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

Abstract

The present paper deals with approximation properties of q-Szász-Mirakyan-Kantorovich operators. We construct new bivariate generalization by 𝑞_𝑅-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-Szász-Mirakyan 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 𝑞-integers and called them bivariate 𝑞-Bernstein operators. Doğ‌ru and Gupta [14] constructed a bivariate generalization of the Meyer-König and Zeller operators based on the 𝑞-integers. 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-Szász-Mirakyan-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 [𝑟]_𝑞=1+𝑞+⋯+𝑞^𝑟−1if𝑟𝑞≠1if[0]𝑞=1,^𝑞=1, (1) where 𝑞 is a positive real number. The 𝑞-factorial is defined as [𝑟]_𝑞[1]!=^𝑞[2]_𝑞⋯[𝑟]_𝑞if1𝑟=1,2,…if[0]𝑟=0,^𝑞!=1. (2) Two 𝑞-analogues of the exponential function 𝑒^𝑥 are given as 𝐸_𝑞(𝑥)=^∞_𝑛=0𝑞^{𝑛(𝑛−1)/2}𝑥^𝑛[𝑛]_𝑞!𝜀,𝑥∈ℝ,_𝑞(𝑥)=^∞_𝑛=0𝑥^𝑛[𝑛]_𝑞!,|𝑥|<1.1−𝑞 (3) The following relation between 𝑞-exponential functions 𝐸_𝑞(𝑥) and 𝜀_𝑞(𝑥) holds:𝐸_𝑞(𝑥)𝜀_𝑞()|𝑥|<1−𝑥=1,.1−𝑞 (4) The 𝑞-derivative of a function 𝑓(𝑥), denoted by 𝐷_𝑞𝑓, is defined by 𝐷_𝑞𝑓(𝑥)=𝑓𝑞𝑥−𝑓(𝑥)𝑥𝐷𝑞−1,𝑥≠0,_𝑞𝑓(0)=lim_𝑥→0𝐷_𝑞𝑓(𝑥). (5) Also, it is known that 𝐷_𝑞𝐸(𝑎𝑥)=𝑎𝐸(𝑞𝑎𝑥).

The 𝑞-integral of the function 𝑓 over the interval [0,𝑏] is defined by ^𝑏₀𝑓(𝑡)𝑑_𝑞𝑡=𝑏1−𝑞^∞_𝑗=0𝑓𝑏𝑞^𝑗𝑞^𝑗,0<𝑞<1. (6) If 𝑓 is integrable over [0,𝑏], then lim_𝑞→1−^𝑏₀𝑓(𝑡)𝑑_𝑞𝑡=^𝑏₀𝑓(𝑡)𝑑𝑡. (7) Generally accepted definition for 𝑞-integral over an interval [𝑎,𝑏] is ^𝑏_𝑎𝑓(𝑡)𝑑_𝑞𝑡=^𝑏₀𝑓(𝑡)𝑑_𝑞𝑡−^𝑎₀𝑓(𝑡)𝑑_𝑞𝑡. (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 𝑞-integral were overcome. The Riemann-type 𝑞-integral [16] in the interval [𝑎,𝑏] was defined as ^𝑏_𝑎𝑓(𝑡)𝑑^𝑅_𝑞𝑡=1−𝑞(𝑏−𝑎)^∞_𝑗=0𝑓𝑎+(𝑏−𝑎)𝑞^𝑗𝑞^𝑗,0<𝑞<1. (9) This definition includes only point inside the interval of the integration.

