Some Approximation Properties of 𝑞-Baskakov-Beta-Stancu Type Operators

Abstract

This paper deals with new type 𝑞-Baskakov-Beta-Stancu operators defined in the paper. First, we have used the properties of 𝑞-integral to establish the moments of these operators. We also obtain some approximation properties and asymptotic formulae for these operators. In the end we have also presented better error estimations for the 𝑞-operators.

1. Introduction

In the recent years, the quantum calculus (𝑞-calculus) has attracted a great deal of interest because of its potential applications in mathematics, mechanics, and physics. Due to the applications of 𝑞-calculus in the area of approximation theory, 𝑞-generalization of some positive operators has attracted much interest, and a great number of interesting results related to these operators have been obtained (see, for instance, [1–3]). In this direction, several authors have proposed the 𝑞-analogues of different linear positive operators and studied their approximation behaviors. Also, Aral and Gupta [4] defined 𝑞-generalization of the Baskakov operators and investigated some approximation properties of these operators. Subsequently, Finta and Gupta [5] obtained global direct error estimates for these operators using the second-order Ditzian Totik modulus of smoothness. To approximate Lebesgue integrable functions on the interval [0,∞), modified Beta operators [6] are defined as 𝐵_𝑛𝑓,𝑥=𝑛−1𝑛^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)^∞₀𝑝_𝑛,𝑘(𝑡)𝑓(𝑡)𝑑𝑡,𝑥∈[0,∞), (1) where 𝑏_𝑛,𝑘(𝑥)=(1/𝐵(𝑘+1,𝑛))(𝑥^𝑘/(1+𝑥)^𝑛+𝑘+1) and 𝑝_𝑛,𝑘(𝑡)=_𝑘^𝑛+𝑘+1(𝑡^𝑘/(1+𝑡)^𝑛+𝑘).

The discrete 𝑞-Beta operators are defined as 𝑉^𝑞_𝑛=1𝑓,𝑥[𝑛]_𝑞^∞_𝑘=0𝑏^𝑞_𝑛,𝑘(𝑥)𝑓[𝑘]_𝑞[𝑛+1]_𝑞𝑞^𝑘−1. (2)

Recently, Maheshwari and Sharma [7] introduced the 𝑞-analogue of the Baskakov-Beta-Stancu operators and studied the rate of approximation and weighted approximation of these operators. Motivated by the Stancu type generalization of 𝑞-Baskakov operators, we propose the 𝑞-analogue of the operators 𝐵_𝑛^(𝛼,𝛽), recently introduced and studied for special values 𝛼=𝛽=0 by Gupta and Kim [8] as 𝐵^𝑞_{𝑛,𝛼,𝛽}[]𝑓,𝑥∶=𝑛−1_𝑞[𝑛]_𝑞^∞_𝑘=0𝑏^𝑞_𝑛,𝑘(𝑥)×₀^∞/𝐴𝑞^𝑘𝑝^𝑞_𝑛,𝑘(𝑡)𝑓[𝑛]_𝑞𝑡+𝛼[𝑛]_𝑞𝑑+𝛽^𝑞𝑡, (3) where 𝑏^𝑞_𝑛,𝑘(𝑥)=(𝑞^{𝑘(𝑘−1)/2}/𝐵_𝑞(𝑘+1,𝑛))(𝑥^𝑘/(1+𝑥)_𝑞^𝑛+𝑘+1) and 𝑝^𝑞_𝑛,𝑘(𝑡)=_𝑘^{𝑛+𝑘−1}_𝑞(𝑞^{𝑘(𝑘−1)/2}/𝐵_𝑞(𝑘+1,𝑛))(𝑡^𝑘/(1+𝑡)_𝑞^𝑛+𝑘+1).

We know that ∑^∞_𝑘=0𝑏^𝑞_𝑛,𝑘(𝑥)=[𝑛]_𝑞 and ∫₀^∞/𝐴𝑞^𝑘𝑝^𝑞_𝑛,𝑘(𝑡)=1/[𝑛−1]_𝑞. We mention that 𝐵^𝑞_𝑛,0,0≡𝐵^𝑞_𝑛 (see [8]).

