-Szász-Mirakyan-Kantorovich Operators of Functions of Two Variables in Polynomial Weighted Spaces

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 823803, 9 pages http://dx.doi.org/10.1155/2013/823803 Research Article 𝑞-Szász-Mirakyan-Kantorovich Operators of Functions of Two Variables in Polynomial Weighted Spaces Mediha Örkcü Department of Mathematics, Faculty of Sciences, Gazi University, Teknikokullar, 06500 Ankara, Turkey Correspondence should be addressed to Mediha Örkcü; Received 20 June 2013; Accepted 10 November 2013 Academic Editor: Sergei V. Pereverzyev Copyright © 2013 Mediha Örkcü. 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. The present paper deals with approximation properties of q-Szász-Mirakyan-Kantorovich operators. We construct new bivariate generalization by 𝑞𝑅 -integral and these operators’ approximation properties in polynomial weighted spaces are investigated. Also, we obtain Voronovskaya-type theorem for the proposed operators in polynomial weighted spaces of functions of two variables. 1. Introduction In the past two decades, 𝑞-calculus has gained popularity in the construction of linear approximation processes. Lupaş [1] and Phillips [2] defined generalizations of the Bernstein operators called 𝑞-Bernstein operators. Then, as Phillips has done for Bernstein operators, the authors introduced modifications of the other important operators based on the 𝑞-integers, for example, 𝑞-Meyer-König operators [3, 4], 𝑞-Bleimann, Butzer, and Hahn operators [5, 6], 𝑞-SzászMirakyan operators [7–9], 𝑞-Baskakov operators [10, 11]. On the other hand, Stancu [12] first introduced new linear positive operators in two- and several dimensional