Details of 𝑞-integers can be found in [17].

2. Construction of the Bivariate Operators

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: 𝑆^𝑞1,𝑞2_𝑚,𝑛=𝑓;𝑥,𝑦_{∞∞𝑙=0𝑘=0}[𝑚]_𝑞^𝑘+1₁𝑥^𝑘[𝑘]_𝑞1![𝑛]_𝑞^𝑙+1₂𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{𝑘(𝑘−1)−1}𝑞₂^{𝑙(𝑙−1)−1}×𝐸_𝑞1−[𝑚]_𝑞1𝑞^𝑘₁𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦×^{[𝑙+1]𝑞2/𝑞2𝑘−1[𝑛]𝑞2}_{[𝑙]𝑞2/𝑞2𝑘−1[𝑛]𝑞2}^{[𝑘+1]𝑞1/𝑞1𝑘−1[𝑚]𝑞1}_{[𝑘]𝑞1/𝑞1𝑘−1[𝑚]𝑞1}𝑓(𝑡,𝑠)𝑑^𝑅_𝑞1𝑡𝑑^𝑅_𝑞2𝑠, (10) where ^𝑑_𝑐^𝑏_𝑎𝑓(𝑡,𝑠)𝑑^𝑅_𝑞1𝑡𝑑^𝑅_𝑞2𝑠=1−𝑞₁1−𝑞₂()×𝑏−𝑎)(𝑐−𝑑_{∞∞𝑗=0𝑖=0}𝑓𝑎+(𝑏−𝑎)𝑞^𝑖₁,𝑐+(𝑐−𝑑)𝑞^𝑗₂𝑞^𝑖₁𝑞^𝑗₂, (11) and 𝑓 is a 𝑞_𝑅-integrable function, so the series in (11) converges. It is clear that the operators given in (10) are linear and positive. For the operator 𝑆^𝑞1,𝑞2_𝑚,𝑛, if 𝑓 is a 𝑞_𝑅-integrable function and 𝑓(𝑥,𝑦)=𝑓₁(𝑥)𝑓₂(𝑦), (𝑥,𝑦)∈ℝ²₊, then 𝑆^𝑞1,𝑞2_𝑚,𝑛𝑓(𝑡,𝑠);𝑥,𝑦=𝑆^𝑞1_𝑚𝑓₁(𝑡),𝑥𝑆^𝑞2_𝑛𝑓₂(𝑠),𝑦. (12) 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 𝑔(𝑡)=^∞_𝑘=0(𝑡−𝑥)^𝑘_𝑞[𝑘]_𝑞!𝐷^𝑘_𝑞𝑔(𝑥), (13) where (𝑡−𝑥)^𝑘_𝑞=^𝑘−1_𝑠=0𝑡−𝑞^𝑠𝑥=^𝑘_𝑠=0𝑘𝑠_𝑞𝑞^{𝑠(𝑠−1)/2}𝑡^𝑘−𝑠(−𝑥)^𝑠. (14) Choosing 𝑡=0 and taking into account (−𝑥)^𝑘_𝑞=(−1)^𝑘𝑥^𝑘𝑞^{𝑘(𝑘−1)/2},𝐷^𝑘_𝑞𝐸_𝑞−[𝑛]_𝑞𝑥=−[𝑛]_𝑞^𝑘𝑞^{𝑘(𝑘−1)/2}𝐸_𝑞−[𝑛]_𝑞𝑞^𝑘𝑥, (15) we get for 𝑔(𝑥)=𝐸_𝑞(−[𝑛]_𝑞𝑥) that 1=𝑔(0)=^∞_𝑘=0(−𝑥)^𝑘_𝑞[𝑘]_𝑞!𝐷^𝑘_𝑞𝑔(𝑥)=^∞_𝑘=0[𝑛]_𝑞𝑥^𝑘[𝑘]_𝑞!𝑞^{𝑘(𝑘−1)}𝐸_𝑞−[𝑛]_𝑞𝑞^𝑘𝑥. (16) Similarly, choosing 𝑡=0 and taking into account (−𝑥)^𝑘_𝑞=(−1)^𝑘𝑥^𝑘𝑞^{𝑘(𝑘−1)/2},𝐷^𝑘_𝑞𝐸_𝑞−[𝑛]_𝑞𝑞𝑥=−[𝑛]_𝑞𝑞^𝑘𝑞^{𝑘(𝑘−1)/2}𝐸_𝑞−[𝑛]_𝑞𝑞^𝑘+1𝑥, (17) we obtain for 𝑔(𝑥)=𝐸_𝑞(−[𝑛]_𝑞𝑞𝑥) that 1=𝑔(0)=^∞_𝑘=0(−𝑥)^𝑘_𝑞[𝑘]_𝑞!𝐷^𝑘_𝑞𝑔(𝑥)=^∞_𝑘=0[𝑛]_𝑞𝑥^𝑘[𝑘]_𝑞!𝑞^𝑘2𝐸_𝑞−[𝑛]_𝑞𝑞^𝑘+1𝑥. (18) Also, using (−𝑥)^𝑘_𝑞=(−1)^𝑘𝑥^𝑘𝑞^{𝑘(𝑘−1)/2},𝐷^𝑘_𝑞𝐸_𝑞−[𝑛]_𝑞𝑞²𝑥=−[𝑛]_𝑞^𝑘𝑞^2𝑘𝑞^{𝑘(𝑘−1)/2}𝐸_𝑞−[𝑛]_𝑞𝑞^𝑘+2𝑥, (19) we have for 𝑔(𝑥)=𝐸_𝑞(−[𝑛]_𝑞𝑞²𝑥) that 1=𝑔(0)=𝑥^∞_𝑘=0(−𝑥)^𝑘_𝑞[𝑘]_𝑞!𝐷^𝑘_𝑞𝑔(𝑥)=^∞_𝑘=0[𝑛]_𝑞𝑥^𝑘[𝑘]_𝑞!𝑞^𝑘2𝐸_𝑞−[𝑛]_𝑞𝑞^𝑘+2𝑥. (20)

