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

Authors: Mediha Örkcü
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The present paper deals with approximation properties ofq-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.


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## Body

## 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. Dog˘ru 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 theq-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 aboutq-integers. For any nonnegative integer r, the q-integer of the number r is defined by
(1)[r]q={1+q+⋯+qr-1ifq≠1rifq=1,[0]q=1,
where q is a positive real number. The q-factorial is defined as
(2)[r]q!={[1]q[2]q⋯[r]qifr=1,2,…1ifr=0,[0]q!=1.
Two q-analogues of the exponential function ex are given as
(3)Eq(x)=∑n=0∞‍qn(n-1)/2xn[n]q!,x∈ℝ,ɛq(x)=∑n=0∞‍xn[n]q!,|x|<11-q.
The following relation between q-exponential functions Eq(x) and ɛq(x) holds:(4)Eq(x)ɛq(-x)=1,|x|<11-q.
The q-derivative of a function f(x), denoted by Dqf, is defined by
(5)(Dqf)(x)=f(qx)-f(x)(q-1)x,x≠0,(Dqf)(0)=limx→0(Dqf)(x).
Also, it is known that DqE(ax)=aE(qax).Theq-integral of the function f over the interval [0,b] is defined by
(6)∫0b‍f(t)dqt=b(1-q)∑j=0∞‍f(bqj)qj,0<q<1.
If f is integrable over [0,b], then
(7)limq→1-∫0b‍f(t)dqt=∫0b‍f(t)dt.
Generally accepted definition for q-integral over an interval [a,b] is (8)∫ab‍f(t)dqt=∫0b‍f(t)dqt-∫0a‍f(t)dqt.
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
(9)∫ab‍f(t)dqRt=(1-q)(b-a)∑j=0∞‍f(a+(b-a)qj)qj,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh0<q<1.
This definition includes only point inside the interval of the integration.Details ofq-integers can be found in [17].


## 2. Construction of the Bivariate Operators

Forq1,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:
(10)Sm,nq1,q2(f;x,y)=∑l=0∞‍∑k=0∞‍[m]q1k+1xk[k]q1![n]q2l+1yl[l]q2!q1k(k-1)-1q2l(l-1)-1×Eq1(-[m]q1q1kx)Eq2(-[n]q2q2ly)×∫[l]q2/q2k-1[n]q2[l+1]q2/q2k-1[n]q2‍∫[k]q1/q1k-1[m]q1[k+1]q1/q1k-1[m]q1‍f(t,s)dq1Rtdq2Rs,
where
(11)∫cd‍∫ab‍f(t,s)dq1Rtdq2Rs=(1-q1)(1-q2)(b-a)(c-d)×∑j=0∞‍∑i=0∞‍f(a+(b-a)q1i,c+(c-d)q2j)q1iq2j,
and f is a qR-integrable function, so the series in (11) converges. It is clear that the operators given in (10) are linear and positive. For the operator Sm,nq1,q2, if f is a qR-integrable function and f(x,y)=f1(x)f2(y), (x,y)∈ℝ+2, then
(12)Sm,nq1,q2(f(t,s);x,y)=Smq1(f1(t),x)Snq2(f2(s),y).
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
(13)g(t)=∑k=0∞‍(t-x)qk[k]q!Dqkg(x),
where
(14)(t-x)qk=∏s=0k-1‍(t-qsx)=∑s=0k‍[ks]qqs(s-1)/2tk-s(-x)s.
Choosing t=0 and taking into account
(15)(-x)qk=(-1)kxkqk(k-1)/2,DqkEq(-[n]qx)=(-[n]q)kqk(k-1)/2Eq(-[n]qqkx),
we get for g(x)=Eq(-[n]qx) that
(16)1=g(0)=∑k=0∞‍(-x)qk[k]q!Dqkg(x)=∑k=0∞‍([n]qx)k[k]q!qk(k-1)Eq(-[n]qqkx).
Similarly, choosing t=0 and taking into account
(17)(-x)qk=(-1)kxkqk(k-1)/2,DqkEq(-[n]qqx)=(-[n]qq)kqk(k-1)/2Eq(-[n]qqk+1x),
we obtain for g(x)=Eq(-[n]qqx) that
(18)1=g(0)=∑k=0∞‍(-x)qk[k]q!Dqkg(x)=∑k=0∞‍([n]qx)k[k]q!qk2Eq(-[n]qqk+1x).
