Abstract and Applied Analysis

Abstract and Applied Analysis / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 680340 | https://doi.org/10.1155/2012/680340

Sezgin Sucu, Gürhan İçöz, Serhan Varma, "On Some Extensions of Szasz Operators Including Boas-Buck-Type Polynomials", Abstract and Applied Analysis, vol. 2012, Article ID 680340, 15 pages, 2012. https://doi.org/10.1155/2012/680340

On Some Extensions of Szasz Operators Including Boas-Buck-Type Polynomials

Academic Editor: Giovanni Galdi
Received25 Apr 2012
Revised19 Jul 2012
Accepted02 Aug 2012
Published30 Aug 2012

Abstract

This paper is concerned with a new sequence of linear positive operators which generalize Szasz operators including Boas-Buck-type polynomials. We establish a convergence theorem for these operators and give the quantitative estimation of the approximation process by using a classical approach and the second modulus of continuity. Some explicit examples of our operators involving Laguerre polynomials, Charlier polynomials, and Gould-Hopper polynomials are given. Moreover, a Voronovskaya-type result is obtained for the operators containing Gould-Hopper polynomials.

1. Introduction

The approximation theory, which is concerned with the approximation of functions by simpler calculated functions, is a branch of mathematical analysis. In 1885, Weierstrass identified the set of continuous functions on a closed and bounded interval through uniform approximation by polynomials. Later, Bernstein gave the first impressive example for these polynomials.

In 1953, Korovkin [1] published his celebrated theorem on the approximation of sequences of linear positive operators. This theorem contains a simple and easily applicable criterion to check if a sequence of linear positive operators converges uniformly to the function. One of the well-known examples of linear positive operators is Szasz operators [2] 𝑆𝑛(𝑓;𝑥)=𝑒𝑛𝑥𝑘=0(𝑛𝑥)𝑘𝑓𝑘𝑘!𝑛,(1.1) where 𝑛, 𝑥0, and 𝑓𝐶[0,) whenever the above sum converges. Many researchers have dealt with the generalization of Szasz operators in a natural way.

Later,  Jakimovski and Leviatan [3] presented a generalization of Szasz operators with Appell polynomials. Let 𝑔(𝑧)=𝑘=0𝑎𝑘𝑧𝑘(𝑎00) be an analytic function in the disc|𝑧|<𝑅(𝑅>1) and assume that 𝑔(1)0. The Appell polynomials 𝑝𝑘(𝑥) have generating functions of the form 𝑔(𝑢)𝑒𝑢𝑥=𝑘=0𝑝𝑘(𝑥)𝑢𝑘.(1.2) Under the assumption 𝑝𝑘(𝑥)0 for 𝑥[0,), Jakimovski and Leviatan introduced the linear positive operators 𝑃𝑛(𝑓;𝑥) via 𝑃𝑛(𝑒𝑓;𝑥)=𝑛𝑥𝑔(1)𝑘=0𝑝𝑘(𝑘𝑛𝑥)𝑓𝑛,for𝑛,(1.3) and gave the approximation properties of these operators.

Remark 1.1. For 𝑔(𝑧)=1, in view of the generating functions (1.2), we easily find 𝑝𝑘(𝑥)=𝑥𝑘/𝑘! and from (1.3) we meet again the Szasz operators given by (1.1).

Then, Ismail [4] obtained another generalization of the Szasz operators (1.1) and also Jakimovski and Leviatan operators (1.3) by means of Sheffer polynomials. Let 𝐴(𝑧)=𝑘=0𝑎𝑘𝑧𝑘(𝑎00) and 𝐻(𝑧)=𝑘=1𝑘𝑧𝑘(10) be analytic functions in the disc|𝑧|<𝑅(𝑅>1) where 𝑎𝑘 and 𝑘 are real. The Sheffer polynomials 𝑝𝑘(𝑥) have generating functions of the type 𝐴(𝑡)𝑒𝑥𝐻(𝑡)=𝑘=0𝑝𝑘(𝑥)𝑡𝑘,|𝑡|<𝑅.(1.4) Using the following assumptions: [(i)for𝑥0,),𝑝𝑘(𝑥)0,(ii)𝐴(1)0,𝐻(1)=1.(1.5) Ismail investigated the approximation properties of linear positive operators given by 𝑇𝑛(𝑒𝑓;𝑥)=𝑛𝑥𝐻(1)𝐴(1)𝑘=0𝑝𝑘(𝑘𝑛𝑥)𝑓𝑛,for𝑛.(1.6)

Remark 1.2. For 𝐻(𝑡)=𝑡, one can observe that the generating functions (1.4) reduce to (1.2) and from this fact, the operators (1.6) return to the operators (1.3).