Lemma 1. Let 𝑞₁,𝑞₂∈(0,1) and (𝑚,𝑛)∈ℕ×ℕ. One has 𝑆^𝑞1,𝑞2_𝑚,𝑛𝑆1;𝑥,𝑦=1,_{𝑞1,𝑞2𝑚,𝑛}()𝑞𝑡;𝑥=𝑥+¹[2]_𝑞1[𝑚]_𝑞1,𝑆^𝑞1,𝑞2_𝑚,𝑛𝑞𝑠;𝑦=𝑦+²[2]_𝑞2[𝑛]_𝑞2,𝑆^𝑞1,𝑞2_𝑚,𝑛𝑡²;𝑥=𝑥²+1[𝑚]_𝑞11+2𝑞₁[2]_𝑞1𝑞𝑥+²¹[3]_𝑞1[𝑚]²_𝑞1,𝑆^𝑞1,𝑞2_𝑚,𝑛𝑠²;𝑦=𝑦²+1[𝑛]_𝑞21+2𝑞₂[2]_𝑞2𝑞𝑦+²²[3]_𝑞2[𝑛]²_𝑞2. (21)

Proof. Using [𝑘+1]_𝑞=[𝑘]_𝑞+𝑞^𝑘 and from ∫^{[𝑙+1]𝑞2/𝑞2𝑘−1[𝑛]𝑞2}_{[𝑙]𝑞2/𝑞2𝑘−1[𝑛]𝑞2}∫^{[𝑘+1]𝑞1/𝑞1𝑘−1[𝑚]𝑞1}_{[𝑘]𝑞1/𝑞1𝑘−1[𝑚]𝑞1}𝑑^𝑅_𝑞1𝑡𝑑^𝑅_𝑞2𝑠=𝑞₁𝑞₂/[𝑚]_𝑞1[𝑛]_𝑞2, we can write 𝑆^𝑞1,𝑞2_𝑚,𝑛1;𝑥,𝑦=^∞∞_𝑙=0𝑘=0[𝑚]^𝑘_𝑞1𝑥^𝑘[𝑘]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{𝑘(𝑘−1)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞^𝑘₁𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦. (22)𝑆^𝑞1,𝑞2_𝑚,𝑛(1;𝑥,𝑦)=1 is obtained from the above identity (16).
Now, taking 𝑓(𝑡,𝑠)=𝑡 and since ∫^{[𝑙+1]𝑞2/𝑞2𝑘−1[𝑛]𝑞2}_{[𝑙]𝑞2/𝑞2𝑘−1[𝑛]𝑞2}∫^{[𝑘+1]𝑞1/𝑞1𝑘−1[𝑚]𝑞1}_{[𝑘]𝑞1/𝑞1𝑘−1[𝑚]𝑞1}𝑡𝑑^𝑅_𝑞1𝑡𝑑^𝑅_𝑞2𝑠=(𝑞₁𝑞₂/[𝑚]_𝑞1[𝑛]_𝑞2)(([𝑘]_𝑞1/𝑞₁^𝑘−1[𝑚]_𝑞1)+(𝑞₁/[2]_𝑞1[𝑚]_𝑞1)), we get from the linearity of 𝑆^𝑞1,𝑞2_𝑚,𝑛 that 𝑆^𝑞1,𝑞2_𝑚,𝑛(𝑡;𝑥)=^∞∞_𝑙=0𝑘=0[𝑚]^𝑘_𝑞1𝑥^𝑘[𝑘]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{𝑘(𝑘−1)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞^𝑘₁𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦[𝑘]_𝑞1𝑞₁^𝑘−1[𝑚]_𝑞1+^∞∞_𝑙=0𝑘=0[𝑚]^𝑘_𝑞1𝑥^𝑘[𝑘]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{𝑘(𝑘−1)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞^𝑘₁𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦𝑞₁[2]_𝑞1[𝑚]_𝑞1. (23) From this, applying (16) and (18), we have 𝑆^𝑞1,𝑞2_𝑚,𝑛()=𝑡;𝑥_{∞∞𝑙=0𝑘=1}[𝑚]_𝑞^𝑘−1₁𝑥^𝑘[𝑘−1]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{𝑘(𝑘−1)}𝑞₁^𝑘−1𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞^𝑘₁𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦+𝑞₁[2]_𝑞1[𝑚]_𝑞1=𝑥^∞∞_𝑙=0𝑘=0[𝑚]^𝑘_𝑞1𝑥^𝑘[𝑘]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞^𝑘2₁𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞₁^𝑘+1𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦+𝑞₁[2]_𝑞1[𝑚]_𝑞1𝑞=𝑥+¹[2]_𝑞1[𝑚]_𝑞1. (24) Similarly, we write that 𝑆^𝑞1,𝑞2_𝑚,𝑛𝑞𝑠;𝑦=𝑦+²[2]_𝑞2[𝑛]_𝑞2. (25) Now, taking 𝑓(𝑡,𝑠)=𝑡² and from ^{[𝑙+1]𝑞2/𝑞2𝑘−1[𝑛]𝑞2}_{[𝑙]𝑞2/𝑞2𝑘−1[𝑛]𝑞2}^{[𝑘+1]𝑞1/𝑞1𝑘−1[𝑚]𝑞1}_{[𝑘]𝑞1/𝑞1𝑘−1[𝑚]𝑞1}𝑡²𝑑^𝑅_𝑞1𝑡𝑑^𝑅_𝑞2𝑠=𝑞₁𝑞₂[𝑚]_𝑞1[𝑛]_𝑞2×[𝑘]²_𝑞1𝑞₁^2𝑘−2[𝑚]²_𝑞1+[𝑘]_𝑞1𝑞₁^𝑘−1[𝑚]_𝑞12𝑞₁[2]_𝑞1[𝑚]_𝑞1+𝑞²₁[3]_𝑞1[𝑚]²_𝑞1, (26) we have 𝑆^𝑞1,𝑞2_𝑚,𝑛𝑡²=;𝑥_{∞∞𝑙=0𝑘=1}[𝑚]_𝑞^𝑘−1₁𝑥^𝑘[𝑘−1]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{𝑘(𝑘−1)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞^𝑘₁𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦[𝑘]_𝑞1𝑞₁^2𝑘−2[𝑚]_𝑞1+2𝑞₁[2]_𝑞1[𝑚]_𝑞1𝑞𝑥+²¹[3]_𝑞1[𝑚]²_𝑞1. (27) From this, using [𝑘]_𝑞=[𝑘−1]_𝑞+𝑞^𝑘−1 we get 𝑆^𝑞1,𝑞2_𝑚,𝑛𝑡²;𝑥=^∞∞_𝑙=0𝑘=2[𝑚]_𝑞^𝑘−2₁𝑥^𝑘[𝑘−2]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{𝑘(𝑘−1)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞^𝑘₁𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦1𝑞₁^2𝑘−2+^∞∞_𝑙=0𝑘=1[𝑚]_𝑞^𝑘−1₁𝑥^𝑘[𝑘−1]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{𝑘(𝑘−1)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞^𝑘₁𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦1[𝑚]_𝑞1𝑞₁^𝑘−1+2𝑞₁[2]_𝑞1[𝑚]_𝑞1𝑞𝑥+²¹[3]_𝑞1[𝑚]²_𝑞1. (28) Replacing 𝑘 by 𝑘+2 and 𝑘 by 𝑘+1 in the abovementioned, we obtain 𝑆^𝑞1,𝑞2_𝑚,𝑛𝑡²;𝑥=𝑥^2∞∞_𝑙=0𝑘=0[𝑚]^𝑘_𝑞1𝑥^𝑘[𝑘]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{(𝑘+2)(𝑘+1)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞₁^𝑘+2𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦1𝑞₁^2𝑘+2+𝑥^∞∞_𝑙=0𝑘=0[𝑚]^𝑘_𝑞1𝑥^𝑘[𝑘]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞^𝑘2₁𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞₁^𝑘+1𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦1[𝑚]_𝑞1+2𝑞₁[2]_𝑞1[𝑚]_𝑞1𝑞𝑥+²¹[3]_𝑞1[𝑚]²_𝑞1. (29) Equation (20) implies 𝑆^𝑞1,𝑞2_𝑚,𝑛𝑡²;𝑥=𝑥²+1[𝑚]_𝑞11+2𝑞₁[2]_𝑞1𝑞𝑥+²¹[3]_𝑞1[𝑚]²_𝑞1. (30) Similarly, we write that 𝑆^𝑞1,𝑞2_𝑚,𝑛𝑠²;𝑦=𝑦²+1[𝑛]_𝑞21+2𝑞₂[2]_𝑞2𝑞𝑦+²²[3]_𝑞2[𝑛]²_𝑞2. (31) So, the proof is completed.