Also, using
(19)(-x)qk=(-1)kxkqk(k-1)/2,DqkEq(-[n]qq2x)=(-[n]q)kq2kqk(k-1)/2Eq(-[n]qqk+2x),
we have for g(x)=Eq(-[n]qq2x) that
(20)1=g(0)=x∑k=0∞‍(-x)qk[k]q!Dqkg(x)=∑k=0∞‍([n]qx)k[k]q!qk2Eq(-[n]qqk+2x).Lemma 1.
Letq1,q2∈(0,1) and (m,n)∈ℕ×ℕ. One has
(21)Sm,nq1,q2(1;x,y)=1,Sm,nq1,q2(t;x)=x+q1[2]q1[m]q1,Sm,nq1,q2(s;y)=y+q2[2]q2[n]q2,Sm,nq1,q2(t2;x)=x2+1[m]q1(1+2q1[2]q1)x+q12[3]q1[m]q12,Sm,nq1,q2(s2;y)=y2+1[n]q2(1+2q2[2]q2)y+q22[3]q2[n]q22.Proof.
Using[k+1]q=[k]q+qk and from ∫[l]q2/q2k-1[n]q2[l+1]q2/q2k-1[n]q2‍∫[k]q1/q1k-1[m]q1[k+1]q1/q1k-1[m]q1‍dq1Rtdq2Rs=q1q2/[m]q1[n]q2, we can write
(22)Sm,nq1,q2(1;x,y)=∑l=0∞‍∑k=0∞‍[m]q1kxk[k]q1![n]q2lyl[l]q2!q1k(k-1)q2l(l-1)×Eq1(-[m]q1q1kx)Eq2(-[n]q2q2ly).Sm,nq1,q2(1;x,y)=1 is obtained from the above identity (16).
Now, takingf(t,s)=t and since ∫[l]q2/q2k-1[n]q2[l+1]q2/q2k-1[n]q2‍∫[k]q1/q1k-1[m]q1[k+1]q1/q1k-1[m]q1‍tdq1Rtdq2Rs=(q1q2/[m]q1[n]q2)(([k]q1/q1k-1[m]q1)+(q1/[2]q1[m]q1)), we get from the linearity of Sm,nq1,q2 that
(23)Sm,nq1,q2(t;x)=∑l=0∞‍∑k=0∞‍[m]q1kxk[k]q1![n]q2lyl[l]q2!q1k(k-1)q2l(l-1)×Eq1(-[m]q1q1kx)Eq2(-[n]q2q2ly)[k]q1q1k-1[m]q1+∑l=0∞‍∑k=0∞‍[m]q1kxk[k]q1![n]q2lyl[l]q2!q1k(k-1)q2l(l-1)×Eq1(-[m]q1q1kx)Eq2(-[n]q2q2ly)q1[2]q1[m]q1.
From this, applying (16) and (18), we have
(24)Sm,nq1,q2(t;x)=∑l=0∞‍∑k=1∞‍[m]q1k-1xk[k-1]q1![n]q2lyl[l]q2!q1k(k-1)q1k-1q2l(l-1)×Eq1(-[m]q1q1kx)Eq2(-[n]q2q2ly)+q1[2]q1[m]q1=x∑l=0∞‍∑k=0∞‍[m]q1kxk[k]q1![n]q2lyl[l]q2!q1k2q2l(l-1)×Eq1(-[m]q1q1k+1x)Eq2(-[n]q2q2ly)+q1[2]q1[m]q1=x+q1[2]q1[m]q1.
Similarly, we write that
(25)Sm,nq1,q2(s;y)=y+q2[2]q2[n]q2.
Now, taking f(t,s)=t2 and from
(26)∫[l]q2/q2k-1[n]q2[l+1]q2/q2k-1[n]q2‍∫[k]q1/q1k-1[m]q1[k+1]q1/q1k-1[m]q1‍t2dq1Rtdq2Rs=q1q2[m]q1[n]q2×([k]q12q12k-2[m]q12+[k]q1q1k-1[m]q12q1[2]q1[m]q1+q12[3]q1[m]q12),
we have
(27)Sm,nq1,q2(t2;x)=∑l=0∞‍∑k=1∞‍[m]q1k-1xk[k-1]q1![n]q2lyl[l]q2!q1k(k-1)q2l(l-1)×Eq1(-[m]q1q1kx)Eq2(-[n]q2q2ly)[k]q1q12k-2[m]q1+2q1[2]q1[m]q1x+q12[3]q1[m]q12.
