Approximation of the Summation-Integral-Type 𝑞-Szász-Mirakjan Operators

Abstract

We introduce summation-integral-type 𝑞-Szász-Mirakjan operators and study approximation properties of these operators. We establish local approximation theorem. We give weighted approximation theorem. Also we estimate the rate of convergence of these operators for functions of polynomial growth on the interval [0,∞).

1. Introduction

First of all, we mention some concepts and notations from 𝑞-calculus. All of the results can be found in [1–5]. In what follows, 𝑞 is a real number satisfying 0<𝑞<1.

For each nonnegative integer 𝑘, the 𝑞-integer [𝑘]_𝑞 and the 𝑞-factorial [𝑘]_𝑞! are defined as [𝑘]_𝑞∶=1−𝑞^𝑘/[𝑘]1−𝑞,𝑞≠1,𝑘,𝑞=1,_𝑞[𝑘]!∶=^𝑞[𝑘−1]_𝑞⋯[1]_𝑞,𝑘≥11,𝑘=0. (1.1)

Then for integers 𝑛,𝑘, 𝑛≥𝑘≥0, we have [𝑘+1]_𝑞=1+𝑞[𝑘]_𝑞,[𝑘]_𝑞+𝑞^𝑘[𝑛−𝑘]_𝑞=[𝑛]_𝑞. (1.2)

For the integers 𝑛,𝑘,𝑛≥𝑘≥0, the 𝑞-binomial coefficient is defined as 𝑛𝑘_𝑞[𝑛]∶=^𝑞![𝑘]_𝑞![𝑛−𝑘]_𝑞!. (1.3)

The two 𝑞-analogues of the exponential function are defined as 𝑒_𝑞(𝑥)∶=^∞_𝑛=0𝑥^𝑛[𝑛]_𝑞!=11−1−𝑞𝑥^∞_𝑞,|𝑥|<1,𝐸1−𝑞_𝑞(𝑥)∶=^∞_𝑛=0𝑞^{𝑛(𝑛−1)/2}𝑥^𝑛[𝑛]_𝑞!=1+1−𝑞𝑥^∞_𝑞, (1.4) where (1+𝑥)^∞_𝑞∶=∏^∞_𝑖=0(1+𝑞^𝑖𝑥).

It is easily observed that 𝑒_𝑞(𝑥)𝐸_𝑞(−𝑥)=𝑒_𝑞(−𝑥)𝐸_𝑞(𝑥)=1. (1.5)

The 𝑞-Jackson integral and the 𝑞-improper integral are defined as ^𝑎₀𝑓(𝑡)𝑑_𝑞𝑡∶=1−𝑞𝑎^∞_𝑛=0𝑓𝑎𝑞^𝑛𝑞^𝑛,𝑎>0,_0∞/𝐴𝑓(𝑡)𝑑_𝑞𝑡∶=1−𝑞^∞_𝑛=−∞𝑓𝑞^𝑛𝐴𝑞^𝑛𝐴,𝐴>0, (1.6) provided the sums converge absolutely.

For 𝑡>0, the 𝑞-Gamma function is given by Γ_𝑞(𝑠)=𝐾(𝐴,𝑠)₀^{∞/𝐴(1−𝑞)}𝑡^𝑠−1𝑒_𝑞(−𝑡)𝑑_𝑞𝑡, (1.7) where 𝐾()=𝐴𝐴,𝑠^𝑠11+𝐴1+𝐴^𝑠_𝑞(1+𝐴)_𝑞^1−𝑠,𝐴>0. (1.8)

It was observed in [5] that 𝐾(𝑥,𝑠) is a 𝑞-constant, that is, 𝐾(𝑞𝑥,𝑠)=𝐾(𝑥,𝑠). In particular, for 𝑠∈ℕ, 𝐾(𝐴,𝑠)=𝑞^{𝑠(𝑠−1)/2} and 𝐾(𝐴,0)=1. Γ_𝑞(𝑠+1)=[𝑠]_𝑞Γ_𝑞(𝑠), Γ_𝑞(1)=1.

In 1997, Phillips [6] firstly introduced and studied 𝑞 analogue of Bernstein polynomials. After this, the applications of 𝑞-calculus in the approximation theory become one of the main areas of research; many authors studied new classes of 𝑞-generalized operators (for instance, see [7–12]). In order to approximate integrable functions, in 2008 Gupta and Heping [13] introduced and studied the 𝑞-Durrmeyer-type operators as follows:𝑀_𝑛,𝑞𝑓;𝑥=[𝑛+1]_𝑞^𝑛_𝑘=1𝑞^1−𝑘𝑝_𝑛,𝑘𝑞;𝑥¹₀𝑝_{𝑛,𝑘−1}𝑞;𝑞𝑡𝑓(𝑡)𝑑_𝑞𝑡+𝑓(0)𝑝_𝑛,0𝑞;𝑥, (1.9) where 𝑓∈𝐶[0,1], 𝑥∈[0,1], 𝑛=1,2,…, 0<𝑞<1, and 𝑝_𝑛,𝑘𝑛𝑘𝑞;𝑥∶=_𝑞𝑥^{𝑘𝑛−𝑘−1}_𝑠=01−𝑞^𝑠𝑥. (1.10)