variables. Recently, Barbosu [13] introduced a Stancu-type generalization of two-dimensional Bernstein operators based on 𝑞integers and called them bivariate 𝑞-Bernstein operators. ̆ and Gupta [14] constructed a bivariate generalization Dogru of the Meyer-König and Zeller operators based on the 𝑞integers. Agratini [15] presented two-dimensional extension of some univariate positive approximation processes expressed by series. All the above mentioned new operators motivate us for current work. In this paper, we firstly extend the 𝑞-SzászMirakyan-Kantorovich operators to the case of bivariate functions. Then these operators’ approximation properties in polynomial weighted spaces are investigated. Also we obtain Voronovskaya-type theorem for the proposed operators in polynomial weighted spaces of functions of two variables. Now we recall some definitions about 𝑞-integers. For any nonnegative integer 𝑟, the 𝑞-integer of the number 𝑟 is defined by 1 + 𝑞 + ⋅ ⋅ ⋅ + 𝑞𝑟−1 [𝑟]𝑞 = { 𝑟 if 𝑞 ≠ 1 if 𝑞 = 1, [0]𝑞 = 1, (1) where 𝑞 is a positive real number. The 𝑞-factorial is defined as [1] [2] ⋅ ⋅ ⋅ [𝑟]𝑞 [𝑟]𝑞 ! = { 𝑞 𝑞 1 if 𝑟 = 1, 2, . . . if 𝑟 = 0, [0]𝑞 ! = 1. (2) Two 𝑞-analogues of the exponential function 𝑒𝑥 are given as ∞ 𝐸𝑞 (𝑥) = ∑ 𝑞𝑛(𝑛−1)/2 𝑛=0 ∞ 𝑥𝑛 𝜀𝑞 (𝑥) = ∑ , 𝑛=0 [𝑛]𝑞 ! 𝑥𝑛 , [𝑛]𝑞 ! 𝑥 ∈ R, 1 . |𝑥| < 1−𝑞 (3) The following relation between 𝑞-exponential functions 𝐸𝑞 (𝑥) and 𝜀𝑞 (𝑥) holds: 𝐸𝑞 (𝑥) 𝜀𝑞 (−𝑥) = 1, |𝑥| < 1 . 1−𝑞 (4) 2 Abstract and Applied Analysis The 𝑞-derivative of a function 𝑓(𝑥), denoted by 𝐷𝑞 𝑓, is defined by (𝐷𝑞 𝑓) (𝑥) = 𝑓 (𝑞𝑥) − 𝑓 (𝑥) , (𝑞 − 1) 𝑥 𝑥 ≠ 0, where 𝑑 𝑏 𝑐 𝑎 ∫ ∫ 𝑓 (𝑡, 𝑠) 𝑑𝑞𝑅1 𝑡𝑑𝑞𝑅2 𝑠 = (1 − 𝑞1 ) (1 − 𝑞2 ) (𝑏 − 𝑎) (𝑐 − 𝑑) (5) ∞ ∞ (𝐷𝑞 𝑓) (0) = lim (𝐷𝑞 𝑓) (𝑥) . Also, it is known that 𝐷𝑞 𝐸(𝑎𝑥) = 𝑎𝐸(𝑞𝑎𝑥). The 𝑞-integral of the function 𝑓 over the interval [0, 𝑏] is defined by 𝑏 ∞ 0 𝑗=0 0 < 𝑞 < 1. (6) If 𝑓 is integrable over [0, 𝑏], then 𝑏 𝑏 0 0 lim− ∫ 𝑓 (𝑡) 𝑑𝑞 𝑡 = ∫ 𝑓 (𝑡) 𝑑𝑡. 