From this, using [k]q=[k-1]q+qk-1 we get
(28)Sm,nq1,q2(t2;x)=∑l=0∞‍∑k=2∞‍[m]q1k-2xk[k-2]q1![n]q2lyl[l]q2!q1k(k-1)q2l(l-1)×Eq1(-[m]q1q1kx)Eq2(-[n]q2q2ly)1q12k-2+∑l=0∞‍∑k=1∞‍[m]q1k-1xk[k-1]q1![n]q2lyl[l]q2!q1k(k-1)q2l(l-1)×Eq1(-[m]q1q1kx)Eq2(-[n]q2q2ly)1[m]q1q1k-1+2q1[2]q1[m]q1x+q12[3]q1[m]q12.
Replacing k by k+2 and k by k+1 in the abovementioned, we obtain
(29)Sm,nq1,q2(t2;x)=x2∑l=0∞‍∑k=0∞‍[m]q1kxk[k]q1![n]q2lyl[l]q2!q1(k+2)(k+1)q2l(l-1)×Eq1(-[m]q1q1k+2x)Eq2(-[n]q2q2ly)1q12k+2+x∑l=0∞‍∑k=0∞‍[m]q1kxk[k]q1![n]q2lyl[l]q2!q1k2q2l(l-1)×Eq1(-[m]q1q1k+1x)Eq2(-[n]q2q2ly)1[m]q1+2q1[2]q1[m]q1x+q12[3]q1[m]q12.
Equation (20) implies
(30)Sm,nq1,q2(t2;x)=x2+1[m]q1(1+2q1[2]q1)x+q12[3]q1[m]q12.
Similarly, we write that
(31)Sm,nq1,q2(s2;y)=y2+1[n]q2(1+2q2[2]q2)y+q22[3]q2[n]q22.
So, the proof is completed.Similarly, given by the proof of Lemma1, we calculate Sm,nq1,q2(t3;x) and Sm,nq1,q2(t4;x), shortly. Since
(32)∫[l]q2/q2k-1[n]q2[l+1]q2/q2k-1[n]q2‍∫[k]q1/q1k-1[m]q1[k+1]q1/q1k-1[m]q1‍t3dq1Rtdq2Rs=q1q2[m]q1[n]q2([k]q13q13k-3[m]q13+3[k]q12q12k-2[m]q12q1[2]q1[m]q1+3[k]q1q1k-1[m]q1q12[3]q1[m]q12+q13[4]q1[m]q13),
we write
(33)Sm,nq1,q2(t3;x)=∑l=0∞‍∑k=1∞‍[m]q1k-1xk[k-1]q1![n]q2lyl[l]q2!q1k(k-1)q2l(l-1)×Eq1(-[m]q1q1kx)Eq2(-[n]q2q2ly)×[k-1]q12+2qk-1[k-1]+q2k-2q13k-3[m]q12+3q1[2]q1[m]q1(x2+1[m]q1x)+3q12[3]q1[m]q12x+q13[4]q1[m]q13.
Using [k]q=[k-1]q+qk-1, we have(34)∑l=0∞‍∑k=1∞‍[m]q1k-1xk[k-1]q1![n]q2lyl[l]q2!q1k(k-1)q2l(l-1)×Eq1(-[m]q1q1kx)Eq2(-[n]q2q2ly)×[k-1]q12+2qk-1[k-1]+q2k-2q13k-3[m]q12=∑l=0∞‍∑k=3∞‍[m]q1k-3xk[k-3]q1![n]q2lyl[l]q2!q1k(k-1)q2l(l-1)×Eq1(-[m]q1q1kx)Eq2(-[n]q2q2ly)1q13k-3+∑l=0∞‍∑k=2∞‍[m]q1k-2xk[k-2]q1![n]q2lyl[l]q2!q1k(k-1)q2l(l-1)×Eq1(-[m]q1q1kx)Eq2(-[n]q2q2ly)q1k-2q13k-3[m]q1+∑l=0∞‍∑k=2∞‍[m]q1k-2xk[k-2]q1![n]q2lyl[l]q2!q1k(k-1)q2l(l-1)×Eq1(-[m]q1q1kx)Eq2(-[n]q2q2ly)2qk-1q13k-3[m]q1+∑l=0∞‍∑k=1∞‍[m]q1k-1xk[k-1]q1![n]q2lyl[l]q2!q1k(k-1)q2l(l-1)×Eq1(-[m]q1q1kx)Eq2(-[n]q2q2ly)q2k-2q13k-3[m]q12.