In 2010, Mahmudov and Kaffaoğlu [14] defined and studied 𝑞-Szász-Durrmeyer operators as follows: 𝐷_𝑛,𝑞𝑓;𝑥=[𝑛]_𝑞^∞_𝑘=0𝑞^𝑘𝑠_𝑛,𝑘𝑞;𝑥₀^{∞/(1−𝑞)}𝑠_𝑛,𝑘𝑞;𝑡𝑓(𝑡)𝑑_𝑞𝑡, (1.11) where 𝑓∈ℝ^[0,∞), 0<𝑞<1, 𝑛∈ℕ, and 𝑠_𝑛,𝑘=1𝑞;𝑥𝐸_𝑞[𝑛]_𝑞𝑥𝑞^{𝑘(𝑘−1)/2}[𝑛]^𝑘_𝑞𝑥^𝑘[𝑘]_𝑞!. (1.12)

In 2011, Aral and Gupta [15] introduced and studied 𝑞-generalized Szász-Durrmeyer operators by means of the 𝑞-integral as follows: 𝒟^𝑞_𝑛𝑓(𝑡)=[𝑛],𝑥^𝑞𝑏_𝑛^∞_𝑘=0𝑠^𝑞_𝑛,𝑘(𝑥)^{𝑞𝑏𝑛/(1−𝑞𝑛)}₀𝑠^𝑞_𝑛,𝑘(𝑡)𝑓(𝑡)𝑑_𝑞𝑡, (1.13) where 𝑓∈𝐶([0,∞)), 0<𝑞<1, 𝑛∈ℕ, {𝑏_𝑛} is a sequence of positive numbers such that lim_𝑛→∞𝑏_𝑛=∞, 0≤𝑥<𝛼_𝑞(𝑛), 𝛼_𝑞(𝑛)∶=𝑏_𝑛/(1−𝑞)[𝑛]_𝑞 and 𝑠^𝑞_𝑛,𝑘(𝑥)=[𝑛]_𝑞𝑥^𝑘𝑞^(𝑘+1)/2[𝑘]_𝑞!𝑏_𝑛^𝑘𝐸_𝑞−[𝑛]_𝑞𝑥𝑏_𝑛. (1.14)

In the present paper, we will introduce a kind of 𝑞-analogue of the summation-integral-type Szász-Mirakjan operators as follows.

Let 𝑓∈𝐶([0,∞)). For every 𝑛∈ℕ, 𝑞∈(0,1), 𝑥∈[0,∞), and 𝐴>0, the summation-integral-type 𝑞-Szász Mirakjan operators are defined as 𝑆_𝑛,𝑞𝑓;𝑥=[𝑛]_𝑞^∞_𝑘=0𝑠_𝑛,𝑘𝑥,𝑞₀^{∞/𝐴(1−𝑞)}𝑞^𝑘2𝑓(𝑡)𝑠_𝑛,𝑘𝑡,𝑞𝑑_𝑞𝑡, (1.15) where 𝑠_𝑛,𝑘=[𝑛]𝑥,𝑞^𝑞𝑥^𝑘[𝑘]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑥. (1.16)

We will study the approximation properties of the operators (1.15). We point out that the operators (1.15) considered in this paper are more general than the operators (1.11) considered in [14]. It is easy to verify when 𝐴=1, the operators (1.15) reduce to the operators (1.11). Thus, the operators (1.11) are the special case of the operators (1.15). The literature [14] studied the local approximation and the Voronovskaja-type theorem for the operators (1.11). In the present paper, besides the local approximation of the operators (1.15), our research work is different from the literature in [14]; we study the rate of convergence, the weighted Korovkin-type theorem, and the weighted approximation error of the operators (1.15) and obtain some new results. As regards the operators (1.13) considered in [15], it is obvious that these operators are quite different from the operators (1.15). In [15], for the operators (1.13), a direct approximation result in weighted function space with the help of a weighted Korovkin type theorem on the finite interval was obtained. The weighted approximation error of these operators in terms of weighted modulus of continuity was given. Also, an asymptotic formula was established. In the present paper, we introduce a new weighted modulus of continuity which is different from that in [15]. We obtain some results of the weighted approximation with the help of the new weighted modulus of continuity on the infinite interval.

2. Two Lemmas

For the operators 𝑆_𝑛,𝑞(𝑓;𝑥) defined by (1.15), we give the following two lemmas.

Lemma 2.1. For 𝑆_𝑛,𝑞(𝑡^𝑖;𝑥),𝑖=0,1,2,3,4, we have (i)𝑆_𝑛,𝑞(1;𝑥)=1; (ii)𝑆_𝑛,𝑞(𝑡;𝑥)=𝑥/𝑞²+1/𝑞[𝑛]_𝑞; (iii)𝑆_𝑛,𝑞(𝑡²;𝑥)=𝑥²/𝑞⁶+([3]_𝑞+𝑞)𝑥/𝑞⁵[𝑛]_𝑞+[2]_𝑞/𝑞³[𝑛]²_𝑞; (iv)𝑆_𝑛,𝑞(𝑡³;𝑥)=𝑥³/𝑞¹²+(𝑥²/𝑞¹¹[𝑛]_𝑞)([5]_𝑞+𝑞[3]_𝑞+𝑞²)+(𝑥/𝑞⁹[𝑛]²_𝑞)([6]_𝑞+2𝑞[4]_𝑞+2𝑞²[2]_𝑞)   +[2]_𝑞[3]_𝑞/𝑞⁶[𝑛]³_𝑞; (v)𝑆_𝑛,𝑞(𝑡⁴;𝑥)=𝑥⁴/𝑞²⁰+(𝑥³/𝑞¹⁹[𝑛]_𝑞)([7]_𝑞+𝑞[5]_𝑞+𝑞²[3]_𝑞+𝑞³)+(𝑥²/𝑞¹⁷[𝑛]²_𝑞)([10]_𝑞+2𝑞[8]_𝑞  +4𝑞²[6]_𝑞+4𝑞³[4]_𝑞+3𝑞⁴[2]_𝑞)+(𝑥/𝑞¹⁴[𝑛]³_𝑞)([10]_𝑞+3𝑞[8]_𝑞+5𝑞²[6]_𝑞+6𝑞³[4]_𝑞+4𝑞⁴[2]_𝑞)  +[2]_𝑞[3]_𝑞[4]_𝑞/𝑞¹⁰[𝑛]⁴_𝑞.

