it suffices to check that (2.4) holds true.
First computations yield 𝐵_𝑛,0,2𝑒₀(𝑥)𝐵=1,_𝑛,0,2𝑒₂(𝑥)=𝑥²,𝐵_𝑛,0,2𝑒₄(𝑥)=2𝑛()𝑥𝑛−1²+4(𝑛−2)𝑛()𝑥𝑛−1³+(𝑛−3)(𝑛−2)𝑛()𝑥𝑛−1⁴,𝐵_𝑛,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,𝐵_𝑛,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. (3.9) Thus for 𝑖=1,2,4 the quantities 𝐵_𝑛,0,2𝑒^𝑥_𝜑,𝑖 appear after some calculations using the following identities: 𝑒^𝑥_𝜑,1=𝑒^𝑥_𝑒2,1=𝑒₂−𝑥²𝑒₀,𝑒^𝑥_𝜑,2=𝑒^𝑥_𝑒2,2=𝑒₄−2𝑥²𝑒₂+𝑥⁴𝑒₀,𝑒^𝑥_𝜑,4=𝑒^𝑥_𝑒2,4=𝑒₈−4𝑥²𝑒₆+6𝑥⁴𝑒₄−4𝑥⁶𝑒₂+𝑥⁸𝑒₀,𝐵_𝑛,0,2𝑒^𝑥_𝜑,1(𝑥)𝐵=0,_𝑛,0,2𝑒^𝑥_𝜑,2(𝑥)=2(1−𝑥)𝑥²()𝑛(),𝐵𝑛−11−3𝑥+2𝑛𝑥_𝑛,0,2𝑒^𝑥_𝜑,4(𝑥)=4(1−𝑥)𝑥²(𝑛−1)³𝑛³𝑛⁴(12−12𝑥)𝑥⁴+𝑛³112−324𝑥+216𝑥²𝑥³+𝑛²159−1041𝑥+1881𝑥²−1011𝑥³𝑥²+𝑛52−759𝑥+2921𝑥²−4089𝑥³+1887𝑥⁴𝑥+2−102𝑥+876𝑥²−2580𝑥³+3060𝑥⁴−1260𝑥⁵. (3.10)
Finally we are in a position to apply Theorem 2.1 and then prove the corollary, since we show below that assumptions in (2.4) are fulfilled: lim_𝑛→∞𝑛𝐵_𝑛,0,2𝑒^𝑥_𝜑,1(𝑥)=0=2𝑥𝑞(𝑥)+2𝑝(𝑥)=𝑞(𝑥)𝜑^′(𝑥)+𝑝(𝑥)𝜑^′′(𝑥),lim_𝑛→∞𝑛𝐵_𝑛,0,2𝑒^𝑥_𝜑,2(𝑥)=4(1−𝑥)𝑥³=8𝑥²𝑝(𝑥)=2𝑝(𝑥)𝜑′(𝑥)²,lim_𝑛→∞𝑛𝐵_𝑛,0,2𝑒^𝑥_𝜑,4(𝑥)=0. (3.11)

3.2. The Modified Meyer-König and Zeller Operators

For 𝑓∈𝐶[0,1], 𝑥∈[0,1], and 𝑛>1, we consider the operators defined as𝑅_𝑛𝑓(𝑥)=^∞_𝑘=0𝑓𝑘(𝑘−1)()𝑘𝑥𝑛+𝑘)(𝑛+𝑘−1𝑛+𝑘_𝑘(1−𝑥)^𝑛+1, (3.12) for 𝑥<1 and 𝑅_𝑛𝑓(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 preserve 𝑒₀ 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 for 𝑅_𝑛𝑒^𝑥_𝑖(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 all 𝑓∈𝐶[0,1] and 𝑥∈(0,1), whenever 𝑓^′′(𝑥) exists, lim_𝑛→∞𝑛𝑅_𝑛𝑓(𝑥)(𝑥)()−𝑓=−1−𝑥²2𝑓^′(𝑥)+𝑥(1−𝑥)²2𝑓^′′(𝑥). (3.13)