Remark 1.3. For 𝐻(𝑡)=𝑡 and 𝐴(𝑡)=1, it is easy to get the Szasz operators from the operators (1.6).

Recently, Varma et al. [5] constructed linear positive operators including Brenke-type polynomials. Brenke-type polynomials [6] have generating functions of the form 𝐴(𝑡)𝐵(𝑥𝑡)=𝑘=0𝑝𝑘(𝑥)𝑡𝑘,(1.7) where 𝐴 and 𝐵 are analytic functions 𝐴(𝑡)=𝑟=0𝑎𝑟𝑡𝑟,𝑎0𝐵0,(𝑡)=𝑟=0𝑏𝑟𝑡𝑟,𝑏𝑟0(𝑟0),(1.8) and have the following explicit expression: 𝑝𝑘(𝑥)=𝑘𝑟=0𝑎𝑘𝑟𝑏𝑟𝑥𝑟,𝑘=0,1,2,.(1.9) Using the following assumptions 𝑎(i)𝐴(1)0,𝑘𝑟𝑏𝑟[𝐴(1)0,0𝑟𝑘,𝑘=0,1,2,,(ii)𝐵0,)(0,),(iii)(1.7)andthepowerseries(1.8)convergefor|𝑡|<𝑅(𝑅>1),(1.10) Varma et al. introduced the following linear positive operators involving the Brenke-type polynomials 𝐿𝑛(1𝑓;𝑥)=𝐴(1)𝐵(𝑛𝑥)𝑘=0𝑝𝑘(𝑘𝑛𝑥)𝑓𝑛,(1.11) where 𝑥0 and 𝑛.

Remark 1.4. Let 𝐵(𝑡)=𝑒𝑡. In this case, the operators (1.11) (resp., (1.7)) reduce to the operators given by (1.3) (resp., (1.2)).

Remark 1.5. Let 𝐵(𝑡)=𝑒𝑡 and 𝐴(𝑡)=1. We meet again the Szasz operators (1.1).

In this paper, our aim is to construct linear positive operators by using Boas-Buck-type polynomials including the Brenke-type polynomials, Sheffer polynomials, and Appell polynomials with special cases. Boas-Buck-type polynomials [7] have generating functions of the type 𝐴(𝑡)𝐵(𝑥𝐻(𝑡))=𝑘=0𝑝𝑘(𝑥)𝑡𝑘,(1.12) where 𝐴, 𝐵, and 𝐻 are analytic functions 𝐴(𝑡)=𝑘=0𝑎𝑘𝑡𝑘𝑎00,𝐵(𝑡)=𝑘=0𝑏𝑘𝑡𝑘𝑏𝑘,𝐻0(𝑡)=𝑘=1𝑘𝑡𝑘1.0(1.13)

We will restrict ourselves to the Boas-Buck-type polynomials satisfying 𝐴(i)(1)0,𝐻(1)=1,𝑝𝑘((𝑥)0,𝑘=0,1,2,,(ii)𝐵(0,),iii)(1.12)andthepowerseries(1.13)convergefor|𝑡|<𝑅(𝑅>1).(1.14)

Now, given the above restrictions, we present a new form of linear positive operators with Boas-Buck-type polynomials as follows: 𝑛(1𝑓;𝑥)=𝐴(1)𝐵(𝑛𝑥𝐻(1))𝑘=0𝑝𝑘(𝑘𝑛𝑥)𝑓𝑛,(1.15) where 𝑥0 and 𝑛.

Remark 1.6. Let 𝐻(𝑡)=𝑡. The operators (1.15) (resp., (1.12)) reduce to the operators given by (1.11) (resp., (1.7)).

Remark 1.7. Let 𝐵(𝑡)=𝑒𝑡. The operators (1.15) (resp., (1.12)) return to the operators given by (1.6) (resp., (1.4)).

Remark 1.8. Let 𝐻(𝑡)=𝑡 and 𝐵(𝑡)=𝑒𝑡. It is obvious that one can get the operators (1.3) from the operators (1.15). In addition, if we choose 𝐴(𝑡)=1, we meet again the well-known Szasz operators (1.1).