Proof. In view of the concepts of the 𝑞-improper integral and 𝑞-exponential function, for nonnegative integer 𝑖, we have ₀^{∞/𝐴(1−𝑞)}𝑡^𝑖𝑠_𝑛,𝑘𝑡,𝑞𝑑_𝑞𝑡=₀^{∞/𝐴(1−𝑞)}𝑡^𝑖[𝑛]_𝑞𝑡^𝑘[𝑘]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑡𝑑_𝑞𝑡=1[𝑛]_𝑞^𝑖+1[𝑘]_𝑞!^{∞/(𝐴/[𝑛]𝑞0)(1−𝑞)}𝑢^𝑘+𝑖𝑒_𝑞(−𝑢)𝑑_𝑞𝑢=Γ_𝑞(𝑘+𝑖+1)[𝑛]_𝑞^𝑖+1[𝑘]_𝑞!𝐾𝐴/[𝑛]_𝑞=[],𝑘+𝑖+1𝑘+𝑖_𝑞![𝑛]_𝑞^𝑖+1[𝑘]_𝑞!𝑞^{(𝑘+𝑖+1)(𝑘+𝑖)/2}. (2.1)(i) When 𝑖=0, by the formulas (1.5) and (2.1), we can get 𝑆_𝑛,𝑞(1;𝑥)=^∞_𝑘=0[𝑛]_𝑞𝑥^𝑘[𝑘]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑥𝑞^{𝑘(𝑘−1)/2}=𝑒_𝑞−[𝑛]_𝑞𝑥𝐸_𝑞[𝑛]_𝑞𝑥=1. (2.2)(ii)When 𝑖=1, using [𝑘+1]_𝑞=[𝑘]_𝑞+𝑞^𝑘, by the formulas (1.5) and (2.1), we can get 𝑆_𝑛,𝑞(𝑡;𝑥)=^∞_𝑘=0[𝑛]_𝑞𝑥^𝑘[𝑘]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑥𝑞^{(𝑘2−3𝑘−2)/2}[𝑘]_𝑞+𝑞^𝑘[𝑛]_𝑞=1[𝑛]_𝑞^∞_𝑘=1[𝑛]_𝑞𝑥^𝑘[𝑘−1]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑥𝑞^{(𝑘2−3𝑘−2)/2}+1𝑞[𝑛]_𝑞=𝑥𝑞²+1𝑞[𝑛]_𝑞. (2.3)(iii)When 𝑖=2, using [𝑘+2]_𝑞[𝑘+1]_𝑞=[𝑘]²_𝑞+𝑞^𝑘(1+[2]_𝑞)[𝑘]_𝑞+𝑞^2𝑘[2]_𝑞, by the formulas (1.5) and (2.1), we can get 𝑆_𝑛,𝑞𝑡²;𝑥=^∞_𝑘=0[𝑛]_𝑞𝑥^𝑘[𝑘]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑥𝑞^{(𝑘2−5𝑘−6)/2}[𝑘]²_𝑞+𝑞^𝑘1+[2]_𝑞[𝑘]_𝑞+𝑞^2𝑘[2]_𝑞[𝑛]²_𝑞=1[𝑛]²_𝑞^∞_𝑘=2[𝑛]_𝑞𝑥^𝑘[𝑘−2]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑥𝑞^{(𝑘2−5𝑘−6)/2}+[3]_𝑞+𝑞[𝑛]²_𝑞^∞_𝑘=1[𝑛]_𝑞𝑥^𝑘[𝑘−1]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑥𝑞^{(𝑘2−3𝑘−8)/2}+[2]_𝑞𝑞³[𝑛]²_𝑞=𝑥²𝑞⁶+[3]_𝑞+𝑞𝑥𝑞⁵[𝑛]_𝑞+[2]_𝑞𝑞³[𝑛]²_𝑞. (2.4)(iv)When 𝑖=3, by the formulas (1.5) and (2.1), we can get 𝑆_𝑛,𝑞𝑡³;𝑥=^∞_𝑘=0[𝑛]_𝑞𝑥^𝑘[𝑘]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑥𝑞^{(𝑘2−7𝑘−12)/2}[𝑘+3]_𝑞[𝑘+2]_𝑞[𝑘+1]_𝑞[𝑛]³_𝑞. (2.5)
Using [𝑘+3]_𝑞[𝑘+2]_𝑞[𝑘+1]_𝑞=[𝑘]³_𝑞+𝑞^𝑘(1+[2]_𝑞+[3]_𝑞)[𝑘]²_𝑞+𝑞^2𝑘([2]_𝑞+[3]_𝑞+[2]_𝑞[3]_𝑞)[𝑘]_𝑞  +𝑞^3𝑘[2]_𝑞[3]_𝑞, we have 𝑆_𝑛,𝑞𝑡³=1;𝑥[𝑛]³_𝑞^∞_𝑘=3[𝑛]_𝑞𝑥^𝑘[𝑘−3]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑥𝑞^{(𝑘2−7𝑘−12)/2}+1[𝑛]³_𝑞^∞_𝑘=2[𝑛]_𝑞𝑥^𝑘[𝑘−2]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑥𝑞^{(𝑘2−5𝑘−16)/2}1+2𝑞+𝑞²1+[2]_𝑞+[3]_𝑞+1[𝑛]³_𝑞^∞_𝑘=1[𝑛]_𝑞𝑥^𝑘[𝑘−1]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑥𝑞^{(𝑘2−3𝑘−16)/2}×1+𝑞1+[2]_𝑞+[3]_𝑞+𝑞²[2]_𝑞+[3]_𝑞+[2]_𝑞[3]_𝑞+[2]_𝑞[3]_𝑞𝑞⁶[𝑛]³_𝑞=𝑥³𝑞¹²+𝑥²𝑞¹¹[𝑛]_𝑞[5]_𝑞+𝑞[3]_𝑞+𝑞²+𝑥𝑞⁹[𝑛]²_𝑞[6]_𝑞+2𝑞[4]_𝑞+2𝑞²[2]_𝑞+[2]_𝑞[3]_𝑞𝑞⁶[𝑛]³_𝑞. (2.6)(v)When 𝑖=4, similar to the case of 𝑖=3, by simple calculation we can get the desired result.