Similarly, given by the proof of Lemma 1, we calculate 𝑆^𝑞1,𝑞2_𝑚,𝑛(𝑡³;𝑥) and 𝑆^𝑞1,𝑞2_𝑚,𝑛(𝑡⁴;𝑥), shortly. Since ^{[𝑙+1]𝑞2/𝑞2𝑘−1[𝑛]𝑞2}_{[𝑙]𝑞2/𝑞2𝑘−1[𝑛]𝑞2}^{[𝑘+1]𝑞1/𝑞1𝑘−1[𝑚]𝑞1}_{[𝑘]𝑞1/𝑞1𝑘−1[𝑚]𝑞1}𝑡³𝑑^𝑅_𝑞1𝑡𝑑^𝑅_𝑞2𝑠=𝑞₁𝑞₂[𝑚]_𝑞1[𝑛]_𝑞2[𝑘]³_𝑞1𝑞₁^3𝑘−3[𝑚]³_𝑞1+3[𝑘]²_𝑞1𝑞₁^2𝑘−2[𝑚]²_𝑞1𝑞₁[2]_𝑞1[𝑚]_𝑞1+3[𝑘]_𝑞1𝑞₁^𝑘−1[𝑚]_𝑞1𝑞²₁[3]_𝑞1[𝑚]²_𝑞1+𝑞³₁[4]_𝑞1[𝑚]³_𝑞1, (32) we write 𝑆^𝑞1,𝑞2_𝑚,𝑛𝑡³=;𝑥_{∞∞𝑙=0𝑘=1}[𝑚]_𝑞^𝑘−1₁𝑥^𝑘[𝑘−1]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{𝑘(𝑘−1)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞^𝑘₁𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦×[𝑘−1]²_𝑞1+2𝑞^𝑘−1[𝑘−1]+𝑞^2𝑘−2𝑞₁^3𝑘−3[𝑚]²_𝑞1+3𝑞₁[2]_𝑞1[𝑚]_𝑞1𝑥²+1[𝑚]_𝑞1𝑥+3𝑞²₁[3]_𝑞1[𝑚]²_𝑞1𝑞𝑥+³¹[4]_𝑞1[𝑚]³_𝑞1. (33) Using [𝑘]_𝑞=[𝑘−1]_𝑞+𝑞^𝑘−1, we have^∞∞_𝑙=0𝑘=1[𝑚]_𝑞^𝑘−1₁𝑥^𝑘[𝑘−1]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{𝑘(𝑘−1)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞^𝑘₁𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦×[𝑘−1]²_𝑞1+2𝑞^𝑘−1[𝑘−1]+𝑞^2𝑘−2𝑞₁^3𝑘−3[𝑚]²_𝑞1=^∞∞_𝑙=0𝑘=3[𝑚]_𝑞^𝑘−3₁𝑥^𝑘[𝑘−3]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{𝑘(𝑘−1)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞^𝑘₁𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦1𝑞₁^3𝑘−3+^∞∞_𝑙=0𝑘=2[𝑚]_𝑞^𝑘−2₁𝑥^𝑘[𝑘−2]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{𝑘(𝑘−1)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞^𝑘₁𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦𝑞₁^𝑘−2𝑞₁^3𝑘−3[𝑚]_𝑞1+^∞∞_𝑙=0𝑘=2[𝑚]_𝑞^𝑘−2₁𝑥^𝑘[𝑘−2]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{𝑘(𝑘−1)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞^𝑘₁𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦2𝑞^𝑘−1𝑞₁^3𝑘−3[𝑚]_𝑞1+^∞∞_𝑙=0𝑘=1[𝑚]_𝑞^𝑘−1₁𝑥^𝑘[𝑘−1]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{𝑘(𝑘−1)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞^𝑘₁𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦𝑞^2𝑘−2𝑞₁^3𝑘−3[𝑚]²_𝑞1. (34) Then, we rewrite =𝑥^3∞∞_𝑙=0𝑘=0[𝑚]^𝑘_𝑞1𝑥^𝑘[𝑘]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{(𝑘+3)(𝑘+2)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞₁^𝑘+3𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦1𝑞₁^3𝑘+6+𝑥^2∞∞_𝑙=0𝑘=0[𝑚]^𝑘_𝑞1𝑥^𝑘[𝑘]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{(𝑘+2)(𝑘+1)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞₁^𝑘+2𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦𝑞^𝑘₁𝑞₁^3𝑘+3[𝑚]_𝑞1+𝑥^2∞∞_𝑙=0𝑘=0[𝑚]^𝑘_𝑞1𝑥^𝑘[𝑘]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{(𝑘+2)(𝑘+1)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞₁^𝑘+2𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦2𝑞^𝑘+1𝑞₁^3𝑘+3[𝑚]_𝑞1+𝑥^∞∞_𝑙=0𝑘=0[𝑚]^𝑘_𝑞1𝑥^𝑘[𝑘]_𝑞1![𝑛]^𝑙_𝑞2𝑦^𝑙[𝑙]_𝑞2!𝑞₁^{(𝑘+1)(𝑘)}𝑞₂^{𝑙(𝑙−1)}×𝐸_𝑞1−[𝑚]_𝑞1𝑞₁^𝑘+1𝑥𝐸_𝑞2−[𝑛]_𝑞2𝑞^𝑙₂𝑦𝑞^2𝑘𝑞₁^3𝑘[𝑚]²_𝑞1. (35) Finally, we have 𝑆^𝑞1,𝑞2_𝑚,𝑛𝑡³;𝑥=𝑥³+1[𝑚]_𝑞11𝑞₁+2𝑥²+1[𝑚]²_𝑞1𝑥+3𝑞₁[2]_𝑞1[𝑚]_𝑞1𝑥²+1[𝑚]_𝑞1𝑥+3𝑞²₁[3]_𝑞1[𝑚]²_𝑞1𝑞𝑥+³¹[4]_𝑞1[𝑚]³_𝑞1. (36) Since ^{[𝑙+1]𝑞2/𝑞2𝑘−1[𝑛]𝑞2}_{[𝑙]𝑞2/𝑞2𝑘−1[𝑛]𝑞2}^{[𝑘+1]𝑞1/𝑞1𝑘−1[𝑚]𝑞1}_{[𝑘]𝑞1/𝑞1𝑘−1[𝑚]𝑞1}𝑡⁴𝑑^𝑅_𝑞1𝑡𝑑^𝑅_𝑞2𝑠=𝑞₁𝑞₂[𝑚]_𝑞1[𝑛]_𝑞2[𝑘]⁴_𝑞1𝑞₁^4𝑘−4[𝑚]⁴_𝑞1+4[𝑘]³_𝑞1𝑞₁^3𝑘−3[𝑚]³_𝑞1𝑞₁[2]_𝑞1[𝑚]_𝑞1+6[𝑘]²_𝑞1𝑞₁^2𝑘−2[𝑚]²_𝑞1𝑞²₁[3]_𝑞1[𝑚]²_𝑞1+4[𝑘]_𝑞1𝑞₁^𝑘−1[𝑚]_𝑞1𝑞³₁[4]_𝑞1[𝑚]³_𝑞1+𝑞⁴₁[5]_𝑞1[𝑚]⁴_𝑞1, (37) we obtain𝑆^𝑞1,𝑞2_𝑚,𝑛𝑡⁴;𝑥=𝑥⁴+1[𝑚]_𝑞11𝑞²₁+2𝑞₁+3𝑥³+1[𝑚]²_𝑞11𝑞²₁+3𝑞₁+3𝑥²+1[𝑚]³_𝑞1𝑥+4𝑞₁[2]_𝑞1[𝑚]_𝑞1𝑥³+1[𝑚]_𝑞11𝑞₁+2𝑥²+1[𝑚]²_𝑞1𝑥+6𝑞²₁[3]_𝑞1[𝑚]²_𝑞1𝑥²+1[𝑚]_𝑞1𝑥+4𝑞³₁[4]_𝑞1[𝑚]³_𝑞1𝑞𝑥+⁴¹[5]_𝑞1[𝑚]⁴_𝑞1. (38)