The paper is divided into three sections. Following the introduction, Section 2 is devoted to obtain qualitative and quantitative results for the operators (1.15). In the last section, we give some significant illustrations with the help of Laguerre, Charlier, and Gould-Hopper polynomials for the operators (1.15). Moreover, we give a Voronovskaya-type theorem for the operators including Gould-Hopper polynomials.

2. Approximation Properties of 𝑛 Operators

In this section, with the help of well-known Korovkin's theorem, we get approximation results by means of 𝑛 linear positive operators. Next, we present quantitative results for estimating the error of approximation using the classical approach and the second modulus of continuity.

Lemma 2.1. For the operators 𝑛, one has 𝑛(1;𝑥)=1,𝑛𝐵(s;𝑥)=(𝑛𝑥𝐻(1))𝐴𝐵(𝑛𝑥𝐻(1))𝑥+(1),𝑛𝐴(1)𝑛𝑠2=𝐵;𝑥(𝑛𝑥𝐻(1))𝑥𝐵(𝑛𝑥𝐻(1))2+2𝐴(1)+1+𝐻𝐵(1)𝐴(1)(𝑛𝑥𝐻(1))𝑥+𝐴𝑛𝐴(1)𝐵(𝑛𝑥𝐻(1))(1)+𝐴(1)𝑛2,𝐴(1)(2.1) for any 𝑥[0,).

Proof. From the generating functions of the Boas-Buck-type polynomials given by (1.12), we obtain 𝑘=0𝑝𝑘(𝑛𝑥)=𝐴(1)𝐵(𝑛𝑥𝐻(1)),𝑘=0𝑘𝑝𝑘(𝑛𝑥)=𝐴(1)𝐵(𝑛𝑥𝐻(1))+𝑛𝑥𝐴(1)𝐵(𝑛𝑥𝐻(1)),𝑘=0𝑘2𝑝𝑘𝐴(𝑛𝑥)=(1)+𝐴(1)𝐵(𝑛𝑥𝐻(1))+2𝐴(1)+𝐴(1)+𝐴(1)𝐻(1)×𝐵(𝑛𝑥𝐻(1))𝑛𝑥+𝐴(1)𝐵(𝑛𝑥𝐻(1))(𝑛𝑥)2.(2.2) With regard to these equalities, we get the assertions of the lemma.

Let us define the class of 𝐸 as follows: [𝐸=𝑓𝑥0,),𝑓(𝑥)1+𝑥2isconvergentas𝑥.(2.3)

Theorem 2.2. Let 𝑓𝐶[0,)𝐸 and assume that lim𝑦𝐵(𝑦)𝐵(𝑦)=1,lim𝑦𝐵(𝑦)𝐵(𝑦)=1.(2.4) Then, lim𝑛𝑛(𝑓;𝑥)=𝑓(𝑥)(2.5) uniformly on each compact subset of [0,).

Proof. According to Lemma 2.1 and taking into account the assumptions (2.4), we find lim𝑛𝑛𝑠𝑖;𝑥=𝑥𝑖,𝑖=0,1,2.(2.6) The above-mentioned convergences are satisfied uniformly in each compact subset of [0,). Applying the universal Korovkin-type property (vi) of Theorem 4.1.4 from [8], we lead to the desired result.

In order to estimate the rate of convergence, we will give some definitions and lemmas.

Definition 2.3. Let 𝑓𝐶[0,) and 𝛿>0. The modulus of continuity 𝜔(𝑓;𝛿) of the function 𝑓 is defined by 𝜔(𝑓;𝛿)=sup[||||𝑥,𝑦0,)𝑥𝑦𝛿||||,𝑓(𝑥)𝑓(𝑦)(2.7) where 𝐶[0,) is the space of uniformly continuous functions on [0,).

Definition 2.4. The second modulus of continuity of the function 𝑓𝐶[𝑎,𝑏] is defined by 𝜔2(𝑓;𝛿)=sup0<𝑡𝛿𝑓(+2𝑡)2𝑓(+𝑡)+𝑓(),(2.8) where 𝑓=max𝑥[𝑎,𝑏]|𝑓(𝑥)|.