Lemma 2.2. Let 𝑞∈(0,1),𝑥∈[0,∞), we have (i)𝑆_𝑛,𝑞(𝑡−𝑥;𝑥)=((1−𝑞²)/𝑞²)𝑥+1/𝑞[𝑛]_𝑞; (ii)𝑆_𝑛,𝑞((𝑡−𝑥)²;𝑥)≤(2/𝑞⁶)(1−𝑞⁴+2/[𝑛]_𝑞)𝑥(1+𝑥)+[2]_𝑞/𝑞³[𝑛]²_𝑞.

Proof. By Lemma 2.1, we have 𝑆_𝑛,𝑞(𝑡−𝑥;𝑥)=𝑆_𝑛,𝑞(𝑡;𝑥)−𝑥=1−𝑞²𝑞²1𝑥+𝑞[𝑛]_𝑞. (2.7)
By Lemma 2.1, we have 𝑆_𝑛,𝑞(𝑡−𝑥)²;𝑥=𝑆_𝑛,𝑞𝑡²;𝑥−2𝑥𝑆_𝑛,𝑞(𝑡;𝑥)+𝑥²=𝑥²1𝑞⁶−2𝑞²[3]+1+𝑥^𝑞+𝑞𝑞⁵[𝑛]_𝑞−2𝑞[𝑛]_𝑞+[2]_𝑞𝑞³[𝑛]²_𝑞≤1𝑞⁶−2𝑞²[3]+1+^𝑞+𝑞𝑞⁵[𝑛]_𝑞−2𝑞[𝑛]_𝑞𝑥()+[2]1+𝑥^𝑞𝑞³[𝑛]²_𝑞≤2𝑞⁶1−𝑞⁴+2[𝑛]_𝑞𝑥()+[2]1+𝑥^𝑞𝑞³[𝑛]²_𝑞. (2.8)

3. Local Approximation

Let 𝐶_𝐵[0,∞) denote the class of all real valued continuous bounded functions 𝑓 on [0,∞) endowed with the norm ‖𝑓‖=sup{|𝑓(𝑥)|∶𝑥∈[0,∞)}. The 𝐾-functional is defined as 𝐾₂𝑓,𝛿=inf_𝑔∈𝑊2𝑓−𝑔+𝛿𝑔^′′, (3.1) where 𝛿>0 and 𝑊²={𝑔∈𝐶_𝐵[0,∞)∶𝑔^′,𝑔^′′∈𝐶_𝐵[0,∞)}. By [16, page 177, Theorem 2.4] there exists an absolute constant 𝐶>0 such that 𝐾₂𝑓,𝛿≤𝐶𝜔₂𝑓,√𝛿, (3.2) where 𝜔₂𝑓,√𝛿=sup_{√0<ℎ≤𝛿}sup_{𝑥∈[0,∞)}𝑓(𝑥+2ℎ)−2𝑓(𝑥+ℎ)+𝑓(𝑥) (3.3) is the second order modulus of smoothness of 𝑓∈𝐶_𝐵[0,∞). By 𝜔𝑓,𝛿=sup_0<ℎ≤𝛿sup_{[)𝑥∈0,∞}𝑓(𝑥+ℎ)−𝑓(𝑥), (3.4) we denote the usual modulus of continuity of 𝑓∈𝐶_𝐵[0,∞).

Theorem 3.1. Let 𝑓∈𝐶_𝐵[0,∞). For every 𝑥∈[0,∞),𝑞∈(0,1), we have 𝑆_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)≤𝐶𝜔₂𝑓,𝛿_𝑛𝑞,𝑥+𝜔𝑓,𝛿_𝑛𝑞,𝑥, (3.5) where 𝐶 is an absolute constant, 𝛿_𝑛(𝑞,𝑥)=((2/𝑞⁶)(1−𝑞⁴+2/[𝑛]_𝑞)𝑥(1+𝑥)+[2]_𝑞/𝑞³[𝑛]²_𝑞)^1/2.