Proof. We will apply the theorem with 𝜑=𝑒₂,𝜆_𝑛=𝑛,𝑞(𝑥)=−(1−𝑥)²/2 and 𝑝(𝑥)=𝑥(1−𝑥)²/2. As the operators 𝑅_𝑛 preserve the constants, it suffices to check that (2.4) holds true.
Direct computations with the use of mathematical software (Mathematica) give 𝑅_𝑛𝑒₀(𝑥)𝑅=1,_𝑛𝑒₂(𝑥)=𝑥²,𝑅_𝑛𝑒₄(𝑥)=4𝑛𝑥³()+𝑛2+𝑛)(3+𝑛²(1+𝑛)𝑥⁴()+2+𝑛)(3+𝑛)(4+𝑛4𝑛³𝑥⁵()𝑛2+𝑛)(3+𝑛)(4+𝑛)(5+𝑛+𝒪⁻²,𝑅_𝑛𝑒₆(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+𝒪𝑛⁻²,𝑅_𝑛𝑒₈(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+𝒪𝑛⁻². (3.14) Now, using again (3.10) we can write 𝑅_𝑛𝑒^𝑥_𝜑,1(𝑥)𝑅=0,_𝑛𝑒^𝑥_𝜑,2(𝑥)=4𝑥³𝑛³(1−𝑥)²()𝑛2+𝑛)(3+𝑛)(4+𝑛)(5+𝑛+𝒪⁻²,𝑅_𝑛𝑒^𝑥_𝜑,4(𝑥)=𝑛²⁵𝑥⁷24+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)³−4𝑛¹²𝑥⁷12+17𝑥+12𝑥²(1+𝑛)(2+𝑛)²(3+𝑛)²(4+𝑛)²(5+𝑛)²(6+𝑛)²(7+𝑛)²+6𝑛³𝑥⁷4+𝑥+4𝑥²()𝑛2+𝑛)(3+𝑛)(4+𝑛)(5+𝑛+𝒪⁻² (3.15) and then the following identities which prove the corollary: lim_𝑛→∞𝑛𝑅_𝑛𝑒^𝑥_𝜑,1(𝑥)=0=2𝑥𝑞(𝑥)+2𝑝(𝑥)=𝑞(𝑥)𝜑^′(𝑥)+𝑝(𝑥)𝜑^′′(𝑥),lim_𝑛→∞𝑛𝑅_𝑛𝑒^𝑥_𝜑,2(𝑥)=4𝑥³(1−𝑥)²=8𝑥²𝑝(𝑥)=2𝑝(𝑥)𝜑′(𝑥)²,lim_𝑛→∞𝑛𝑀_𝑛,4𝑒^𝑥_𝜑,4(𝑥)=0. (3.16)

3.3. The Modified Bernstein Operators Which Preserve a General Function 𝜏

Let 𝜏 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 𝐵_𝑛:𝐵^𝜏_𝑛𝑓(𝑥)=^𝑛_𝑘=0𝑛𝑘𝜏(𝑥)^𝑘(1−𝜏(𝑥))^𝑛−𝑘𝑓∘𝜏⁻¹𝑘𝑛. (3.17) 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 all 𝑓∈𝐶[0,1] and 𝑥∈(0,1), whenever 𝑓^′′(𝑥) exists, lim_𝑛→+∞𝑛𝐵^𝜏_𝑛𝑓(𝑥)(𝑥)𝜏(𝑥(𝑥𝜏−𝑓=−)(1−𝜏))^′′(𝑥)2𝜏′(𝑥)³𝑓^′(𝑥)+𝜏(𝑥)(1−𝜏(𝑥))2𝜏^′(𝑥)²𝑓^′′(𝑥). (3.18)

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., 𝐵_𝑛𝑒^𝑥_𝑖(𝑥),𝑖=1,2,4; see, e.g., [2]): 𝐵^𝜏_𝑛𝑒^𝑥_𝜏,1(𝑥)𝐵=0,_𝜏𝑛𝑒^𝑥_𝜏,2(𝑥)=𝜏(𝑥)(1−𝜏(𝑥))𝑛,𝐵^𝜏_𝑛𝑒^𝑥_𝜏,4(𝑥)=3(1−𝜏(𝑥))²𝜏(𝑥)²𝑛²+𝜏(𝑥)(1−𝜏(𝑥))1−6𝜏(𝑥)+6𝜏(𝑥)²𝑛³. (3.19)