Lemma 2.5 (Gavrea and Raşa [9]). Let 𝑔𝐶2[0,𝑎] and (𝐾𝑛)𝑛0 be a sequence of linear positive operators with the property 𝐾𝑛(1;𝑥)=1. Then, ||𝐾𝑛||𝑔(𝑔;𝑥)𝑔(𝑥)𝐾𝑛(𝑠𝑥)2+1;𝑥2𝑔𝐾𝑛(𝑠𝑥)2.;𝑥(2.9)

Lemma 2.6 (Zhuk [10]). Let 𝑓𝐶[𝑎,𝑏] and (0,(𝑏𝑎)/2). Let 𝑓 be the second-order Steklov function attached to the function 𝑓. Then, the following inequalities are satisfied: 𝑓(i)3𝑓4𝜔2𝑓(𝑓;),(ii)322𝜔2(𝑓;).(2.10)

Lemma 2.7. For 𝑥[0,), one has 𝑛(𝑠𝑥)2=𝐵;𝑥(𝑛𝑥𝐻(1))2𝐵(𝑛𝑥𝐻(1))+𝐵(𝑛𝑥𝐻(1))𝑥𝐵(𝑛𝑥𝐻(1))2+𝐻𝐴(1)𝐵(1)+1(𝑛𝑥𝐻(1))+2𝐴𝐵(1)(𝑛𝑥𝐻(1))𝐵(𝑛𝑥𝐻(1))𝑥+𝐴𝑛𝐴(1)𝐵(𝑛𝑥𝐻(1))(1)+𝐴(1)𝑛2.𝐴(1)(2.11)

Proof. Using the linearity property of 𝑛 operators, one can write 𝑛(𝑠𝑥)2;𝑥=𝑛𝑠2;𝑥2𝑥𝑛(𝑠;𝑥)+𝑥2𝑛(1;𝑥).(2.12) Applying Lemma 2.1, we obtain the equality stated in the lemma.

Generally, we use the modulus of continuity and second modulus of continuity to obtain quantitative error estimation for convergence by linear positive operators. Now, we will calculate the rate of convergence in the following two theorems.

Theorem 2.8. Let 𝑓𝐶[0,)𝐸. 𝑛 operators verify the following inequality: ||𝑛||(𝑓;𝑥)𝑓(𝑥)2𝜔𝑓;𝜗𝑛(𝑥),(2.13) where 𝜗=𝜗𝑛(𝑥)=𝑛(𝑠𝑥)2=𝐵;𝑥(𝑛𝑥𝐻(1))2𝐵(𝑛𝑥𝐻(1))+𝐵(𝑛𝑥𝐻(1))𝑥𝐵(𝑛𝑥𝐻(1))2+𝐻𝐴(1)(𝐵1)+1(𝑛𝑥𝐻(1))+2𝐴(𝐵1)(𝑛𝑥𝐻(1))𝐵(𝑛𝑥𝐻(1))𝑥+𝐴𝑛𝐴(1)𝐵(𝑛𝑥𝐻(1))(1)+𝐴(1)𝑛2.𝐴(1)(2.14)

Proof. Making use of Lemma 2.1 and the property of modulus of continuity, we deduce ||𝑛(||1𝑓;𝑥)𝑓(𝑥)𝐴(1)𝐵(𝑛𝑥𝐻(1))𝑘=0𝑝𝑘(|||𝑓𝑘𝑛𝑥)𝑛|||1𝑓(𝑥)1+1𝐴(1)𝐵(𝑛𝑥𝐻(1))𝛿𝑘=0𝑝𝑘|||𝑘(𝑛𝑥)𝑛|||𝑥𝜔(𝑓;𝛿).(2.15) Taking into account the Cauchy-Schwarz inequality and then by using Lemma 2.7, we get 𝑘=0𝑝𝑘|||𝑘(𝑛𝑥)𝑛|||𝑥𝑘=0𝑝𝑘(𝑛𝑥)1/2𝑘=0𝑝𝑘|||𝑘(𝑛𝑥)𝑛|||𝑥21/2=𝐴(1)𝐵(𝑛𝑥𝐻(1))𝑛(𝑠𝑥)2.;𝑥(2.16) Considering the last inequality in (2.15), we obtain ||𝑛||1(𝑓;𝑥)𝑓(𝑥)1+𝛿𝜗𝑛(𝑥)𝜔(𝑓;𝛿),(2.17) where 𝜗𝑛(𝑥) is given by (2.14). In inequality (2.17), by choosing 𝛿=𝜗𝑛(𝑥), we get the desired result.