Then, we rewrite
(35)=x3∑l=0∞‍∑k=0∞‍[m]q1kxk[k]q1![n]q2lyl[l]q2!q1(k+3)(k+2)q2l(l-1)×Eq1(-[m]q1q1k+3x)Eq2(-[n]q2q2ly)1q13k+6+x2∑l=0∞‍∑k=0∞‍[m]q1kxk[k]q1![n]q2lyl[l]q2!q1(k+2)(k+1)q2l(l-1)×Eq1(-[m]q1q1k+2x)Eq2(-[n]q2q2ly)q1kq13k+3[m]q1+x2∑l=0∞‍∑k=0∞‍[m]q1kxk[k]q1![n]q2lyl[l]q2!q1(k+2)(k+1)q2l(l-1)×Eq1(-[m]q1q1k+2x)Eq2(-[n]q2q2ly)2qk+1q13k+3[m]q1+x∑l=0∞‍∑k=0∞‍[m]q1kxk[k]q1![n]q2lyl[l]q2!q1(k+1)(k)q2l(l-1)×Eq1(-[m]q1q1k+1x)Eq2(-[n]q2q2ly)q2kq13k[m]q12.
Finally, we have
(36)Sm,nq1,q2(t3;x)=x3+1[m]q1(1q1+2)x2+1[m]q12x+3q1[2]q1[m]q1(x2+1[m]q1x)+3q12[3]q1[m]q12x+q13[4]q1[m]q13.
Since
(37)∫[l]q2/q2k-1[n]q2[l+1]q2/q2k-1[n]q2‍∫[k]q1/q1k-1[m]q1[k+1]q1/q1k-1[m]q1‍t4dq1Rtdq2Rs=q1q2[m]q1[n]q2([k]q14q14k-4[m]q14+4[k]q13q13k-3[m]q13q1[2]q1[m]q1+6[k]q12q12k-2[m]q12q12[3]q1[m]q12+4[k]q1q1k-1[m]q1q13[4]q1[m]q13+q14[5]q1[m]q14),
we obtain(38)Sm,nq1,q2(t4;x)=x4+1[m]q1(1q12+2q1+3)x3+1[m]q12(1q12+3q1+3)x2+1[m]q13x+4q1[2]q1[m]q1(x3+1[m]q1(1q1+2)x2+1[m]q12x)+6q12[3]q1[m]q12(x2+1[m]q1x)+4q13[4]q1[m]q13x+q14[5]q1[m]q14.


## 3. Approximation Properties in Polynomial Weighted Spaces

For bivariate operators, the space is considered as follows:(39)Cp,q(ℝ+2)={f∈C(ℝ+2):ωp,qfisuniformlycontinuousandboundedonℝ+2=[0,∞)×[0,∞)}
associated with the weighted function ωp,q(x,y)=ωp(x)ωq(y), (p,q)∈ℕ0×ℕ0. The weight ωp is defined as ω0(t)=1, ωp(x)=(1+xp)-1. The norm of this space is denoted by ∥·∥p,q and is defined by
(40)∥f∥p,q=sup(x,y)∈ℝ+2ωp,q(x,y)|f(x,y)|.
Now, we give some useful results given by Agratini [15].For eachz∈ℝ+, define the function φz by φzr(t)=(t-z)r, t∈ℝ+, r∈ℕ. For the one-dimensional operator Ls, s∈ℕ, and for each r∈ℕ, a polynomial Γr exists such that
(41)sr/2(Lsφzr)(t)≤Γr(t),deg(Γr)≤r.Theorem 2 (see [15]).