3. Approximation Properties in Polynomial Weighted Spaces

For bivariate operators, the space is considered as follows: 𝐶_𝑝,𝑞ℝ²₊=𝑓∈𝐶ℝ²₊∶𝜔_𝑝,𝑞𝑓isuniformlycontinuousandboundedonℝ²₊=[0,∞)×[0,∞) (39) associated with the weighted function 𝜔_𝑝,𝑞(𝑥,𝑦)=𝜔_𝑝(𝑥)𝜔_𝑞(𝑦), (𝑝,𝑞)∈ℕ₀×ℕ₀. The weight 𝜔_𝑝 is defined as 𝜔₀(𝑡)=1, 𝜔_𝑝(𝑥)=(1+𝑥^𝑝)⁻¹. The norm of this space is denoted by ‖⋅‖_𝑝,𝑞 and is defined by 𝑓_𝑝,𝑞=sup_{𝑥,𝑦∈ℝ2+}𝜔_𝑝,𝑞𝑥,𝑦𝑓𝑥,𝑦. (40) Now, we give some useful results given by Agratini [15].

For each 𝑧∈ℝ₊, define the function 𝜑_𝑧 by 𝜑^𝑟_𝑧(𝑡)=(𝑡−𝑧)^𝑟, 𝑡∈ℝ₊, 𝑟∈ℕ. For the one-dimensional operator 𝐿_𝑠, 𝑠∈ℕ, and for each 𝑟∈ℕ, a polynomial Γ_𝑟 exists such that 𝑠^𝑟/2𝐿_𝑠𝜑^𝑟_𝑧(𝑡)≤Γ_𝑟(𝑡),degΓ_𝑟≤𝑟. (41)

Theorem 2 (see [15]). Consider (𝑝,𝑞)∈ℕ₀×ℕ₀. For any (𝑚,𝑛)∈ℕ×ℕ, given by 𝐿_𝑚,𝑛𝑓=𝑥,𝑦_{∞∞𝑖=0𝑗=0}𝑎_𝑚,𝑖(𝑥)𝑏_𝑛,𝑗𝑦𝑓𝑥_𝑚,𝑖,𝑦_𝑛,𝑗,𝑥,𝑦∈ℝ²₊ (42) the operator 𝐿_𝑚,𝑛 verifies 𝐿_𝑚,𝑛1𝜔_𝑝,𝑞;⋅_𝑝,𝑞,𝐿≤𝑐𝑝,𝑞_𝑚,𝑛𝑓;⋅_𝑝,𝑞≤𝑐𝑝,𝑞𝑓_𝑝,𝑞,𝑓∈𝐶_𝑝,𝑞ℝ²₊. (43)

Theorem 3 (see [15]). Let (𝑝,𝑞)∈ℕ₀×ℕ₀. For any (𝑚,𝑛)∈ℕ×ℕ, the operator 𝐿_𝑚,𝑛 given by (42) satisfies𝜔_𝑝,𝑞𝑥,𝑦𝐿_𝑚,𝑛𝑓𝜔𝑥,𝑦−𝑓𝑥,𝑦≤𝑐𝑝,𝑞_𝑓𝜙(𝑥)√𝑚,𝜙𝑦√𝑛, (44)(𝑥,𝑦)∈ℝ²₊, where 𝜙 is given by 𝜙(𝑡)=Γ₂(𝑡)+Γ₂(𝑡)+^𝑝_𝑘=0𝑝𝑘Γ_𝑘+2(𝑡), (45) with the polynomials Γ_𝜈(𝑡), 𝜈=2,𝑝+2, being indicated at (41), and 𝑐(𝑝,𝑞) is a suitable constant.

Theorem 4 (see [15]). Let (𝑝,𝑞)∈ℕ₀×ℕ₀. Let the operator 𝐿_𝑚,𝑛,(𝑚,𝑛)∈ℕ×ℕ, be defined by (42). For any (𝑥,𝑦)∈ℝ²₊ the pointwise convergence takes place lim_{𝑚,𝑛→∞}𝐿_𝑚,𝑛𝑓𝑥,𝑦=𝑓𝑥,𝑦,𝑓∈𝐶_𝑝,𝑞ℝ²₊. (46) If 𝐾₁, 𝐾₂ are compact intervals included in ℝ₊, then (46) holds uniformly on the domain 𝐾₁×𝐾₂.

In the latter paper, we use the weight function 𝜌(𝑥,𝑦)=(1+𝑥²+𝑦²)⁻¹ instead of 𝜔_𝑝,𝑞(𝑥,𝑦) and instead of the space 𝐶_𝑝,𝑞(ℝ²₊), the space 𝐶₂(ℝ²₊) associated with the weighted function 𝜌(𝑥,𝑦) is used. We denote the norm of this space by ‖⋅‖₂.

Lemma 5. The operator 𝑆^𝑞1,𝑞2_𝑚,𝑛, (𝑚,𝑛)∈ℕ×ℕ, 𝑞₁,𝑞₂∈(0,1), given by (10) verifies 𝑆^𝑞1,𝑞2_𝑚,𝑛1𝜌(𝑡,𝑠);⋅₂≤103,𝑆^𝑞1,𝑞2_𝑚,𝑛𝑓;⋅₂≤103𝑓₂,𝑓∈𝐶₂ℝ²₊. (47)(48)

Proof. Since 𝑞₁,𝑞₂∈(0,1) and by Lemma 1, we have 𝜌𝑥,𝑦𝑆^𝑞1,𝑞2_𝑚,𝑛1𝜌(𝑡,𝑠);⋅≤𝜌𝑥,𝑦1+𝑥²+𝑦²+73𝑥+𝑦. (49) So, inequality (47) is proved. Since the operator 𝑆^𝑞1,𝑞2_𝑚,𝑛 is linear and positive and by using (47) 𝜌𝑥,𝑦𝑆^𝑞1,𝑞2_𝑚,𝑛𝑓;⋅≤𝜌𝑥,𝑦𝑆^𝑞1,𝑞2_𝑚,𝑛𝜌(𝑓1𝑡,𝑠)𝜌()≤𝑡,𝑠;⋅103𝑓₂, (50) we obtain inequality (48).

The Steklov function associated with 𝑓∈𝐶(ℝ²₊) is given as follows: 𝑓_ℎ,𝛿=1𝑥,𝑦ℎ𝛿^ℎ₀𝑑𝑢^𝛿₀𝑓𝑥+𝑢,𝑦+𝑣𝑑𝑣,𝑥,𝑦∈ℝ²₊, (51) where ℎ,𝛿>0.