Very recently, Gupta et al. [9] introduced some direct results in simultaneous approximation for Baskakov-Durrmeyer-Stancu operators. The aim of this paper is to study the approximation properties of a new generalization of the Baskakov type Beta Stancu operators based on 𝑞-integers. We estimate moments for these operators. Also, we study asymptotic formula for these operators. Finally, we give better error estimations for the operator (3). First, we recall some definitions and notations of 𝑞-calculus. Such notations can be found in [10, 11]. We consider 𝑞 as a real number satisfying 𝑞>0. For 𝑛∈ℕ, [𝑛]_𝑞∶=1−𝑞^𝑛[𝑛]1−𝑞,𝑞≠1,𝑛,𝑞=1,_𝑞[𝑛]!∶=^𝑞[𝑛−1]_𝑞[𝑛−2]_𝑞,…,[1]_𝑞(),𝑛=1,2,…,1,𝑛=0,1+𝑥^𝑛_𝑞∶=^𝑛−1_𝑗=01+𝑞^𝑗𝑥. (4) The 𝑞-binomial coefficients are given by 𝑛𝑘_𝑞=[𝑛]_𝑞![𝑘]_𝑞![𝑛−𝑘]_𝑞!,0≤𝑘≤𝑛. (5) The 𝑞-derivative 𝒟_𝑞𝑓 of a function 𝑓 is given by 𝒟_𝑞𝑓(𝑥)=𝑓(𝑥)−𝑓𝑞𝑥𝑥,1−𝑞if𝑥≠0. (6) The 𝑞-analogues of product and quotient rules are defined as 𝒟_𝑞𝑓(𝑥)𝑔(𝑥)=𝑔(𝑥)𝒟_𝑞𝑓(𝑥)+𝑓𝑞𝑥𝒟_𝑞𝑔(𝑥),𝒟_𝑞𝑓(𝑥)𝑔(𝑥)=𝑔(𝑥)𝒟_𝑞𝑓(𝑥)−𝑓(𝑥)𝒟_𝑞𝑔(𝑥)𝑔(𝑥)𝑔.𝑞𝑥 (7) The 𝑞-Jackson integrals and the 𝑞-improper integrals are defined as [12, 13] ^𝑎₀𝑓(𝑥)𝑑_𝑞𝑥=1−𝑞𝑎^∞_𝑛=0𝑓𝑎𝑞^𝑛𝑞^𝑛,𝑎>0,_0∞/𝐴𝑓(𝑥)𝑑_𝑞𝑥=1−𝑞^∞_𝑛=−∞𝑓𝑞^𝑛𝐴𝑞^𝑛𝐴,𝐴>0, (8)(9) provided that the sums converge absolutely. Using (9), De Sole and Kac [14] defined the 𝑞-analogue of Beta functions of second kind 𝐵(𝑡,𝑠)=∫^∞₀(𝑥^𝑡−1/(1+𝑥)^𝑡+𝑠)𝑑𝑥 as follows: 𝐵_𝑞(𝑡,𝑠)=𝐾(𝐴,𝑡)₀^∞/𝐴𝑥^𝑡−1(1+𝑥)_𝑞^𝑡+𝑠𝑑_𝑞𝑥, (10) where 𝐾(𝑥,𝑡)=(1/(1+𝑥))𝑥^𝑡(1+1/𝑥)^𝑡_𝑞(1+𝑥)_𝑞^1−𝑡. This function is 𝑞-constant in 𝑥; that is, 𝐾(𝑞𝑥,𝑡)=𝐾(𝑥,𝑡). It was observed in [14] that 𝐵_𝑞(𝑡,𝑠) is independent of 𝐴; this is because from the integral and 𝐾(𝐴,𝑡) the term 𝐴 cancels out. In particular for any positive integer 𝑛, we have 𝐾(𝑥,𝑛)=𝑞^{𝑛(𝑛−1)/2},𝐾(𝑥,0)=1. (11) Also, we have 𝐵_𝑞()=[]𝑡,𝑠𝑡−1_𝑞![𝑠−1]_𝑞![𝑡+𝑠−1]_𝑞!. (12) In [8], Gupta and Kim obtained recurrence formula for the moments of the operators as follows.

Theorem 1 (see [8]). If one defined the central moments as 𝑇_𝑛,𝑚(𝑥)∶=𝐵^𝑞_𝑛𝑡^𝑚=[];𝑥𝑛−1_𝑞[𝑛]_𝑞^∞_𝑘=0𝑏^𝑞_𝑛,𝑘(𝑥)×₀^∞/𝐴𝑞^𝑘𝑝^𝑞_𝑛,𝑘(𝑡)𝑡^𝑚𝑑_𝑞𝑡, (13) then, for 𝑛>𝑚+2, one has the following recurrence relation: [𝑛]_𝑞−[𝑚+2]_𝑞𝑇_𝑛,𝑚+1()𝐷𝑞𝑥=𝑞𝑥1+𝑥_𝑞𝑇_𝑛,𝑚(𝑥[])+𝑞𝑚+1_𝑞+[𝑛+1]_𝑞𝑥𝑇_𝑛,𝑚𝑞𝑥. (14)

2. Moment Estimates

Lemma 2 (see [8]). The following equalities hold.(i)𝐵^𝑞_𝑛(1,𝑥)=1.(ii)𝐵^𝑞_𝑛(𝑡,𝑥)=([𝑛+1]_𝑞/𝑞²[𝑛−2]_𝑞)𝑥+(1/𝑞[𝑛−2]_𝑞).(iii)𝐵^𝑞_𝑛(𝑡²,𝑥)=([𝑛+1]_𝑞[𝑛+2]_𝑞/𝑞⁶[𝑛−2]_𝑞[𝑛−3]_𝑞)𝑥²+([𝑛+1]_𝑞[2]²_𝑞/𝑞⁵[𝑛−2]_𝑞[𝑛−3]_𝑞)𝑥+([2]_𝑞/𝑞³[𝑛−2]_𝑞[𝑛−3]_𝑞), for 𝑛>3.