Theorem 2.9. For 𝑓𝐶[0,𝑎], the following estimate ||𝑛||2(𝑓;𝑥)𝑓(𝑥)𝑎𝑓2+34𝑎+2+2𝜔2(𝑓;)(2.18) holds, where =𝑛(𝑥)=4𝑛(𝑠𝑥)2.;𝑥(2.19)

Proof. Let 𝑓 be the second-order Steklov function attached to the function 𝑓. With regard to the identity 𝑛(1;𝑥)=1, we have ||𝑛||||(𝑓;𝑥)𝑓(𝑥)𝑛𝑓𝑓||+||;𝑥𝑛𝑓;𝑥𝑓||+||𝑓(𝑥)||𝑓(𝑥)𝑓(𝑥)2+||𝑓𝑛𝑓;𝑥𝑓(||.𝑥)(2.20) Taking account of the fact that 𝑓𝐶2[0,𝑎], it follows from Lemma 2.5||𝑛𝑓;𝑥𝑓||𝑓(𝑥)𝑛(𝑠𝑥)2+1;𝑥2𝑓𝑛(𝑠𝑥)2.;𝑥(2.21) If one combines Landau inequality with Lemma 2.6, we can write 𝑓2𝑎𝑓+𝑎2𝑓2𝑎𝑓+3𝑎412𝜔2(𝑓;).(2.22) From the last inequality and Lemma 2.6, (2.21) becomes by taking =4𝑛((𝑠𝑥)2;𝑥)||𝑛𝑓;𝑥𝑓||2(𝑥)𝑎𝑓2+3𝑎4𝜔23(𝑓;)+42𝜔2(𝑓;).(2.23) Substituting (2.23) in (2.20), hence Lemma 2.6 gives the proof of the theorem.

Remark 2.10. In Theorem 2.9, we present the proof for (0,𝑎/2). For the special case 𝐵(𝑡)=𝑒𝑡,𝐻(𝑡)=𝑡,𝐴(𝑡)=1 and 𝑥=0, one can get =0 from the equality =𝑛(𝑥)=4𝑛((𝑠𝑥)2;𝑥). The inequality obtained in Theorem 2.9 still holds true when =0.

Remark 2.11. Note that in Theorems 2.82.9 when 𝑛, respectively, 𝜗 and tend to zero under the assumptions (2.4).

3. Examples

Example 3.1. Laguerre polynomials are one of the most important classical orthogonal polynomials in the literature. Such polynomials are used in every area of mathematics. In addition, these polynomials have served with several interesting properties for physicists. For example, Laguerre polynomials arise as solutions of the Coulomb potential in quantum mechanics.
Laguerre polynomials have generating functions of the form 1(1𝑡)𝛼+1exp𝑥𝑡=1𝑡𝑘=0𝐿𝑘(𝛼)(𝑥)𝑡𝑘,|𝑡|<1,(3.1) and explicit expressions 𝐿𝑘(𝛼)(𝑥)=𝑘𝑚=0(𝛼+𝑘)!(𝑘𝑚)!(𝛼+𝑚)!𝑚!(𝑥)𝑚,𝛼>1.(3.2) It is clear that Laguerre polynomials are Boas-Buck-type polynomials. Note that when 𝑥(,0], 𝐿𝑘(𝛼)(𝑥) are positive. For ensuring the restrictions (1.14) and the assumptions (2.4), we have to modify the generating functions (3.1) as follows: 1(1(𝑡/2))𝛼+1exp𝑥𝑡2=(2𝑡)𝑘=0𝐿𝑘(𝛼)(𝑥/2)2𝑘𝑡𝑘,|𝑡|<2.(3.3) With the help of generating functions (3.3), we find the following linear positive operators including Laguerre polynomials from the operators (1.15) 𝐿𝑛(𝑓;𝑥)=𝑒(𝑛𝑥/2)𝑘=0𝐿𝑘(𝛼)((𝑛𝑥/2))2𝛼+𝑘+1𝑓𝑘𝑛,(3.4) where 𝛼>1 and 𝑥[0,).

Remark 3.2. It is worthy to note that we obtain new linear positive operators different from the one given in [4].