The modulus of smoothness function associated with any function 𝑓∈𝐶₂(ℝ²₊) is given by 𝜔_𝑓(ℎ,𝛿)=sup_{0≤𝑢≤ℎ0≤𝑣≤𝛿}𝑓𝑥+𝑢,𝑦+𝑣−𝑓𝑥,𝑦₂,𝑥,𝑦∈ℝ²₊. (52) One can see that 𝑓−𝑓_ℎ,𝛿2=sup_{𝑥,𝑦∈ℝ2+}𝜌1𝑥,𝑦ℎ𝛿^ℎ₀𝑑𝑢^𝛿₀𝑓𝑥+𝑢,𝑦+𝑣−𝑓𝑥,𝑦𝑑𝑣≤sup_{𝑥,𝑦∈ℝ2+}𝜌𝑥,𝑦sup_{0≤𝑢≤ℎ0≤𝑣≤𝛿}𝑓𝑥+𝑢,𝑦+𝑣−𝑓𝑥,𝑦=sup_{0≤𝑢≤ℎ0≤𝑣≤𝛿}𝑓𝑥+𝑢,𝑦+𝑣−𝑓𝑥,𝑦₂=𝜔_𝑓(ℎ,𝛿). (53) The following inequalities verify 𝜕𝑓𝜕𝑥^ℎ,𝛿₂≤2ℎ𝜔_𝑓(),𝜕ℎ,𝛿𝑓𝜕𝑦^ℎ,𝛿₂≤2𝛿𝜔_𝑓(ℎ,𝛿), (54) where ℎ,𝛿>0. In order to justify these inequalities, one can see that 𝜕𝑓𝜕𝑥^ℎ,𝛿₂=sup_{𝑥,𝑦∈ℝ2+}𝜌1𝑥,𝑦ℎ𝛿^𝛿₀𝑓𝑥+ℎ,𝑦+𝑣−𝑓𝑥,𝑦+𝑣𝑑𝑣=sup_{𝑥,𝑦∈ℝ2+}𝜌1𝑥,𝑦ℎ𝛿^𝛿₀𝑓≤1𝑥+ℎ,𝑦+𝑣−𝑓𝑥,𝑦+𝑓𝑥,𝑦−𝑓𝑥,𝑦+𝑣𝑑𝑣ℎ𝛿sup_{𝑥,𝑦∈ℝ2+}𝜌𝑥,𝑦^𝛿₀𝑓+1𝑥+ℎ,𝑦+𝑣−𝑓𝑥,𝑦𝑑𝑣ℎ𝛿sup_{𝑥,𝑦∈ℝ2+}𝜌𝑥,𝑦^𝛿₀𝑓≤2𝑥,𝑦+𝑣−𝑓𝑥,𝑦𝑑𝑣ℎsup_{𝑥,𝑦∈ℝ2+}𝜌𝑥,𝑦sup_{0≤𝑢≤ℎ0≤𝑣≤𝛿}𝑓=2𝑥+𝑢,𝑦+𝑣−𝑓𝑥,𝑦ℎ𝜔_𝑓(ℎ,𝛿). (55)

Theorem 6. For any (𝑚,𝑛)∈ℕ×ℕ and 𝑓∈𝐶₂(ℝ²₊), the operator 𝑆^𝑞1,𝑞2_𝑚,𝑛𝑞₁,𝑞₂∈(0,1) given by (10) satisfies 𝜌𝑥,𝑦𝑆^𝑞1,𝑞2_𝑚,𝑛𝑓≤𝑥,𝑦−𝑓𝑥,𝑦103𝜔_𝑓𝜙𝑥,𝑞₁[𝑚]_𝑞1,𝜙𝑦,𝑞₂[𝑛]_𝑞2, (56) where (𝑥,𝑦)∈ℝ²₊, 𝜙(𝑥,𝑞)=𝑞/[2]_𝑞+Γ(𝑥,𝑞).