Lemma 3. The following equalities hold.(i)𝐵^𝑞_{𝑛,𝛼,𝛽}(1,𝑥)=1.(ii)𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑡,𝑥)=([𝑛+1]_𝑞[𝑛]_𝑞/𝑞²[𝑛−2]_𝑞([𝑛]_𝑞+𝛽))𝑥+([𝑛]_𝑞/𝑞[𝑛−2]_𝑞+𝛼)(1/([𝑛]_𝑞+𝛽)), for 𝑛>2.(iii)𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑡²,𝑥)=([𝑛+1]_𝑞[𝑛+2]_𝑞[𝑛]²_𝑞/𝑞⁶[𝑛−2]_𝑞[𝑛−3]_𝑞([𝑛]_𝑞+𝛽)²)𝑥²+([𝑛]_𝑞[2]²_𝑞/𝑞³[𝑛−3]_𝑞+2𝛼)([𝑛+1]_𝑞[𝑛]_𝑞/𝑞²[𝑛−2]([𝑛]_𝑞+𝛽)²)𝑥+[2]_𝑞[𝑛]²_𝑞/𝑞³[𝑛−2]_𝑞[𝑛−3]_𝑞([𝑛]_𝑞+𝛽)²+2𝛼[𝑛]_𝑞/𝑞[𝑛−2]_𝑞([𝑛]_𝑞+𝛽)²+𝛼²/([𝑛]_𝑞+𝛽)², for 𝑛>3.

Proof. The operators 𝐵^𝑞_{𝑛,𝛼,𝛽} are well defined on function 1, 𝑡, 𝑡². By Lemma 2, for every 𝑛>0 and 𝑥∈[0,∞), we have 𝐵^𝑞_{𝑛,𝛼,𝛽}(1,𝑥)=𝐵^𝑞_𝑛()𝐵1,𝑥=1,_{𝑞𝑛,𝛼,𝛽}()=[]𝑡,𝑥𝑛−1_𝑞[𝑛]_𝑞^∞_𝑘=0𝑏^𝑞_𝑛,𝑘(𝑥)×₀^∞/𝐴𝑞^𝑘𝑝^𝑞_𝑛,𝑘(𝑡[𝑛])^𝑞𝑡+𝛼[𝑛]_𝑞𝑑+𝛽^𝑞𝑡=[𝑛]_𝑞[𝑛]_𝑞𝐵+𝛽^𝑞𝑛()+𝛼𝑡,𝑥[𝑛]_𝑞𝐵+𝛽^𝑞𝑛()=[]𝑡,𝑥𝑛+1_𝑞[𝑛]_𝑞𝑞²[𝑛−2]_𝑞[𝑛]_𝑞𝑥+[𝑛]+𝛽_𝑞𝑞[𝑛−2]_𝑞1+𝛼[𝑛]_𝑞.+𝛽 (15) Similarly, 𝐵^𝑞_{𝑛,𝛼,𝛽}𝑡²=[],𝑥𝑛−1_𝑞[𝑛]_𝑞^∞_𝑘=0𝑏^𝑞_𝑛,𝑘(𝑥)×₀^∞/𝐴𝑞^𝑘𝑝^𝑞_𝑛,𝑘(𝑡)[𝑛]_𝑞𝑡+𝛼[𝑛]_𝑞+𝛽²𝑑_𝑞𝑡=[𝑛]_𝑞[𝑛]_𝑞+𝛽²𝐵^𝑞_𝑛𝑡²+[𝑛],𝑥2𝛼_𝑞[𝑛]_𝑞+𝛽²𝐵^𝑞_𝑛()+𝛼𝑡,𝑥[𝑛]_𝑞+𝛽²𝐵^𝑞_𝑛()=[]1,𝑥𝑛+1_𝑞[𝑛+2]_𝑞[𝑛]²_𝑞𝑞⁶[𝑛−2]_𝑞[𝑛−3]_𝑞[𝑛]_𝑞+𝛽²𝑥²+[𝑛]_𝑞[2]²_𝑞𝑞³[𝑛−3]_𝑞×[]+2𝛼𝑛+1_𝑞[𝑛]_𝑞𝑞²[𝑛−2][𝑛]_𝑞+𝛽²𝑥+[2]_𝑞[𝑛]²_𝑞𝑞³[𝑛−2]_𝑞[𝑛−3]_𝑞[𝑛]_𝑞+𝛽²+2𝛼[𝑛]_𝑞𝑞[𝑛−2]_𝑞[𝑛]_𝑞+𝛽²+𝛼²[𝑛]_𝑞+𝛽². (16)

Remark 4. For all 𝑚∈{0,1,2,…}, 0≤𝛼≤𝛽, we have the following recursive relation for the images of the monomials 𝑡^𝑚 under 𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑡^𝑚,𝑥) in terms of 𝐵^𝑞_𝑛(𝑡^𝑟,𝑥), 𝑟=0,1,2,…,𝑚, as 𝐵^𝑞_{𝑛,𝛼,𝛽}𝑡^𝑚,𝑥=^𝑚_𝑟=0𝑚𝑟𝑛^𝑟𝛼^𝑚−𝑟𝑛+𝛽^𝑚𝐵^𝑞_𝑛𝑡^𝑟,𝑥. (17)

Remark 5. If we put 𝑞=1 and 𝛼=𝛽=0, we get the moments of the modified Beta operators [6] as 𝐵¹_𝑛()=()𝑡,𝑥𝑛+1𝑥+1𝐵𝑛−2,𝑛>2,_1𝑛𝑡²=()𝑥,𝑥𝑛+1)(𝑛+2²+4(𝑛+1)𝑥+2(𝑛−2)(𝑛−3),𝑛>3. (18)