Example 3.3. Varma and Taşdelen [11] gave the following linear positive operators involving Charlier polynomials as a generalization of the Szasz operators: 𝐿𝑛(𝑓;𝑥,𝑎)=𝑒111𝑎(𝑎1)𝑛𝑥𝑘=0𝐶𝑘(𝑎)((𝑎1)𝑛𝑥)𝑓𝑘𝑘!𝑛,(3.5) where 𝑎>1, 𝑥[0,) and 𝐶𝑘(𝑎)(𝑥) Charlier polynomials have the generating functions of the type 𝑒𝑡𝑡1𝑎𝑥=𝑘=0𝐶𝑘(𝑎)(𝑥)𝑡𝑘!𝑘,|𝑡|<𝑎.(3.6) Note that one can get the operators (3.5) as an example of the operators given by the equality (1.6) defined in [4].
On the other hand, Charlier polynomials are also the Boas-Buck-type polynomials by choosing 𝐴(𝑡)=𝑒𝑡,𝐵(𝑡)=𝑒𝑡𝑡,𝐻(𝑡)=ln1𝑎.(3.7) For ensuring the restrictions (1.14) and the assumptions (2.4), we have to change the generating functions (3.6) by 𝑒𝑡𝑒(𝑎1)𝑥ln(1(𝑡/𝑎))=𝑘=0𝐶𝑘(𝑎)((𝑎1)𝑥)𝑡𝑘!𝑘,|𝑡|<𝑎.(3.8) In view of the generating functions (3.8), we have the linear positive operators (3.5) investigated in [11] from the operators (1.15).

Example 3.4. Gould-Hopper polynomials [12] have the generating functions of the form 𝑒𝑡𝑑+1exp(𝑥𝑡)=𝑘=0𝑔𝑘𝑑+1(𝑡𝑥,)𝑘𝑘!(3.9) and the explicit representations 𝑔𝑘𝑑+1(𝑥,)=[𝑘/(𝑑+1)]𝑠=0𝑘!𝑠!(𝑘(𝑑+1)𝑠)!𝑠𝑥𝑘(𝑑+1)𝑠,(3.10) where, as usual, [] denotes the integer part. Gould-Hopper polynomials 𝑔𝑘𝑑+1(𝑥,) are 𝑑-orthogonal polynomial set of Hermite type [13]. Van Iseghem [14] and Maroni [15] discovered the notion of 𝑑-orthogonality. Gould-Hopper polynomials are Boas-Buck-type polynomials with 𝐴(𝑡)=𝑒𝑡𝑑+1,𝐵(𝑡)=𝑒𝑡,𝐻(𝑡)=𝑡.(3.11) Under the assumption 0; the restrictions (1.14) and condition (2.4) for the operators 𝑛 given by (1.15) are satisfied. With the help of the generating functions (3.9), we obtain the explicit form of 𝑛 operators including Gould-Hopper polynomials by 𝐿𝑛(𝑓;𝑥)=𝑒𝑛𝑥𝑘=0𝑔𝑘𝑑+1(𝑛𝑥,)𝑓𝑘𝑘!𝑛,(3.12) where 𝑥[0,).

Remark 3.5. First time, the operators 𝐿𝑛 are given in [5] as an explicit example of the operators (1.11). For =0, we obtain that 𝑔𝑘𝑑+1(𝑛𝑥,0)=(𝑛𝑥)𝑘.(3.13) Substituting =0 and 𝑔𝑘𝑑+1(𝑛𝑥,0)=(𝑛𝑥)𝑘 in the operators (3.12), we get the well-known Szasz operators. By the help of 𝐿𝑛 operators, we present an attractive generalization of the Szasz operators with Gould-Hopper polynomials.

Next, we give a Voronovskaya-type theorem for the operators 𝐿𝑛. In order to prove this theorem, we need some auxiliary results.

Lemma 3.6. For the operators 𝐿𝑛, one has 𝐿𝑛𝐿(1;𝑥)=1,𝑛(𝑠;𝑥)=𝑥+(𝑑+1)𝑛,𝐿𝑛𝑠2;𝑥=𝑥2+2(𝑑+1)+1𝑛𝑥+(+1)(𝑑+1)2𝑛2,𝐿𝑛𝑠3;𝑥=𝑥3+3(𝑑+1)+3𝑛𝑥2+32(𝑑+1)2+3(𝑑+1)(𝑑+2)+1𝑛2𝑥+2(+3+1𝑑+1)3𝑛3,𝐿𝑛𝑠4;𝑥=𝑥4+4(𝑑+1)+6𝑛𝑥3+62(𝑑+1)2+6(𝑑+1)(𝑑+3)+7𝑛2𝑥2+62(𝑑+1)2(2𝑑+3)+2(𝑑+1)2𝑑2+7𝑑+7+43(𝑑+1)3+1𝑛3𝑥+3+62+7+1(𝑑+1)4𝑛4.(3.14)

Proof. By virtue of the generating functions (3.9) for Gould-Hopper polynomials, we obtain the above equalities.