Proof. For any (𝑚,𝑛)∈ℕ×ℕ, we can write 𝜌𝑥,𝑦𝑆^𝑞1,𝑞2_𝑚,𝑛𝑓𝑆𝑥,𝑦−𝑓𝑥,𝑦≤𝜌𝑥,𝑦^𝑞1,𝑞2_𝑚,𝑛𝑓−𝑓_ℎ,𝛿𝑆;𝑥,𝑦+𝜌𝑥,𝑦^𝑞1,𝑞2_𝑚,𝑛𝑓_ℎ,𝛿𝑥,𝑦−𝑓_ℎ,𝛿𝑓𝑥,𝑦+𝜌𝑥,𝑦_ℎ,𝛿𝑥,𝑦−𝑓𝑥,𝑦. (57) Inequalities (48) and (53) imply that 𝜌𝑥,𝑦𝑆^𝑞1,𝑞2_𝑚,𝑛𝑓−𝑓_ℎ,𝛿≤𝑆;𝑥,𝑦_{𝑞1,𝑞2𝑚,𝑛}𝑓−𝑓_ℎ,𝛿;𝑥,𝑦₂≤103𝑓−𝑓_ℎ,𝛿2≤103𝜔_𝑓(ℎ,𝛿). (58) Let 𝐶¹₂(ℝ²₊) be the class of all functions in which partial derivatives belong to 𝐶₂(ℝ²₊). Since 𝑓_ℎ,𝛿∈𝐶¹₂ℝ²₊ and 𝑆^𝑞1,𝑞2_𝑚,𝑛 given by (10) is linear and monotone, we get 𝜌𝑥,𝑦𝑆^𝑞1,𝑞2_𝑚,𝑛𝑓_ℎ,𝛿𝑥,𝑦−𝑓_ℎ,𝛿𝑆𝑥,𝑦=𝜌𝑥,𝑦^𝑞1,𝑞2_𝑚,𝑛^𝑡_𝑥𝜕𝑓𝜕𝑢^ℎ,𝛿()𝑆𝑢,𝑠𝑑𝑢;𝑥,𝑦+𝜌𝑥,𝑦^𝑞1,𝑞2_𝑚,𝑛^𝑠_𝑦𝜕𝑓𝜕𝑣^ℎ,𝛿()𝑆𝑥,𝑠𝑑𝑣;𝑥,𝑦≤𝜌𝑥,𝑦^𝑞1,𝑞2_𝑚,𝑛^𝑡_𝑥𝜕𝑓𝜕𝑢^ℎ,𝛿()𝑆𝑢,𝑠𝑑𝑢;𝑥,𝑦+𝜌𝑥,𝑦^𝑞1,𝑞2_𝑚,𝑛^𝑠_𝑦𝜕𝑓𝜕𝑣^ℎ,𝛿(𝑥,𝑠)𝑑𝑣;𝑥,𝑦. (59) Then, by definition of norm ‖⋅‖₂ and the first mean value theorem for integration, we have ^𝑡_𝑥𝜕𝑓𝜕𝑢^ℎ,𝛿(𝑢,𝑠)𝑑𝑢≤^𝑡_𝑥𝜌(𝜕𝑢,𝑠)𝑓𝜕𝑢^ℎ,𝛿(𝑢,𝑠)𝑑𝑢𝜌()≤𝜕𝑢,𝑠𝑓𝜕𝑥^ℎ,𝛿₂^𝑡_𝑥𝑑𝑢𝜌()=𝜕𝑢,𝑠𝑓𝜕𝑥^ℎ,𝛿₂||1𝑡−𝑥𝜌()()≤𝜕𝜉,𝑠,𝜉∈𝑥,𝑡𝑓𝜕𝑥^ℎ,𝛿₂||1𝑡−𝑥𝜌()+1𝑥,𝑠𝜌().𝑡,𝑠 (60) Following the same way, one finds ^𝑠_𝑦𝜕𝑓𝜕𝑣^ℎ,𝛿()≤𝜕𝑥,𝑠𝑑𝑣𝑓𝜕𝑦^ℎ,𝛿₂1𝑠−𝑦𝜌+1𝑥,𝑦𝜌().𝑥,𝑠 (61) Using inequalities (59), (61), and (60), since 𝑆^𝑞1,𝑞2_𝑚,𝑛 is linear and monotone, we get 𝜌𝑥,𝑦𝑆^𝑞1,𝑞2_𝑚,𝑛𝑓_ℎ,𝛿𝑥,𝑦−𝑓_ℎ,𝛿𝜕𝑥,𝑦≤𝜌𝑥,𝑦𝑓𝜕𝑥^ℎ,𝛿₂𝑆^𝑞1,𝑞2_𝑚,𝑛|𝑡−𝑥|𝜌()𝜕𝑥,𝑠;𝑥,𝑦+𝜌𝑥,𝑦𝑓𝜕𝑥^ℎ,𝛿₂𝑆^𝑞1,𝑞2_𝑚,𝑛|𝑡−𝑥|𝜌()𝜕𝑡,𝑠;𝑥,𝑦+𝜌𝑥,𝑦𝑓𝜕𝑦^ℎ,𝛿₂𝑆^𝑞1,𝑞2_𝑚,𝑛𝑠−𝑦𝜌𝜕𝑥,𝑦;𝑥,𝑦+𝜌𝑥,𝑦𝑓𝜕𝑦^ℎ,𝛿₂𝑆^𝑞1,𝑞2_𝑚,𝑛𝑠−𝑦𝜌().𝑥,𝑠;𝑥,𝑦 (62) From Lemmas 1 and 5 and (36), we can write the following: 𝜌𝑥,𝑦𝑆^𝑞1,𝑞2_𝑚,𝑛𝑓_ℎ,𝛿𝑥,𝑦−𝑓_ℎ,𝛿≤𝜕𝑥,𝑦𝑓𝜕𝑥^ℎ,𝛿₂103𝑞₁[2]_𝑞1[𝑚]_𝑞1+𝜕𝑓𝜕𝑥^ℎ,𝛿₂103Γ𝑥,𝑞₁[𝑚]_𝑞1+𝜕𝑓𝜕𝑦^ℎ,𝛿₂𝑞₂[2]_𝑞2[𝑛]_𝑞2+𝜕𝑓𝜕𝑦^ℎ,𝛿₂103Γ𝑦,𝑞₂[𝑛]_𝑞2, (63) where Γ=1𝑥,𝑞𝑞𝑞+1+[2]_𝑞𝑥²+1+3𝑞[2]_𝑞+2𝑞²[3]_𝑞𝑞𝑥+³[4]_𝑞 (64) is a polynomial of degree 2. Then, by inequalities (54), we have 𝜌𝑥,𝑦𝑆^𝑞1,𝑞2_𝑚,𝑛𝑓_ℎ,𝛿𝑥,𝑦−𝑓_ℎ,𝛿≤2𝑥,𝑦ℎ103𝑞₁[2]_𝑞1[𝑚]_𝑞1+Γ𝑥,𝑞₁[𝑚]_𝑞1+2𝛿𝑞₂[2]_𝑞2[𝑛]_𝑞2+103Γ𝑦,𝑞₂[𝑛]_𝑞2𝜔_𝑓(ℎ,𝛿). (65) Finally, we write from (53) 𝜌𝑥,𝑦𝑓_ℎ,𝛿𝑥,𝑦−𝑓𝑥,𝑦≤𝑓_ℎ,𝛿−𝑓₂≤𝜔_𝑓(ℎ,𝛿). (66) If we go back to (57) and take ℎ=𝜙(𝑥,𝑞₁)/[𝑚]_𝑞1, 𝛿=𝜙(𝑦,𝑞₂)/[𝑛]_𝑞2, then the proof is completed.

We replace 𝑞₁ and 𝑞₂ in (10) by sequences (𝑞_1,𝑚), (𝑞_2,𝑛) so that lim_𝑚→∞𝑞_1,𝑚=1,lim_𝑛→∞𝑞_2,𝑛=1,lim_𝑚→∞1[𝑛]_𝑞1,𝑚=0,lim_𝑛→∞1[𝑛]_𝑞2,𝑛=0. (67) So, (ℎ,𝛿)→(0⁺,0⁺) as 𝑛,𝑚→∞. Knowing that modulus of smoothness function 𝜔_𝑓 satisfies the property lim_{(ℎ,𝛿)→(0}⁺_,0⁺₎𝜔_𝑓(ℎ,𝛿)=0, from Theorem 6, we deduce the following result.

Theorem 7. Let (𝑚,𝑛)∈ℕ×ℕ and let (𝑞_1,𝑚), (𝑞_2,𝑛) be sequences in the interval (0,1) satisfying (67). Let the operator 𝑆^𝑞1,𝑞2_𝑚,𝑛 given by (10) and 𝑓∈𝐶₂(ℝ²₊)𝑞_𝑅-integrable function. For any (𝑥,𝑦)∈ℝ²₊ the pointwise convergence takes place lim_{𝑚,𝑛→∞}𝑆^{𝑞1,𝑚,𝑞2,𝑛}_𝑚,𝑛𝑓𝑥,𝑦=𝑓𝑥,𝑦. (68) If 𝐾₁, 𝐾₂ are compact intervals included in ℝ₊, then (68) holds uniformly on the domain 𝐾₁×𝐾₂.