Remark 6. From Lemma 3, we have 𝐷_{𝑛,𝛼,𝛽}(𝑥)=𝐵^𝑞_{𝑛,𝛼,𝛽}))=[]((𝑡−𝑥,𝑥𝑛+1_𝑞[𝑛]_𝑞𝑞²[𝑛−2]_𝑞[𝑛]_𝑞𝑥+[𝑛]+𝛽−1_𝑞𝑞[𝑛−2]_𝑞1+𝛼[𝑛]_𝑞,+𝛽for𝐸𝑛>2,_{𝑛,𝛼,𝛽}(𝑥)=𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑡−𝑥)²,𝑥=𝐵^𝑞_{𝑛,𝛼,𝛽}𝑡²,𝑥−2𝑥𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑡,𝑥)+𝑥²=[𝑛+1]_𝑞[𝑛+2]_𝑞[𝑛]²_𝑞𝑞⁶[𝑛−2]_𝑞[𝑛−3]_𝑞[𝑛]_𝑞+𝛽²−2[𝑛+1]_𝑞[𝑛]_𝑞𝑞²[𝑛−2]_𝑞[𝑛]_𝑞𝑥+𝛽+1²+[𝑛+1]_𝑞[𝑛]²_𝑞[2]²_𝑞𝑞⁵[𝑛−2]_𝑞[𝑛−3]_𝑞[𝑛]_𝑞+𝛽²+2[𝑛+1]_𝑞[𝑛]_𝑞𝛼𝑞²[𝑛−2]_𝑞[𝑛]_𝑞+𝛽²[𝑛]−2^𝑞𝑞[𝑛−2]_𝑞1+𝛼[𝑛]_𝑞𝑥+[2]+𝛽_𝑞[𝑛]²_𝑞𝑞³[𝑛−2]_𝑞[𝑛−3]_𝑞[𝑛]_𝑞+𝛽²+2𝛼[𝑛]_𝑞𝑞[𝑛−2]_𝑞[𝑛]_𝑞+𝛽²+𝛼²[𝑛]_𝑞+𝛽²,for𝑛>3. (19)

3. Direct Result and Asymptotic Formula

Let the space 𝐶_𝐵[0,∞) of all real-valued continuous bounded functions be endowed with the norm ‖𝑓‖=sup{|𝑓(𝑡)|∶𝑥∈[0,∞)}. Further, let us consider the following 𝐾-functional: 𝐾₂𝑓;𝛿=inf_{𝑔∈𝐶2[)0,∞}𝑓−𝑔+𝛿𝑔^′′, (20) where 𝛿>0 and 𝐶²_𝐵={𝑔∈𝐶_𝐵[0,∞)∶𝑔^′,𝑔^′′∈𝐶_𝐵[0,∞)}. By [15, page 177, Theorem  2.4], there exists an absolute constant 𝑀>0 such that 𝐾₂𝑓;𝛿≤𝑀𝜔₂𝑓;√𝛿, (21) where 𝜔₂𝑓;√𝛿=sup_{√0<ℎ≤𝛿}sup_{𝑥∈[0,∞)}𝑓(𝑥+2ℎ)−2𝑓(𝑥+ℎ)+𝑓(𝑥) (22) is the second-order modulus of smoothness of 𝑓∈𝐶_𝐵[0,∞). Also we set 𝜔𝑓;𝛿=sup_{√0<ℎ≤𝛿}sup_{𝑥∈[0,∞)}𝑓(𝑥+ℎ)−𝑓(𝑥). (23)

Theorem 7. Let 𝑓∈𝐶_𝐵[0,∞) and 𝑞=𝑞_𝑛∈(0,1) such that 𝑞_𝑛→1 as 𝑛→∞. Then for all 𝑥∈[0,∞) and 𝑛>3, there exists an absolute constant 𝐶>0 such that 𝐵^𝑞_{𝑛,𝛼,𝛽}𝑓;𝑥−𝑓(𝑥)≤𝐶𝜔₂𝑓;𝐸_{𝑛,𝛼,𝛽}(𝑥)+𝐷²_{𝑛,𝛼,𝛽}(𝑥)+𝜔𝑓;𝐷_{𝑛,𝛼,𝛽}(𝑥). (24)