Lemma 3.7. For 𝑥0, the following equalities hold: 𝐿𝑛(𝑠𝑥)2=𝑥;𝑥𝑛+(+1)(𝑑+1)2𝑛2,𝐿𝑛(𝑠𝑥)4=3;𝑥𝑛2𝑥2+62(𝑑+1)2+2(𝑑+1)(3𝑑+5)+1𝑛3𝑥+3+62+7+1(𝑑+1)4𝑛4.(3.15)

Proof. According to Lemma 3.6, it is easy to get the above equalities.

Theorem 3.8. Let 𝑓𝐶2[0,𝑎]. Then, one has lim𝑛𝑛𝐿𝑛(𝑓;𝑥)𝑓(𝑥)=(𝑑+1)𝑓(𝑥)+𝑥𝑓(𝑥)2.(3.16)

Proof. In view of Taylor formula for the function 𝑓, we find 𝑓(𝑠)=𝑓(𝑥)+(𝑠𝑥)𝑓(𝑥)+(𝑠𝑥)2𝑓2!(𝑥)+(𝑠𝑥)2𝜂(𝑠;𝑥),(3.17) where 𝜂(𝑠;𝑥)𝐶[0,𝑎] and lim𝑠𝑥𝜂(𝑠;𝑥)=0. Applying 𝐿𝑛 to the both sides of (3.17), we get 𝐿𝑛(𝑓;𝑥)=𝑓(𝑥)+𝑓(𝑥)𝐿𝑛𝑓(𝑠𝑥;𝑥)+(𝑥)2𝐿𝑛(𝑠𝑥)2;𝑥+𝐿𝑛(𝑠𝑥)2.𝜂(𝑠;𝑥);𝑥(3.18) According to Lemmas 3.63.7, (3.18) becomes 𝐿𝑛(𝑓;𝑥)=𝑓(𝑥)+𝑓(𝑥)(𝑑+1)𝑛+𝑓(𝑥)2𝑥𝑛+(+1)(𝑑+1)2𝑛2+𝐼,(3.19) where 𝐼=𝑒𝑛𝑥𝑘=0𝑔𝑘𝑑+1(𝑛𝑥,)𝑘𝑘!𝑛𝑥2𝜂𝑘𝑛.;𝑥(3.20) Now, we consider the sum 𝐼 as follows: 𝐼=𝑒𝑛𝑥||||(𝑘/𝑛)𝑥𝛿𝑔𝑘𝑑+1(𝑛𝑥,)𝑘𝑘!𝑛𝑥2𝜂𝑘𝑛;𝑥+𝑒𝑛𝑥||||(𝑘/𝑛)𝑥>𝛿𝑔𝑘𝑑+1(𝑛𝑥,)𝑘𝑘!𝑛𝑥2𝜂𝑘𝑛.;𝑥(3.21) From the continuity of function 𝜂, it results that for all 𝜀>0, there exists a positive 𝛿 such that if |(𝑘/𝑛)𝑥|𝛿, then |𝜂(𝑘/𝑛;𝑥)|<𝜀. Furthermore, since the function 𝜂 is bounded, we can write |𝜂(𝑘/𝑛;𝑥)|<𝑀 for |(𝑘/𝑛)𝑥|>𝛿. In view of these facts, (3.21) leads to 𝐼𝜀𝐿𝑛(𝑠𝑥)2;𝑥+𝑀𝑒𝑛𝑥||||(𝑘/𝑛)𝑥>𝛿𝑔𝑘𝑑+1(𝑛𝑥,)𝑘𝑘!𝑛𝑥2.(3.22) Taking into account the fact 𝑒𝑛𝑥||||(𝑘/𝑛)𝑥>𝛿𝑔𝑘𝑑+1(𝑛𝑥,)𝑘𝑘!𝑛𝑥21𝛿2𝐿𝑛(𝑠𝑥)4;𝑥(3.23) in the last inequality, we have 𝐼𝜀𝐿𝑛(𝑠𝑥)2+𝑀;𝑥𝛿2𝐿𝑛(𝑠𝑥)4.;𝑥(3.24) Substituting the inequality (3.24) in the equality (3.19), then from Lemma 3.7, we obtain 𝐿𝑛(𝑓;𝑥)𝑓(𝑥)𝑓(𝑥)(𝑑+1)𝑛+𝑓𝜀+(𝑥)2𝑥𝑛+(+1)(𝑑+1)2𝑛2+𝑀𝛿23𝑛2𝑥2+62(𝑑+1)2+2(𝑑+1)(3𝑑+5)+1𝑛3𝑥+3+62+7+1(𝑑+1)4𝑛4.(3.25) Equivalently, we can write 𝐿𝑛1(𝑓;𝑥)𝑓(𝑥)=𝒪𝑛𝑓𝑓(𝑥)(𝑑+1)+𝜀+(𝑥)2𝑥+(+1)(𝑑+1)2𝑛+𝑀𝛿23𝑛𝑥2+62(𝑑+1)2+2(𝑑+1)(3𝑑+5)+1𝑛2𝑥+3+62+7+1(𝑑+1)4𝑛3.(3.26) Taking limits for 𝑛, (3.26) becomes lim𝑛𝑛𝐿𝑛(𝑓;𝑥)𝑓(𝑥)=(𝑑+1)𝑓(𝑥)+𝑥𝑓(𝑥)2,(3.27) which completes the proof.