Proof. For 𝑓∈𝐶_𝐵[0,∞), 𝑥∈[0,∞), we define 𝑆_𝑛,𝑞𝑓;𝑥=𝑆_𝑛,𝑞𝑥𝑓;𝑥−𝑓𝑞²+1𝑞[𝑛]_𝑞+𝑓(𝑥). (3.6)
By Lemma 2.1, we get 𝑆_𝑛,𝑞(1;𝑥)=1,𝑆_𝑛,𝑞(𝑡;𝑥)=𝑥. Let 𝑔∈𝑊², 𝑥,𝑡∈[0,∞), by Taylor’s formula 𝑔(𝑡)=𝑔(𝑥)+(𝑡−𝑥)𝑔^′(𝑥)+^𝑡_𝑥(𝑡−𝑢)𝑔^′′(𝑢)𝑑𝑢, (3.7) we obtain 𝑆_𝑛,𝑞𝑔;𝑥=𝑔(𝑥)+𝑆_𝑛,𝑞^𝑡_𝑥(𝑡−𝑢)𝑔^′′(𝑢)𝑑𝑢;𝑥. (3.8)
By the definition given by (3.6), for 𝑥∈[0,∞), we have 𝑆_𝑛,𝑞𝑔;𝑥−𝑔(𝑥)≤𝑆_𝑛,𝑞^𝑡_𝑥(𝑡−𝑢)𝑔^′′(𝑢)+𝑑𝑢;𝑥_{𝑥/𝑞2+1/𝑞[𝑛]𝑞𝑥}𝑥𝑞²+1𝑞[𝑛]_𝑞−𝑢𝑔^′′(𝑢)𝑑𝑢≤𝑆_𝑛,𝑞^𝑡_𝑥|𝑡−𝑢|𝑔^′′(𝑢+)𝑑𝑢;𝑥_{𝑥/𝑞2+1/𝑞[𝑛]𝑞𝑥}𝑥𝑞²+1𝑞[𝑛]_𝑞−𝑢𝑔^′′(𝑢≤𝑆)𝑑𝑢_𝑛,𝑞(𝑡−𝑥)²+𝑥;𝑥𝑞²+1𝑞[𝑛]_𝑞−𝑥²𝑔^′′. (3.9)
Since (𝑥/𝑞²+1/𝑞[𝑛]_𝑞−𝑥)²≤𝛿²_𝑛(𝑞,𝑥), so, by Lemma 2.2, we have 𝑆_𝑛,𝑞𝑔;𝑥−𝑔(𝑥)≤2𝛿²_𝑛𝑞,𝑥𝑔^′′. (3.10)
By the definition given by (1.15) and Lemma 2.1, we have 𝑆_𝑛,𝑞𝑓;𝑥≤𝑆_𝑛,𝑞(1;𝑥)𝑓=𝑓. (3.11)
So, by the definition given by (3.6), we obtain 𝑆_𝑛,𝑞𝑓;𝑥≤𝑆_𝑛,𝑞𝑓;𝑥+2𝑓≤3𝑓. (3.12)
Thus, for 𝑥∈[0,∞), we have 𝑆_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)≤𝑆_𝑛,𝑞𝑓−𝑔;𝑥+𝑆_𝑛,𝑞(𝑥+𝑔(𝑥)(𝑥+𝑓𝑥𝑔;𝑥−𝑔)−𝑓)𝑞²+1𝑞[𝑛]_𝑞−𝑓(𝑥)≤4𝑓−𝑔+2𝛿²_𝑛𝑞,𝑥𝑔^′′+𝜔𝑓,𝛿_𝑛𝑞,𝑥. (3.13)
Hence, taking infimum on the right hand side over all 𝑔∈𝑊², we can get 𝑆_𝑛,𝑞𝑓;𝑞;𝑥−𝑓(𝑥)≤4𝐾₂𝑓,𝛿²_𝑛𝑞,𝑥+𝜔𝑓,𝛿_𝑛𝑞,𝑥. (3.14)
By inequality (3.2), for every 𝑞∈(0,1), we have 𝑆_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)≤𝐶𝜔₂𝑓,𝛿_𝑛𝑞,𝑥+𝜔𝑓,𝛿_𝑛𝑞,𝑥. (3.15)

4. Rate of Convergence

Let 𝐻_𝑥𝑚[0,∞) be the set of all functions 𝑓 defined on [0,∞) satisfying the condition |𝑓(𝑥)|≤𝑀_𝑓(1+𝑥^𝑚), where 𝑚>0, 𝑀_𝑓 is a constant depending only on 𝑓. Let 𝐶_𝑥𝑚[0,∞) denote the subspace of all continuous functions in 𝐻_𝑥𝑚[0,∞). Also let 𝐶^∗_𝑥𝑚[0,∞) be the subspace of all functions 𝑓∈𝐶_𝑥𝑚[0,∞), for which lim_|𝑥|→∞(𝑓(𝑥)/(1+𝑥^𝑚)) is finite. The norm on 𝐶^∗_𝑥𝑚[0,∞) is ‖𝑓‖_𝑥𝑚=sup_{𝑥∈[0,∞)}(|𝑓(𝑥)|/(1+𝑥^𝑚)). We denote the modulus of continuity of 𝑓 on the closed interval [0,𝑎], 𝑎>0 by 𝑥∈[0,∞) as 𝜔_𝑎𝑓,𝛿=sup_{|𝑡−𝑥|≤𝛿}sup_{𝑥,𝑡∈[0,𝑎]}𝑓(𝑡)−𝑓(𝑥),𝛿>0. (4.1) We observe that for 𝑓∈𝐶^∗_𝑥𝑚[0,∞), the modulus of continuity 𝜔_𝑎(𝑓,𝛿)→0 as 𝛿→0⁺.