Proof. We are introducing the auxiliary operators as follows: 𝐵^𝑞_{𝑛,𝛼,𝛽}𝑓;𝑥=𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑥)[]𝑓;𝑥+𝑓−𝑓𝑛+1_𝑞[𝑛]_𝑞𝑞²[𝑛−2]_𝑞[𝑛]_𝑞×𝑞+𝛽𝑥+²𝛼[𝑛−2]_𝑞[𝑛]_𝑞[𝑛+1]_𝑞+𝑞[𝑛+1]_𝑞. (25) From (25) and Lemma 3, we have 𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑡−𝑥;𝑥)=0. (26) Let 𝑥∈[0,∞) and 𝑔∈𝐶_𝐵²[0,∞). Using Taylor’s formula 𝑔(𝑡)−𝑔(𝑥)=(𝑡−𝑥)𝑔^′(𝑥)+^𝑡_𝑥(𝑡−𝑢)𝑔^′′(𝑢)𝑑𝑢, (27) applying 𝐵^𝑞_{𝑛,𝛼,𝛽}, and by (26), we get 𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑥)=𝐵𝑔;𝑥−𝑔_{𝑞𝑛,𝛼,𝛽}(𝑡−𝑥)𝑔^′(𝑥)+𝐵;𝑥_{𝑞𝑛,𝛼,𝛽𝑡𝑥}(𝑡−𝑢)𝑔^′′(𝑢)𝑑𝑢;𝑥=𝑔^′(𝑥)𝐵^𝑞_{𝑛,𝛼,𝛽}((𝑡−𝑥);𝑥)+𝐵^𝑞_{𝑛,𝛼,𝛽}^𝑡_𝑥(𝑡−𝑢)𝑔^′′(𝑢)−𝑑𝑢;𝑥_{([𝑛+1]𝑞[𝑛]𝑞/𝑞2[𝑛−2]𝑞([𝑛]𝑞+𝛽))𝑥+([𝑛]𝑞/𝑞[𝑛−2]𝑞+𝛼)(1/([𝑛]𝑞𝑥+𝛽))}[𝑛+1]_𝑞[𝑛]_𝑞𝑞²[𝑛−2]_𝑞[𝑛]_𝑞𝑥+[𝑛]+𝛽_𝑞𝑞[𝑛−2]_𝑞1+𝛼[𝑛]_𝑞𝑔+𝛽−𝑢^′′(𝑢)𝑑𝑢=𝐵^𝑞_{𝑛,𝛼,𝛽}^𝑡_𝑥(𝑡−𝑢)𝑔^′′(𝑢)−𝑑𝑢;𝑥_{[𝑛+1]𝑞[𝑛]𝑞𝑥/𝑞2[𝑛−2]𝑞([𝑛]𝑞+𝛽)+([𝑛]𝑞/𝑞[𝑛−2]𝑞+𝛼)(1/([𝑛]𝑞𝑥+𝛽))}[𝑛+1]_𝑞[𝑛]_𝑞𝑥𝑞²[𝑛−2]_𝑞[𝑛]_𝑞+[𝑛]+𝛽_𝑞𝑞[𝑛−2]_𝑞1+𝛼[𝑛]_𝑞𝑔+𝛽−𝑢^′′(𝑢)𝑑𝑢. (28) On the other hand, since ^𝑡_𝑥|𝑡−𝑢|𝑔^′′(𝑢)𝑑𝑢≤𝑔^′′(𝑡−𝑥)²,_{[𝑛+1]𝑞[𝑛]𝑞𝑥/𝑞2[𝑛−2]𝑞([𝑛]𝑞+𝛽)+([𝑛]𝑞/𝑞[𝑛−2]𝑞+𝛼)(1/([𝑛]𝑞𝑥+𝛽))}[𝑛+1]_𝑞[𝑛]_𝑞𝑥𝑞²[𝑛−2]_𝑞[𝑛]_𝑞+[𝑛]+𝛽𝑞[𝑛−^𝑞2]_𝑞×1+𝛼[𝑛]_𝑞𝑔+𝛽−𝑢^′′(𝑢)≤𝑔𝑑𝑢_′′[𝑛+1]_𝑞[𝑛]_𝑞𝑥𝑞²[𝑛−2]_𝑞[𝑛]_𝑞+[𝑛]+𝛽_𝑞𝑞[𝑛−2]_𝑞1+𝛼[𝑛]_𝑞+𝛽−𝑥²=𝐷²_{𝑛,𝛼,𝛽}(𝑥)𝑔^′′. (29) We conclude by Remark 6 that 𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑥=𝐵𝑔;𝑥−𝑔)_{𝑞𝑛,𝛼,𝛽𝑡𝑥}(𝑡−𝑢)𝑔^′′(𝑢)+𝑑𝑢;𝑥_{[𝑛+1]𝑞[𝑛]𝑞𝑥/𝑞2[𝑛−2]𝑞([𝑛]𝑞+𝛽)+([𝑛]𝑞/𝑞[𝑛−2]𝑞+𝛼)(1/([𝑛]𝑞𝑥+𝛽))}[𝑛+1]_𝑞[𝑛]_𝑞𝑥𝑞²[𝑛−2]_𝑞[𝑛]_𝑞+[𝑛]+𝛽_𝑞𝑞[𝑛−2]_𝑞1+𝛼[𝑛]_𝑞𝑔+𝛽−𝑢^′′(𝑢)𝑑𝑢≤𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑡−𝑥)²𝑔^′′+[];𝑥𝑛+1_𝑞[𝑛]_𝑞𝑥𝑞²[𝑛−2]_𝑞[𝑛]_𝑞+[𝑛]+𝛽_𝑞𝑞[𝑛−2]_𝑞1+𝛼([𝑛]_𝑞+𝛽)−𝑥²𝑔^′′𝐵^𝑞_{𝑛,𝛼,𝛽}𝑔;𝑥−𝑔(𝑥)=𝐸_{𝑛,𝛼,𝛽}(𝑥)𝑔^′′+𝐷²_{𝑛,𝛼,𝛽}(𝑥)𝑔^′′. (30)(31) From (25), we can write that 𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑥≤𝐵𝑓;𝑥−𝑓)_{𝑞𝑛,𝛼,𝛽}(𝑥+𝑓(𝑥)[]𝑓;𝑥−𝑓)−𝑓𝑛+1_𝑞[𝑛]_𝑞𝑞²[𝑛−2]_𝑞[𝑛]_𝑞×𝑞+𝛽𝑥+²𝛼[𝑛−2]_𝑞[𝑛]_𝑞[𝑛+1]_𝑞+𝑞[𝑛+1]_𝑞≤𝐵^𝑞_{𝑛,𝛼,𝛽}𝑓−𝑔;𝑥−𝑓−𝑔(𝑥)+𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑥+𝑓(𝑥)[]𝑔;𝑥−𝑔)−𝑓𝑛+1_𝑞[𝑛]_𝑞𝑞²[𝑛−2]_𝑞[𝑛]_𝑞×𝑞+𝛽𝑥+²𝛼[𝑛−2]_𝑞[𝑛]_𝑞[𝑛+1]_𝑞+𝑞[𝑛+1]_𝑞. (32) Now, taking into account the boundedness of 𝐵^𝑞_{𝑛,𝛼,𝛽} and from (31), we get 𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑥+𝐸𝑓;𝑥−𝑓)≤4𝑓−𝑔_{𝑛,𝛼,𝛽}(𝑥)+𝐷²_{𝑛,𝛼,𝛽}(𝑥)𝑔^′′+𝑓(𝑥)[]−𝑓𝑛+1_𝑞[𝑛]_𝑞𝑞²[𝑛−2]_𝑞[𝑛]_𝑞×𝑞+𝛽𝑥+²𝛼[𝑛−2]_𝑞[𝑛]_𝑞[𝑛+1]_𝑞+𝑞[𝑛+1]_𝑞≤4𝑓−𝑔+𝜔𝑓;𝐷_{𝑛,𝛼,𝛽}(𝑥+𝐸_{𝑛,𝛼,𝛽})(𝑥)+𝐷²_{𝑛,𝛼,𝛽}(𝑥)𝑔^′′. (33) Now, taking infimum on the right-hand side over all 𝑔∈𝐶_𝐵²[0,∞) and from (21), we get 𝐵^𝑞_{𝑛,𝛼,𝛽}𝑓;𝑥−𝑓(𝑥)≤4𝐾₂𝑓;𝐸_{𝑛,𝛼,𝛽}(𝑥)+𝐷²_{𝑛,𝛼,𝛽}(𝑥)+𝜔𝑓;𝐷_{𝑛,𝛼,𝛽}(𝑥)=𝐶𝜔₂𝑓;𝐸_{𝑛,𝛼,𝛽}(𝑥)+𝐷²_{𝑛,𝛼,𝛽}(𝑥)+𝜔𝑓;𝐷_{𝑛,𝛼,𝛽}(𝑥), (34) where 4𝐶₁=𝐶>0. This proves the theorem.