4. Voronovskaya-Type Theorem

We will prove the Voronovskaya-type theorem.

Theorem 8. Let (𝑞_1,𝑛), (𝑞_2,𝑛) be sequences in the interval (0,1) satisfying (67). Suppose that 𝑓∈𝐶²₂(ℝ²₊) is the class of all functions in which the second partial derivatives belong to 𝐶₂(ℝ²₊) and 𝑞_𝑅-integrable function. Then, for every (𝑥,𝑦)∈ℝ²₊, one has lim_𝑛→∞𝑛𝑆^{𝑞1,𝑛,𝑞2,𝑛}_𝑛,𝑛=𝑥𝑓;𝑥,𝑦−𝑓𝑥,𝑦2𝑓_𝑥𝑥+𝑦𝑥,𝑦2𝑓_𝑦𝑦+1𝑥,𝑦2𝑓_𝑥+1𝑥,𝑦2𝑓_𝑦𝑥,𝑦. (69)

Proof. Let 𝑓∈𝐶²₂(ℝ²₊) and 𝑞_𝑅-integrable function and let (𝑥₀,𝑦₀)∈ℝ²₊ be fixed point. Then, by the Taylor formula, we can write 𝑓()𝑥𝑡,𝑠=𝑓₀,𝑦₀+𝑓_𝑥𝑥₀,𝑦₀𝑡−𝑥₀+𝑓_𝑦𝑥₀,𝑦₀𝑠−𝑦₀+12𝑓_𝑥𝑥𝑥₀,𝑦₀𝑡−𝑥₀²+2𝑓_𝑥𝑦𝑥₀,𝑦₀×𝑡−𝑥₀𝑠−𝑦₀+𝑓_𝑦𝑦𝑥₀,𝑦₀𝑠−𝑦₀²+𝜑𝑡,𝑠;𝑥₀,𝑦₀𝑡−𝑥₀⁴+𝑠−𝑦₀⁴, (70) where (𝑡,𝑠)∈ℝ²₊, 𝜑(𝑡,𝑠)=𝜑(𝑡,𝑠;𝑥₀,𝑦₀) belongs to 𝐶₂(ℝ²₊), and lim_{(𝑡,𝑠)→(𝑥0,𝑦0)}𝜑(𝑡,𝑠)=0 for 𝑛∈ℕ. From the linearity of 𝑆^𝑞1,𝑞2_𝑛,𝑛, we have 𝑆^𝑞1,𝑞2_𝑛,𝑛𝑓(𝑡,𝑠);𝑥₀,𝑦₀=𝑓𝑥₀,𝑦₀+𝑓_𝑥𝑥₀,𝑦₀𝑆^𝑞1,𝑞2_𝑛,𝑛𝑡−𝑥₀;𝑥₀+𝑓_𝑦𝑥₀,𝑦₀𝑆^𝑞1,𝑞2_𝑛,𝑛𝑠−𝑦₀;𝑦₀+12𝑓_𝑥𝑥𝑥₀,𝑦₀𝑆^𝑞1,𝑞2_𝑛,𝑛𝑡−𝑥₀²;𝑥₀+2𝑓_𝑥𝑦𝑥₀,𝑦₀𝑆^𝑞1,𝑞2_𝑛,𝑛𝑡−𝑥₀;𝑥₀𝑆^𝑞1,𝑞2_𝑛,𝑛𝑠−𝑦₀;𝑦₀+𝑓_𝑦𝑦𝑥₀,𝑦₀𝑆^𝑞1,𝑞2_𝑛,𝑛𝑠−𝑦₀²;𝑦₀+𝑆^𝑞1,𝑞2_𝑛,𝑛𝜑(𝑡,𝑠)𝑡−𝑥₀⁴+𝑠−𝑦₀⁴;𝑥₀,𝑦₀. (71) By Lemma 1 and since the sequences 𝑞_1,𝑛, 𝑞_2,𝑛 satisfy (67), we obtain lim_𝑛→∞𝑛𝑆^{𝑞1,𝑛,𝑞2,𝑛}_𝑛,𝑛𝑡−𝑥₀;𝑥₀=12=lim_𝑛→∞𝑛𝑆^{𝑞1,𝑛,𝑞2,𝑛}_𝑛,𝑛𝑠−𝑦₀;𝑦₀,lim_𝑛→∞𝑛𝑆^{𝑞1,𝑛,𝑞2,𝑛}_𝑛,𝑛𝑡−𝑥₀²;𝑥₀=𝑥₀,lim_𝑛→∞𝑛𝑆^{𝑞1,𝑛,𝑞2,𝑛}_𝑛,𝑛𝑠−𝑦₀²;𝑦₀=𝑦₀. (72) By the Hölder inequality, we have 𝑆^𝑞1,𝑞2_𝑛,𝑛𝜑(𝑡,𝑠)𝑡−𝑥₀⁴+𝑠−𝑦₀⁴;𝑥₀,𝑦₀≤𝑆^𝑞1,𝑞2_𝑛,𝑛𝜑²(𝑡,𝑠);𝑥₀,𝑦₀^1/2×𝑆^𝑞1,𝑞2_𝑛,𝑛𝑡−𝑥₀⁴;𝑥₀+𝑆^𝑞1,𝑞2_𝑛,𝑛𝑠−𝑦₀⁴;𝑦₀^1/2. (73) By properties of 𝜑 and Theorem 7, we get lim_𝑛→∞𝑆^{𝑞1,𝑛,𝑞2,𝑛}_𝑛,𝑛𝜑²(𝑡,𝑠);𝑥₀,𝑦₀=𝜑²𝑥₀,𝑦₀=0. (74) From the foregoing facts and using (12) and (38), we obtain lim_𝑛→∞𝑛𝑆^{𝑞1,𝑛,𝑞2,𝑛}_𝑛,𝑛𝜑(𝑡,𝑠)𝑡−𝑥₀⁴+𝑠−𝑦₀⁴;𝑥₀,𝑦₀=0. (75) Then, using (72) and (75), we reproduce from (71) lim_𝑛→∞𝑛𝑆^{𝑞1,𝑛,𝑞2,𝑛}_𝑛,𝑛𝑓(𝑡,𝑠);𝑥₀,𝑦₀−𝑓𝑥₀,𝑦₀=𝑥₀2𝑓_𝑥𝑥𝑥₀,𝑦₀+𝑦₀2𝑓_𝑦𝑦𝑥₀,𝑦₀+12𝑓_𝑥+1𝑥,𝑦2𝑓_𝑦𝑥,𝑦. (76) Thus, the proof is completed for 𝑓∈𝐶²₂(ℝ²₊).