𝑞→1 𝑏 𝑎 𝑎 0 0 (7) and 𝑓 is a 𝑞𝑅 -integrable function, so the series in (11) converges. It is clear that the operators given in (10) are linear 𝑞1 ,𝑞2 , if 𝑓 is a 𝑞𝑅 -integrable and positive. For the operator 𝑆𝑚,𝑛 function and 𝑓(𝑥, 𝑦) = 𝑓1 (𝑥)𝑓2 (𝑦), (𝑥, 𝑦) ∈ R2+ , then 𝑞1 ,𝑞2 𝑞1 (𝑓 (𝑡, 𝑠) ; 𝑥, 𝑦) = 𝑆𝑚 (𝑓1 (𝑡) , 𝑥) 𝑆𝑛𝑞2 (𝑓2 (𝑠) , 𝑦) . 𝑆𝑚,𝑛 (8) In order to generalize and spread the existing inequalities, Marinković et al. considered new type of the 𝑞-integral. So, the problems which ensue from the general definition of 𝑞integral were overcome. The Riemann-type 𝑞-integral [16] in the interval [𝑎, 𝑏] was defined as ∞ 𝑎 𝑗=0 ∫ 𝑓 (𝑡) 𝑑𝑞𝑅 𝑡 = (1 − 𝑞) (𝑏 − 𝑎) ∑𝑓 (𝑎 + (𝑏 − 𝑎) 𝑞𝑗 ) 𝑞𝑗 , (9) Now, in order to obtain approximation properties of proposed operators, we give some auxiliary results. For a fixed 𝑥 ∈ R+ , by the 𝑞-Taylor theorem [18], we write ∞ (𝑡 − 𝑥)𝑘 𝑞 𝑔 (𝑡) = ∑ 𝑘=0 𝑘−1 𝑞1 ,𝑞2 (𝑓; 𝑥, 𝑦) 𝑆𝑚,𝑛 ∞ ∞ [𝑚]𝑘+1 𝑥𝑘 [𝑛]𝑙+1 𝑦𝑙 𝑞1 𝑞2 = ∑∑ 𝑞1𝑘(𝑘−1)−1 𝑞2𝑙(𝑙−1)−1 ! ! [𝑘] [𝑙] 𝑞1 𝑞2 𝑙=0 𝑘=0 × 𝐸𝑞1 (−[𝑚]𝑞1 𝑞1𝑘 𝑥) 𝐸𝑞2 (−[𝑛]𝑞2 𝑞2𝑙 𝑦) ∫ [𝑘]𝑞1 /𝑞1𝑘−1 [𝑚]𝑞1 𝑓 (𝑡, 𝑠) 𝑑𝑞𝑅1 𝑡𝑑𝑞𝑅2 𝑠, (14) Choosing 𝑡 = 0 and taking into account (−𝑥)𝑘𝑞 = (−1)𝑘 𝑥𝑘 𝑞𝑘(𝑘−1)/2 , 𝑘 𝐷𝑞𝑘 𝐸𝑞 (−[𝑛]𝑞 𝑥) = (−[𝑛]𝑞 ) 𝑞𝑘(𝑘−1)/2 𝐸𝑞 (−[𝑛]𝑞 𝑞𝑘 𝑥) , (15) we get for 𝑔(𝑥) = 𝐸𝑞 (−[𝑛]𝑞 𝑥) that [𝑘]𝑞 ! 𝑘=0 𝐷𝑞𝑘 𝑔 (𝑥) 𝑘 ∞ ([𝑛] 𝑥) 𝑞 𝑘(𝑘−1) =∑ [𝑘]𝑞 ! 