Our next result in this section is an asymptotic formula.

Theorem 8. Let 𝑓 be bounded and integrable function on the interval [0,∞); the second derivative of 𝑓 exists at a fixed point 𝑥∈[0,∞) and 𝑞=𝑞_𝑛∈(0,1) such that 𝑞_𝑛→1 as 𝑛→∞. Consider lim_𝑛→∞[𝑛]_𝑞𝑛𝐵^𝑞𝑛_{𝑛,𝛼,𝛽}(𝑥=()𝑓𝑓;𝑥−𝑓)3−𝛽𝑥+1+𝛼^′(𝑥)+𝑥(1+𝑥)𝑓^′′(𝑥). (35)

Proof. Using Taylor’s expansion of 𝑓, we can write 𝑓(𝑡)−𝑓(𝑥)=(𝑡−𝑥)𝑓^′(𝑥)+(𝑡−𝑥)²𝑓2!^′′(𝑥)+𝜀(𝑡−𝑥)(𝑡−𝑥)², (36)
where 𝜀 is bounded and lim_𝑡→0𝜀(𝑡)=0. Applying the operator 𝐵^𝑞𝑛_{𝑛,𝛼,𝛽} to the above relation, we get 𝐵^𝑞𝑛_{𝑛,𝛼,𝛽}𝑓;𝑥−𝑓(𝑥)=𝑓^′(𝑥)𝐵^𝑞𝑛_{𝑛,𝛼,𝛽}))+𝑓((𝑡−𝑥;𝑥_′′(𝑥)𝐵2!^{𝑞𝑛𝑛,𝛼,𝛽}(𝑡−𝑥)²;𝑥+𝐵^𝑞𝑛_{𝑛,𝛼,𝛽}𝜀(𝑡−𝑥)(𝑡−𝑥)²;𝑥=𝑓^′(𝑥)𝐷_{𝑛,𝛼,𝛽}(𝑥)+𝑓^′′(𝑥)𝐸2!^{𝑛,𝛼,𝛽}(𝑥)+𝐵^𝑞𝑛_{𝑛,𝛼,𝛽}𝜀(𝑡−𝑥)(𝑡−𝑥)²;𝑥, (37) where 𝐷_{𝑛,𝛼,𝛽}(𝑥) and 𝐸_{𝑛,𝛼,𝛽}(𝑥) are defined in Remark 6.
Using Cauchy-Schwarz inequality, we have [𝑛]_𝑞𝑛𝐵^𝑞𝑛_{𝑛,𝛼,𝛽}𝜀(𝑡−𝑥)(𝑡−𝑥)²;𝑥≤𝐵^𝑞𝑛_{𝑛,𝛼,𝛽}𝜀²(𝑡−𝑥);𝑥^1/2[𝑛]²_𝑞𝑛𝐵^𝑞𝑛_{𝑛,𝛼,𝛽}(𝑡−𝑥)⁴;𝑥^1/2. (38) Using Theorem 1 with the help of Remark 4, we can easily find that lim_𝑛→∞[𝑛]²_𝑞𝑛𝐵^𝑞𝑛_{𝑛,𝛼,𝛽}(𝑡−𝑥)⁴;𝑥=0. (39) Also, since lim_𝑛→∞[𝑛]_𝑞𝑛𝐷_{𝑛,𝛼,𝛽}(𝑥)=3−𝛽𝑥+(1+𝛼),lim_𝑛→∞[𝑛]_𝑞𝑛𝐸_{𝑛,𝛼,𝛽}=2𝑥(1+𝑥). (40) Thus, lim_𝑛→∞[𝑛]_𝑞𝑛𝐵^𝑞𝑛_{𝑛,𝛼,𝛽}(𝑥=()𝑓𝑓;𝑥−𝑓)3−𝛽𝑥+1+𝛼^′(𝑥)+𝑥(1+𝑥)𝑓^′′(𝑥), (41) which completes the proof.