Theorem 4.1. Let 𝑓∈𝐶_𝑥2[0,∞), 𝑞∈(0,1), and 𝜔_𝑎+1(𝑓,𝛿) be the modulus of continuity of 𝑓 on the finite interval [0,𝑎+1]⊂[0,∞), where 𝑎>0. Then we have 𝑆_𝑛,𝑞(𝑓;⋅)−𝑓_𝐶[0,𝑎]≤5𝑀_𝑓1+𝑎²𝜂²_𝑛𝑞+2𝜔_𝑎+1𝑓,𝜂_𝑛𝑞, (4.2) where 𝜂_𝑛(𝑞)=((2/𝑞⁶)(1−𝑞⁴+2/[𝑛]_𝑞)𝑎(1+𝑎)+[2]_𝑞/(𝑞³[𝑛]²_𝑞))^1/2, ‖𝑓‖_𝐶[0,𝑎]=max{|𝑓(𝑥)|, 𝑥∈[0,𝑎]}.

Proof. For 𝑥∈[0,𝑎] and 𝑡>𝑎+1, since 𝑡−𝑥>1, we have |𝑓(𝑡)−𝑓(𝑥)|≤𝑀_𝑓(2+𝑥²+𝑡²)≤𝑀_𝑓[2+3𝑥²+2(𝑡−𝑥)²]≤5𝑀_𝑓(1+𝑎²)(𝑡−𝑥)².
For 𝑥∈[0,𝑎] and 0≤𝑡≤𝑎+1, we have |𝑓(𝑡)−𝑓(𝑥)|≤𝜔_𝑎+1(𝑓,|𝑡−𝑥|)≤(1+|𝑡−𝑥|/𝛿)𝜔_𝑎+1(𝑓,𝛿) with 𝛿>0.
So, for 𝑥∈[0,𝑎] and 𝑡≥0, we may write 𝑓(𝑡)−𝑓(𝑥)≤5𝑀_𝑓1+𝑎²(𝑡−𝑥)²+||1+𝑡−𝑥𝛿𝜔_𝑎+1𝑓,𝛿. (4.3)
Thus, by Cauchy-Schwartz inequality, we obtain 𝑆_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)≤𝑆_𝑛,𝑞𝑓(𝑡)−𝑓(𝑥);𝑥≤5𝑀_𝑓1+𝑎²𝑆_𝑛,𝑞(𝑡−𝑥)²;𝑥+𝜔_𝑎+11𝑓,𝛿1+𝛿𝑆_𝑛,𝑞(𝑡−𝑥)²;𝑥^1/2. (4.4)
By Lemma 2.2, for 𝑥∈[0,𝑎] and 𝜂_𝑛(𝑞)=((2/𝑞⁶)(1−𝑞⁴+2/[𝑛]_𝑞)𝑎(1+𝑎)+[2]_𝑞/𝑞³[𝑛]²_𝑞)^1/2, we have 𝑆_𝑛,𝑞((𝑡−𝑥)²;𝑥)≤𝜂²_𝑛(𝑞). Hence, for every 𝑥∈[0,𝑎], 𝑞∈(0,1), we obtain 𝑆_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)≤5𝑀_𝑓1+𝑎²𝜂²_𝑛𝑞+𝜔_𝑎+1𝜂𝑓,𝛿1+^𝑛𝑞𝛿. (4.5)
By taking 𝛿=𝜂_𝑛(𝑞), we immediately get the desired result.

Corollary 4.2. Assume that 𝑓∈𝐶_𝑥2[0,∞), 𝑞∈(0,1). Let 𝑀>0,𝛼∈(0,1], 𝑓∈Lip^𝛼_𝑀 on [0,𝑎+1], where 𝑎>0. Then we have 𝑆_𝑛,𝑞𝑓;⋅−𝑓_𝐶[0,𝑎]≤5𝑀_𝑓1+𝑎²𝜂²_𝑛𝑞+𝑀𝜂^𝛼_𝑛𝑞, (4.6) where 𝜂_𝑛(𝑞) and ‖𝑓‖_𝐶[0,𝑎] are given in Theorem 4.1.

Proof. Let 𝑞∈(0,1), 𝑀>0,𝛼∈(0,1], 𝑓∈Lip^𝛼_𝑀 on [0,𝑎+1]. Then for any 𝑥∈[0,𝑎],𝑡∈[0,𝑎+1], we have |𝑓(𝑡)−𝑓(𝑥)|≤𝑀|𝑡−𝑥|^𝛼. So, according to the proof of Theorem 4.1, for 𝑥∈[0,𝑎], 𝑡≥0, we have 𝑓(𝑡)−𝑓(𝑥)≤5𝑀_𝑓1+𝑎²𝑆_𝑛,𝑞(𝑡−𝑥)²;𝑥+𝑀|𝑡−𝑥|^𝛼. (4.7)
Using the Hölder inequality with 𝑚=2/𝛼,𝑛=2/(2−𝛼), for any 𝑥∈[0,𝑎],𝑡≥0, we get 𝑆_𝑛,𝑞𝑀|𝑡−𝑥|^𝛼;𝑥≤𝑀𝑆_𝑛,𝑞(𝑡−𝑥)²;𝑥^𝛼/2. (4.8)
So, for any 𝑥∈[0,𝑎],𝑡≥0, we have 𝑆_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)≤5𝑀_𝑓1+𝑎²𝑆_𝑛,𝑞(𝑡−𝑥)²;𝑥+𝑀𝑆_𝑛,𝑞(𝑡−𝑥)²;𝑥^𝛼/2≤5𝑀_𝑓1+𝑎²𝜂²_𝑛𝑞+M𝜂^𝛼_𝑛𝑞. (4.9) The desired result follows immediately.

5. Weighted Approximation

Now we give the weighted approximation result for the operators 𝑆_𝑛,𝑞(𝑓;𝑥).