3.4. The Modified Bernstein Operators Which Preserve 𝑥^𝑗

This section deals with the family of sequences of positive linear polynomial operators 𝐵_𝑛,0,𝑗,𝑗=1,2,…, presented in [4] and defined for 𝑓∈𝐶[0,1] and 𝑛≥𝑗 as𝐵_𝑛,0,𝑗𝑓(𝑥)=^𝑛_𝑘=0𝑓𝑘(𝑘−1)⋯𝑘−𝑗+1𝑛(𝑛−1)⋯𝑛−𝑗+1^1/𝑗𝑛𝑘𝑥^𝑘(1−𝑥)^𝑛−𝑘. (3.20)

The first two elements 𝐵_𝑛,0,1 and 𝐵_𝑛,0,2 are, respectively, the Bernstein operators and those ones studied in Section 3.1. The operator 𝐵_𝑛,0,𝑗 holds fixed the functions 𝑒₀ and 𝑒_𝑗.

The next corollary deals with 𝐵_𝑛,0,3 and shows an application of Theorem 2.1 with 𝜑=𝑒₃.

Corollary 3.5. For all 𝑓∈𝐶[0,1] and 𝑥∈(0,1), whenever 𝑓^′′(𝑥) exists, lim_𝑛→+∞𝑛𝐵_𝑛,0,3𝑓(𝑥)(𝑥)=𝑥()−𝑓1−𝑥2𝑓^′′(𝑥)−(1−𝑥)𝑓^′(𝑥). (3.21)

Proof. It follows the same pattern as Corollary 3.2 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 (𝑛−3)^(𝑖)=()⋯(),𝐵𝑛−3)(𝑛−4𝑛−𝑖−2_𝑛,0,3𝑒₀(𝑥)𝐵=1,_𝑛,0,3𝑒₃(𝑥)=𝑥³,𝐵_𝑛,0,3𝑒₆(𝑥)=(𝑛−3)⁽³⁾𝑛()𝑥𝑛−1)(𝑛−2⁶+9(𝑛−3)⁽²⁾𝑛()𝑥𝑛−1)(𝑛−2⁵+18(𝑛−3)𝑛()𝑥𝑛−1)(𝑛−2⁴+6𝑛()𝑥𝑛−1)(𝑛−2³,𝐵_𝑛,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,𝐵_𝑛,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. (3.22) For 𝑖=1,2,4 the quantities 𝐵_𝑛,0,3𝑒^𝑥_𝜑,𝑖=𝐵_𝑛,0,3𝑒^𝑥_𝑒3,𝑖, required to apply Theorem 2.1, appear after some calculations using the following identities: 𝑒^𝑥_𝜑,1=𝑒^𝑥_𝑒3,1=𝑒₃−𝑥³𝑒₀,𝑒^𝑥_𝜑,2=𝑒^𝑥_𝑒3,2=𝑒₆−2𝑥³𝑒₃+𝑥⁶𝑒₀,𝑒^𝑥_𝜑,4=𝑒^𝑥_𝑒3,4=𝑒₁₂−4𝑥³𝑒₉+6𝑥⁶𝑒₆−4𝑥⁹𝑒₃+𝑥¹²𝑒₀. (3.23)

Finally, motivated by the well-known Voronovskaja formula for the classical Bernstein operators and by the results in Corollaries 3.2 and 3.5, we close this section and the paper stating the following conjecture.

Conjecture 3.6. For all 𝑓∈𝐶[0,1],𝑥∈(0,1), and 𝑗∈{4,5,…}, whenever 𝑓^′′(𝑥) exists, lim_𝑛→+∞𝑛𝐵_𝑛,0,𝑗𝑓(𝑥)(𝑥)=𝑥()−𝑓1−𝑥2𝑓^′′(𝑥)−()𝑗−11−𝑥2𝑓^′(𝑥). (3.24)

Acknowledgments

The authors thank the referees for their suggestions. The first two authors are partially supported by the Junta de Andalucía (Research Group FQM-0178) and by Universidad de Jaén and Caja Rural de Jaén (Project UJA 2009/12/07).