4. Better Estimation

It is well known that the operators preserve constant as well as linear functions. To make the convergence faster, King [16] proposed an approach to modify the classical Bernstein polynomials, so that this sequence preserves two test functions: 𝑒₀ and 𝑒₂. After this, several researchers have studied that many approximating operators, 𝐿, possess these properties; that is, 𝐿(𝑒_𝑖;𝑥)=𝑒_𝑖(𝑥), where 𝑒_𝑖(𝑥)=𝑥^𝑖  (𝑖=0,1) or 𝑥^𝑖  (𝑖=0,2), for example, Bernstein, Baskakov, and Baskakov-Durrmeyer-Stancu operators (see [4, 5, 17–19]).

As the operators 𝐵^𝑞_{𝑛,𝛼,𝛽} introduced in (3) preserve only the constant functions, further modification of these operators is proposed to be made so that the modified operators preserve the constant as well as linear functions. For this purpose, the modification of 𝐵^𝑞_{𝑛,𝛼,𝛽} is as follows: 𝐵^𝑞_{𝑛,𝛼,𝛽}𝑓;𝑥=𝐵^𝑞_{𝑛,𝛼,𝛽}𝑓;𝑟_𝑛,𝑞(𝑥=[])𝑛−1_𝑞[𝑛]_𝑞^∞_𝑘=0𝑏^𝑞_𝑛,𝑘𝑟_𝑛,𝑞(𝑥)×₀^∞/𝐴𝑞^𝑘𝑝^𝑞_𝑛,𝑘(𝑡)𝑓[𝑛]_𝑞𝑡+𝛼[𝑛]_𝑞𝑑+𝛽^𝑞𝑡, (42) where 𝑟_𝑛,𝑞(𝑥)=−𝑞[𝑛]_𝑞−𝛼𝑞²[𝑛−2]_𝑞+𝑞²𝑥[𝑛−2]_𝑞[𝑛]_𝑞+𝛽[𝑛]_𝑞[𝑛+1]_𝑞,𝑥∈𝐼_𝑛,𝑞=[𝑛]_𝑞𝑞[𝑛−2]_𝑞[𝑛]_𝑞+𝛼+𝛽[𝑛]_𝑞.+𝛽,∞ (43)

Lemma 9. For each 𝑥∈𝐼_𝑛,𝑞, one has 𝐵^𝑞_{𝑛,𝛼,𝛽}(1,𝑥)=1,𝐵^𝑞_{𝑛,𝛼,𝛽}()𝐵𝑡,𝑥=𝑥,_{𝑞𝑛,𝛼,𝛽}𝑡²=[],𝑥𝑛−2_𝑞[𝑛+2]_𝑞𝑞²[𝑛+1]_𝑞[𝑛−3]_𝑞𝑥²+1𝑞³[𝑛−3][𝑛]_𝑞×[𝑛]+𝛽_𝑞[2]²_𝑞+2𝑞³𝛼[𝑛−3]_𝑞−2[𝑛+2]_𝑞[𝑛]_𝑞+𝑞𝛼[𝑛−2]_𝑞[𝑛+1]_𝑞𝑥+1𝑞⁴[𝑛−2]_𝑞[𝑛−3]_𝑞[𝑛]_𝑞+𝛽²×[𝑛+2]_𝑞[𝑛]_𝑞+𝑞𝛼[𝑛−2]_𝑞²[𝑛+1]_𝑞−[𝑛]²_𝑞[2]_𝑞−𝑞𝛼[2]²_𝑞[𝑛]_𝑞[𝑛−2]_𝑞−𝑞⁴𝛼²[𝑛−3]_𝑞[𝑛−2]_𝑞,𝑓𝑜𝑟𝑛>3. (44)