Theorem 5.1. Let the sequence 𝑞={𝑞_𝑛} satisfies 𝑞_𝑛∈(0,1) and 𝑞_𝑛→1 as 𝑛→∞. For 𝑓∈𝐶^∗_𝑥2[0,∞), we have lim_𝑛→∞𝑆_{𝑛,𝑞𝑛}(𝑓;⋅)−𝑓_𝑥2=0. (5.1)

Proof. Using the Theorem in [17], we see that it is sufficient to verify the following three conditions: lim_𝑛→∞𝑆_{𝑛,𝑞𝑛}(𝑒_𝑖;⋅)−𝑒_𝑖𝑥2=0, (5.2) where 𝑒_𝑖(𝑥)=𝑥^𝑖,𝑖=0,1,2.
Since 𝑆_{𝑛,𝑞𝑛}(𝑒₀;𝑥)=1, it is clear that lim_𝑛→∞‖𝑆_{𝑛,𝑞𝑛}(𝑒₀;⋅)−𝑒₀‖_𝑥2=0.
By Lemma 2.2, we have 𝑆_{𝑛,𝑞𝑛}(𝑒₁;⋅)−𝑒_1𝑥2=sup_{𝑥∈[0,∞)}𝑆_{𝑛,𝑞𝑛}(𝑡;𝑥)−𝑥1+𝑥²≤1−𝑞²_𝑛𝑞²_𝑛sup_{𝑥∈[0,∞)}𝑥1+𝑥²+1𝑞_𝑛[𝑛]_𝑞𝑛≤1−𝑞²_𝑛𝑞²_𝑛+1𝑞_𝑛[𝑛]_𝑞𝑛. (5.3)
Since 𝑞_𝑛∈(0,1) and lim_𝑛→∞𝑞_𝑛=1, we have [𝑛]_𝑞𝑛→∞ as 𝑛→∞ (see [18]), so, we can obtain lim_𝑛→∞‖𝑆_{𝑛,𝑞𝑛}(𝑒₁;⋅)−𝑒₁‖_𝑥2=0.
For 𝑖=2, by Lemma 2.1, we have 𝑆_{𝑛,𝑞𝑛}𝑒₂;⋅−𝑒_2𝑥2=sup_{𝑥∈[0,∞)}𝑆_{𝑛,𝑞𝑛}𝑡²;𝑥−𝑥²1+𝑥²≤1−𝑞⁶_𝑛𝑞⁶_𝑛sup_{𝑥∈[0,∞)}𝑥²1+𝑥²+[3]_𝑞𝑛+𝑞_𝑛𝑞⁵_𝑛[𝑛]_𝑞𝑛sup_{𝑥∈[0,∞)}𝑥1+𝑥²+[2]_𝑞𝑛𝑞³_𝑛[𝑛]²_𝑞𝑛≤1−𝑞⁶_𝑛𝑞⁶_𝑛+[3]_𝑞𝑛+𝑞_𝑛𝑞⁵_𝑛[𝑛]_𝑞𝑛+[2]_𝑞𝑛𝑞³_𝑛[𝑛]²_𝑞𝑛, (5.4) which implies that lim_𝑛→∞‖𝑆_{𝑛,𝑞𝑛}(𝑒₂;⋅)−𝑒₂‖_𝑥2=0. In a word, we complete the proof.

It is known that, if 𝑓 is not uniformly continuous on the interval [0,∞), then the usual first modulus of continuity 𝜔(𝑓,𝛿) does not tend to zero as 𝛿→0⁺. For every 𝑓∈𝐶^∗_𝑥2[0,∞), we would like to take a weighted modulus of continuity Ω(𝑓,𝛿) which tends to zero as 𝛿→0⁺.

For every 𝑓∈𝐶^∗_𝑥2[0,∞), let Ω𝑓,𝛿=sup_{0<ℎ≤𝛿,𝑥≥0}𝑓(𝑥+ℎ)−𝑓(𝑥)1+ℎ²1+𝑥². (5.5) The weighed modulus of continuity Ω(𝑓,𝛿) was defined by İspir in [19]. It is known that Ω(𝑓,𝛿) has the following properties.

Lemma 5.2 (see [19]). Let 𝑓∈𝐶^∗_𝑥2[0,∞). Then (i)Ω(𝑓,𝛿) is a monotone increasing function of 𝛿; (ii)for each 𝑓∈𝐶^∗_𝑥2[0,∞), lim_𝛿→0⁺Ω(𝑓,𝛿)=0; (iii)for each 𝑚∈ℕ, Ω(𝑓,𝑚𝛿)≤𝑚Ω(𝑓,𝛿); (iv)for each 𝜆∈ℝ⁺, Ω(𝑓,𝜆𝛿)≤(1+𝜆)Ω(𝑓,𝛿).

Theorem 5.3. Let 𝑓∈𝐶^∗_𝑥2[0,∞), 𝑞∈(0,1). Then we have the inequality 𝑆_𝑛,𝑞(𝑓;⋅)−𝑓_𝑥5≤𝑀𝑞Ω𝑓,1−𝑞⁴+3[𝑛]_𝑞^1/2, (5.6) where 𝑀(𝑞) is a positive constant independent 𝑓 and 𝑛.