Remark 3.9. Theorem 3.8 is an explicit example of Gonska's result given in [16]. It is worthy to note that this Voronovskaya-type result is given for the 𝐿𝑛 operators which contain Gould-Hopper polynomials.

Remark 3.10. Taking =0 in Theorem 3.8, we get a Voronovskaya-type result for the Szasz operators.

References

  1. P. P. Korovkin, “On convergence of linear positive operators in the space of continuous functions,” Doklady Akademii Nauk SSSR, vol. 90, pp. 961–964, 1953 (Russian). View at: Google Scholar
  2. O. Szasz, “Generalization of S. Bernstein's polynomials to the infinite interval,” Journal of Research of the National Bureau of Standards, vol. 45, pp. 239–245, 1950. View at: Google Scholar
  3. A. Jakimovski and D. Leviatan, “Generalized Szász operators for the approximation in the infinite interval,” Mathematica, vol. 11, pp. 97–103, 1969. View at: Google Scholar
  4. M. E. H. Ismail, “On a generalization of Szász operators,” Mathematica, vol. 39, no. 2, pp. 259–267, 1974. View at: Google Scholar | Zentralblatt MATH
  5. S. Varma, S. Sucu, and G. İçöz, “Generalization of Szasz operators involving Brenke type polynomials,” Computers & Mathematics with Applications, vol. 64, no. 2, pp. 121–127, 2012. View at: Publisher Site | Google Scholar
  6. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, NY, USA, 1978.
  7. M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge, UK, 2005.
  8. F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and Its Applications, vol. 17 of de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, Germany, 1994. View at: Publisher Site
  9. I. Gavrea and I. Raşa, “Remarks on some quantitative Korovkin-type results,” Revue d'Analyse Numérique et de Théorie de l'Approximation, vol. 22, no. 2, pp. 173–176, 1993. View at: Google Scholar | Zentralblatt MATH
  10. V. V. Zhuk, “Functions of the Lip1 class and S. N. Bernstein's polynomials,” Vestnik Leningradskogo Universiteta. Matematika, Mekhanika, Astronomiya, vol. 1, pp. 25–30, 1989. View at: Google Scholar
  11. S. Varma and F. Taşdelen, “Szász type operators involving Charlier polynomials,” Mathematical and Computer Modelling, vol. 56, no. 5-6, pp. 118–122, 2012. View at: Publisher Site | Google Scholar
  12. H. W. Gould and A. T. Hopper, “Operational formulas connected with two generalizations of Hermite polynomials,” Duke Mathematical Journal, vol. 29, pp. 51–63, 1962. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. K. Douak, “The relation of the d-orthogonal polynomials to the Appell polynomials,” Journal of Computational and Applied Mathematics, vol. 70, no. 2, pp. 279–295, 1996. View at: Publisher Site | Google Scholar
  14. J. Van Iseghem, “Vector orthogonal relations. Vector QD-algorithm,” Journal of Computational and Applied Mathematics, vol. 19, no. 1, pp. 141–150, 1987. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  15. P. Maroni, “L'orthogonalité et les récurrences de polynômes d'ordre supérieur à deux,” Annales Faculté des Sciences de Toulouse Série 5, vol. 10, no. 1, pp. 105–139, 1989. View at: Google Scholar | Zentralblatt MATH
  16. H. Gonska, “On the degree of approximation in Voronovskaja's theorem,” Studia. Universitatis Babeş-Bolyai. Mathematica, vol. 52, no. 3, pp. 103–115, 2007. View at: Google Scholar

Copyright © 2012 Sezgin Sucu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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