Lemma 10. For each 𝑥∈𝐼_𝑛,𝑞, the following equalities hold: 𝐵^𝑞_{𝑛,𝛼,𝛽}))𝐹((𝑡−𝑥,𝑥=0,_{𝑛,𝛼,𝛽}=𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑡−𝑥)²=[],𝑥𝑛−2_𝑞[𝑛+2]_𝑞𝑞²[𝑛+1]_𝑞[𝑛−3]_𝑞−1𝑥²+1𝑞³[𝑛−3][𝑛]_𝑞×[𝑛]+𝛽_𝑞[2]²_𝑞−2[𝑛+2]_𝑞[𝑛]_𝑞+𝑞𝛼[𝑛−2]_𝑞[𝑛+1]_𝑞+2𝑞³𝛼[𝑛−3]_𝑞𝑥+1𝑞⁴[𝑛−2]_𝑞[𝑛−3]_𝑞[𝑛]_𝑞+𝛽²×[𝑛+2]_𝑞[𝑛]_𝑞+𝑞𝛼[𝑛−2]_𝑞²[𝑛+1]_𝑞−[𝑛]²_𝑞[2]_𝑞−𝑞𝛼[2]²_𝑞[𝑛]_𝑞[𝑛−2]_𝑞−𝑞⁴𝛼²[𝑛−3]_𝑞[𝑛−2]_𝑞,𝑓𝑜𝑟𝑛>3. (45)

Theorem 11. Let 𝑓∈𝐶_𝐵(𝐼_𝑛,𝑞) and 𝑞=𝑞_𝑛∈(0,1) such that 𝑞_𝑛→1 as 𝑛→∞. Then for all 𝑥∈𝐼_𝑛,𝑞 and 𝑛>3, there exists an absolute constant 𝐶>0 such that 𝐵^𝑞_{𝑛,𝛼,𝛽}𝑓;𝑥−𝑓(𝑥)≤𝐶𝜔₂𝑓;𝐹_{𝑛,𝛼,𝛽}(𝑥). (46)

Proof. Let 𝑥,𝑡∈𝐼_𝑛,𝑞 and 𝑔∈𝐶²_𝐵(𝐼_𝑛,𝑞). Using Taylor’s formula, we get 𝑔(𝑡)−𝑔(𝑥)=(𝑡−𝑥)𝑔^′(𝑥)+^𝑡_𝑥(𝑡−𝑢)𝑔^′′(𝑢)𝑑𝑢. (47) Applying 𝐵^𝑞_{𝑛,𝛼,𝛽}, we get 𝐵^𝑞_{𝑛,𝛼,𝛽}𝑔;𝑥−𝑔(𝑥)=𝑔^′(𝑥)𝐵^𝑞_{𝑛,𝛼,𝛽}))+𝐵((𝑡−𝑥;𝑥_{𝑞𝑛,𝛼,𝛽𝑡𝑥}(𝑡−𝑢)𝑔^′′(𝑢)𝑑𝑢;𝑥. (48) Obviously, we have ^𝑡_𝑥|𝑡−𝑢|𝑔^′′(𝑢)𝑑𝑢≤𝑔^′′(𝑡−𝑥)². (49) Therefore, 𝐵^𝑞_{𝑛,𝛼,𝛽}𝑔;𝑥−𝑔(𝑥)≤𝑔^′′𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑡−𝑥)²;𝑥=𝐹_{𝑛,𝛼,𝛽}𝑔^′′. (50) Since |𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑓;𝑥)−𝑓(𝑥)|≤2‖𝑓‖, thus 𝐵^𝑞_{𝑛,𝛼,𝛽}(𝑥≤𝐵𝑓;𝑥−𝑓)_{𝑞𝑛,𝛼,𝛽}−(𝑥+𝐵𝑓−𝑔;𝑥𝑓−𝑔)_{𝑞𝑛,𝛼,𝛽}𝑔;𝑥−𝑔(𝑥)≤2𝑓−𝑔+𝐹_{𝑛,𝛼,𝛽}(𝑥)𝑔^′′. (51)
Now, taking infimum on the right-hand side over all 𝑔∈𝐶_𝐵²(𝐼_𝑛,𝑞) and from (21), we get 𝐵^𝑞_{𝑛,𝛼,𝛽}𝑓;𝑥−𝑓(𝑥)≤𝐶𝜔₂𝑓;𝐹_{𝑛,𝛼,𝛽}(𝑥). (52) which proves the theorem.

Theorem 12. Let 𝑓 be bounded and integrable function on the interval 𝐼_𝑛,𝑞; the second derivative of 𝑓 exists at a fixed point 𝑥∈𝐼_𝑛,𝑞 and 𝑞=𝑞_𝑛∈(0,1) such that 𝑞_𝑛→1 as 𝑛→∞; then lim_𝑛→∞[𝑛]_𝑞𝑛𝐵^𝑞𝑛_{𝑛,𝛼,𝛽}𝑓;𝑥−𝑓(𝑥)=𝑥(1+𝑥)𝑓^′′(𝑥). (53)

The proof follows along the same lines of Theorem 8.

Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgments

The authors thank the anonymous learned reviewers for their valuable suggestions, which substantially improved the standard of the paper. Special thanks are due to Professor Adam Kowalewski, for kind cooperation and smooth behavior during communication and for his efforts to send the reports of the paper timely.