Proof. By the definition of Ω(𝑓,𝛿) and Lemma 5.2 (iv), for every 𝛿>0, we have 𝑓(𝑡)−𝑓(𝑥)≤1+(𝑡−𝑥)²1+𝑥²||1+𝑡−𝑥𝛿Ω𝑓,𝛿. (5.7)
Then, we obtain 𝑆_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)≤1+𝑥²Ω𝑓,𝛿𝑆_𝑛,𝑞1+(𝑡−𝑥)²||1+𝑡−𝑥𝛿≤;𝑥1+𝑥²Ω𝑓,𝛿𝑆_𝑛,𝑞1+(𝑡−𝑥)²;𝑥+𝑆_𝑛,𝑞1+(𝑡−𝑥)²|𝑡−𝑥|𝛿;𝑥. (5.8)
Also by the Cauchy-Schwartz inequality, we have 𝑆_𝑛,𝑞1+(𝑡−𝑥)²|𝑡−𝑥|𝛿;𝑥≤𝑆_𝑛,𝑞1+(𝑡−𝑥)²²;𝑥^1/2𝑆_𝑛,𝑞|𝑡−𝑥|²𝛿²;𝑥^1/2. (5.9)
Consequently, 𝑆_𝑛,𝑞(𝑥≤𝑓;𝑥−𝑓)1+𝑥²Ω×𝑆𝑓,𝛿_𝑛,𝑞1+(𝑡−𝑥)²;𝑥+𝑆_𝑛,𝑞1+(𝑡−𝑥)²²;𝑥^1/2𝑆_𝑛,𝑞|𝑡−𝑥|²𝛿²;𝑥^1/2. (5.10)
In view of Lemmas 2.1 and 2.2, we get 𝑆_𝑛,𝑞1+(𝑡−𝑥)²2;𝑥≤1+𝑞⁶1−𝑞⁴+2[𝑛]_𝑞𝑥()+[2]1+𝑥^𝑞𝑞³[𝑛]²_𝑞≤2𝑞⁶2−𝑞⁴+3[𝑛]_𝑞(1+𝑥)²≤𝑀₁𝑞(1+𝑥)². (5.11)
Also, 𝑆_𝑛,𝑞1+(𝑡−𝑥)²²;𝑥=1+2𝑆_𝑛,𝑞(𝑡−𝑥)²;𝑥+𝑆_𝑛,𝑞𝑡⁴;𝑥−4𝑥𝑆_𝑛,𝑞𝑡³;𝑥+6𝑥²𝑆_𝑛,𝑞𝑡²;𝑥−4𝑥³𝑆_𝑛,𝑞(𝑡;x)+𝑥⁴=𝑥⁴1𝑞²⁰−4𝑞¹²+6𝑞⁶−4𝑞²+1+𝑥³1𝑞¹⁹[𝑛]_𝑞[7]_𝑞+𝑞[5]_𝑞+𝑞²[3]_𝑞+𝑞³−4𝑞¹¹[𝑛]_𝑞[5]_𝑞+𝑞[3]_𝑞+𝑞²+6𝑞⁵[𝑛]_𝑞[3]_𝑞−4+𝑞𝑞[𝑛]_𝑞+𝑥²2𝑞⁶−4𝑞²1+2+𝑞¹⁷[𝑛]²_𝑞[10]_𝑞+2𝑞[8]_𝑞+4𝑞²[6]_𝑞+4𝑞³[4]_𝑞+3𝑞⁴[2]_𝑞−4𝑞⁹[𝑛]²_𝑞[6]_𝑞+2𝑞[4]_𝑞+2𝑞²[2]_𝑞+6[2]_𝑞𝑞³[𝑛]²_𝑞23]+𝑥[_𝑞+𝑞𝑞⁵[𝑛]_𝑞−4𝑞[𝑛]_𝑞+1𝑞¹⁴[𝑛]³_𝑞[10]_𝑞+3𝑞[8]_𝑞+5𝑞²[6]_𝑞+6𝑞³[4]_𝑞+4𝑞⁴[2]_𝑞−4[2]_𝑞[3]_𝑞𝑞⁶[𝑛]³_𝑞+2[2]1+^𝑞𝑞³[𝑛]²_𝑞+[2]_𝑞[3]_𝑞[4]_𝑞𝑞¹⁰[𝑛]⁴_𝑞≤16𝑞²⁰𝑥⁴+80𝑞¹⁹𝑥³+164𝑞¹⁷𝑥²+84𝑞¹⁴𝑥+1+28𝑞¹⁰≤𝑀₂𝑞1+𝑥²²,𝑆_𝑛,𝑞|𝑡−𝑥|²𝛿²;𝑥^1/2≤1𝛿2𝑞⁶1−𝑞⁴+2[𝑛]_𝑞𝑥()+[2]1+𝑥^𝑞𝑞³[𝑛]²_𝑞^1/2≤1𝛿𝑀₃𝑞1−𝑞⁴+3[𝑛]_𝑞^1/2(1+𝑥). (5.12)
Now from inequalities (5.10)–(5.12), we have 𝑆_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)≤Ω𝑓,𝛿𝑀₁𝑞1+𝑥²(1+𝑥)²+𝑀₂𝑞𝑀₃𝑞1𝛿1−𝑞⁴+3[𝑛]_𝑞^1/21+𝑥²²()𝑀1+𝑥≤Ω𝑓,𝛿₁𝑞1+𝑥²(1+𝑥)²+𝑀₂𝑞𝑀₃𝑞𝑀₄1𝛿1−𝑞⁴+3[𝑛]_𝑞^1/21+𝑥⁵, (5.13) where 𝑀₄=sup_𝑥≥0((1+𝑥²)²(1+𝑥)/(1+𝑥⁵)).
Taking 𝛿=(1−𝑞⁴+3/[𝑛]_𝑞)^1/2, from the above inequality we can obtain the desired result.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 61170324), the Class A Science and Technology Project of Education Department of Fujian Province of China (Grant no. JA12324), and the Natural Science Foundation of Fujian Province of China (Grant no. 2010J01012). The authors thank the associate editor and the referees for several important comments and suggestions which improve the quality of the paper.