𝑞 (16) 𝐸𝑞 (−[𝑛]𝑞 𝑞𝑘 𝑥) . Similarly, choosing 𝑡 = 0 and taking into account For 𝑞1 , 𝑞2 ∈ (0, 1) and (𝑚, 𝑛) ∈ N × N, we now define new operators that we call the 𝑞-Szász-Mirakyan-Kantorovich operators of functions of two variables as follows: [𝑙]𝑞2 /𝑞2𝑘−1 [𝑛]𝑞2 (13) 𝑘 𝑘=0 2. Construction of the Bivariate Operators ×∫ 𝐷𝑞𝑘 𝑔 (𝑥) , 𝑘 (𝑡 − 𝑥)𝑘𝑞 = ∏ (𝑡 − 𝑞𝑠 𝑥) = ∑[ ] 𝑞𝑠(𝑠−1)/2 𝑡𝑘−𝑠 (−𝑥)𝑠 . 𝑠 𝑞 𝑠=0 𝑠=0 1 = 𝑔 (0) = ∑ This definition includes only point inside the interval of the integration. Details of 𝑞-integers can be found in [17]. [𝑘+1]𝑞1 /𝑞1𝑘−1 [𝑚]𝑞1 [𝑘]𝑞 ! ∞ (−𝑥)𝑘 𝑞 0 < 𝑞 < 1. [𝑙+1]𝑞2 /𝑞2𝑘−1 [𝑛]𝑞2 (12) where ∫ 𝑓 (𝑡) 𝑑𝑞 𝑡 = ∫ 𝑓 (𝑡) 𝑑𝑞 𝑡 − ∫ 𝑓 (𝑡) 𝑑𝑞 𝑡. 𝑏 𝑗 𝑗=0 𝑖=0 Generally accepted definition for 𝑞-integral over an interval [𝑎, 𝑏] is 𝑏 𝑗 × ∑ ∑𝑓 (𝑎 + (𝑏 − 𝑎) 𝑞1𝑖 , 𝑐 + (𝑐 − 𝑑) 𝑞2 ) 𝑞1𝑖 𝑞2 , 𝑥→0 ∫ 𝑓 (𝑡) 𝑑𝑞 𝑡 = 𝑏 (1 − 𝑞) ∑ 𝑓 (𝑏𝑞𝑗 ) 𝑞𝑗 , (11) (−𝑥)𝑘𝑞 = (−1)𝑘 𝑥𝑘 𝑞𝑘(𝑘−1)/2 , 𝑘 𝐷𝑞𝑘 𝐸𝑞 (−[𝑛]𝑞 𝑞𝑥) = (−[𝑛]𝑞 𝑞) 𝑞𝑘(𝑘−1)/2 𝐸𝑞 (−[𝑛]𝑞 𝑞𝑘+1 𝑥) , (17) we obtain for 𝑔(𝑥) = 𝐸𝑞 (−[𝑛]𝑞 𝑞𝑥) that (10) ∞ (−𝑥)𝑘 𝑞 1 = 𝑔 (0) = ∑ [𝑘]𝑞 ! 𝑘=0 𝑘 ∞ ([𝑛] 𝑥) 𝑞 𝑘2 =∑ 𝑘=0 [𝑘]𝑞 ! 𝐷𝑞𝑘 𝑔 (𝑥) 𝑞 𝐸𝑞 (−[𝑛]𝑞 𝑞𝑘+1 𝑥) . (18) Abstract and Applied Analysis 3 (([𝑘]𝑞1 /𝑞1𝑘−1 [𝑚]𝑞1 ) + (𝑞1 /[2]𝑞1 [𝑚]𝑞1 )), we get from the 𝑞1 ,𝑞2 that linearity of 𝑆𝑚,𝑛 Also, using (−𝑥)𝑘𝑞 = (−1)𝑘 𝑥𝑘 𝑞𝑘(𝑘−1)/2 , 𝑘 𝐷𝑞𝑘 𝐸𝑞 (−[𝑛]𝑞 𝑞2 𝑥) = (−[𝑛]𝑞 ) 𝑞2𝑘 𝑞𝑘(𝑘−1)/2 𝐸𝑞 (−[𝑛]𝑞 𝑞𝑘+2 𝑥) , (19) ∞ ∞ [𝑚]𝑘 𝑥𝑘 [𝑛]𝑙 𝑦𝑙 𝑞1 𝑞2 𝑞1𝑘(𝑘−1) 𝑞2𝑙(𝑙−1) ! ! [𝑘] [𝑙] 𝑞 𝑞 𝑙=0 𝑘=0 1 2 𝑞1 ,𝑞2 𝑆𝑚,𝑛 (𝑡; 𝑥) = ∑ ∑ × 𝐸𝑞1 (−[𝑚]𝑞1 𝑞1𝑘 𝑥) 𝐸𝑞2 (−[𝑛]𝑞2 𝑞2𝑙 𝑦) we have for 𝑔(𝑥) = 𝐸𝑞 (−[𝑛]𝑞 𝑞2 𝑥) that ∞ (−𝑥)𝑘 𝑞 1 = 𝑔 (0) = 𝑥 ∑ 𝑘=0 [𝑘]𝑞 ! ∞ ∞ [𝑚]𝑘 𝑥𝑘 [𝑛]𝑙 𝑦𝑙 𝑞1 𝑞2 𝑞1𝑘(𝑘−1) 𝑞2𝑙(𝑙−1) ! ! [𝑘] [𝑙] 𝑞 𝑞 𝑙=0 𝑘=0 1 2 + ∑∑ × 𝐸𝑞1 (−[𝑚]𝑞1 𝑞1𝑘 𝑥) 𝐸𝑞2 (−[𝑛]𝑞2 𝑞2𝑙 𝑦) 𝐷𝑞𝑘 𝑔 (𝑥) (20) 𝑘 ∞ ([𝑛] 𝑥) 𝑞 𝑘2 =∑ 𝑘=0 [𝑘]𝑞 ! 𝑞 𝐸𝑞 (−[𝑛]𝑞 𝑞𝑘+2 𝑥) . 𝑞1 ,𝑞2 𝑆𝑚,𝑛 (𝑡; 𝑥) ∞ ∞ [𝑚]𝑘−1 𝑥𝑘 [𝑛]𝑙 𝑦𝑙 𝑘(𝑘−1) 𝑞1 𝑞1 𝑞2 𝑞2𝑙(𝑙−1) =∑∑ 𝑘−1 − 1] ! ! [𝑘 [𝑙] 𝑞 𝑞1 𝑞2 1 𝑙=0 𝑘=1 𝑞1 , (...truncated)