Consider(p,q)∈ℕ0×ℕ0. For any (m,n)∈ℕ×ℕ, given by
(42)(Lm,nf)(x,y)=∑i=0∞‍∑j=0∞‍am,i(x)bn,j(y)f(xm,i,yn,j),(x,y)∈ℝ+2
the operator Lm,n verifies
(43)∥Lm,n(1ωp,q;·)∥p,q≤c(p,q),∥Lm,n(f;·)∥p,q≤c(p,q)∥f∥p,q,f∈Cp,q(ℝ+2).Theorem 3 (see [15]).
Let(p,q)∈ℕ0×ℕ0. For any (m,n)∈ℕ×ℕ, the operator Lm,n given by (42) satisfies(44)ωp,q(x,y)|(Lm,nf)(x,y)-f(x,y)|≤c(p,q)ωf(ϕ(x)m,ϕ(y)n),(x,y)∈ℝ+2, where ϕ is given by
(45)ϕ(t)=Γ2(t)+Γ2(t)+∑k=0p‍(pk)Γk+2(t),
with the polynomials Γν(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 Lm,n,(m,n)∈ℕ×ℕ, be defined by (42). For any (x,y)∈ℝ+2 the pointwise convergence takes place
(46)limm,n→∞(Lm,nf)(x,y)=f(x,y),f∈Cp,q(ℝ+2).
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 functionρ(x,y)=(1+x2+y2)-1 instead of ωp,q(x,y) and instead of the space Cp,q(ℝ+2), the space C2(ℝ+2) associated with the weighted function ρ(x,y) is used. We denote the norm of this space by ∥·∥2.Lemma 5.
The operatorSm,nq1,q2, (m,n)∈ℕ×ℕ, q1,q2∈(0,1), given by (10) verifies
(47)∥Sm,nq1,q2(1ρ(t,s);·)∥2≤103,(48)∥Sm,nq1,q2(f;·)∥2≤103∥f∥2,f∈C2(ℝ+2).Proof.
Sinceq1,q2∈(0,1) and by Lemma 1, we have
(49)ρ(x,y)Sm,nq1,q2(1ρ(t,s);·)≤ρ(x,y)(1+x2+y2+73(x+y)).
So, inequality (47) is proved. Since the operator Sm,nq1,q2 is linear and positive and by using (47)
(50)ρ(x,y)Sm,nq1,q2(f;·)≤ρ(x,y)Sm,nq1,q2(ρ(t,s)|f|1ρ(t,s);·)≤103∥f∥2,
we obtain inequality (48).The Steklov function associated withf∈C(ℝ+2) is given as follows:
(51)fh,δ(x,y)=1hδ∫0h‍du∫0δ‍f(x+u,y+v)dv,(x,y)∈ℝ+2,
where h,δ>0.The modulus of smoothness function associated with any functionf∈C2(ℝ+2) is given by
(52)ωf(h,δ)=sup0≤u≤h0≤v≤δ∥f(x+u,y+v)-f(x,y)∥2,(x,y)∈ℝ+2.
One can see that
(53)∥f-fh,δ∥2=sup(x,y)∈ℝ+2ρ(x,y)|1hδ∫0h‍du∫0δ‍f(x+u,y+v)1hδ∫0h‍-f(x,y)dv|≤sup(x,y)∈ℝ+2ρ(x,y)sup0≤u≤h0≤v≤δf(x+u,y+v)-f(x,y)=sup0≤u≤h0≤v≤δ∥f(x+u,y+v)-f(x,y)∥2=ωf(h,δ).
The following inequalities verify
(54)∥∂∂xfh,δ∥2≤2hωf(h,δ),∥∂∂yfh,δ∥2≤2δωf(h,δ),
where h,δ>0. In order to justify these inequalities, one can see that
(55)∥∂∂xfh,δ∥2=sup(x,y)∈ℝ+2ρ(x,y)1hδ|∫0δ‍f(x+h,y+v)∫0δ‍-f(x,y+v)dv|=sup(x,y)∈ℝ+2ρ(x,y)1hδ|∫0δ‍f(x+h,y+v)-f(x,y)∫0δ‍+f(x,y)-f(x,y+v)dv|≤1hδsup(x,y)∈ℝ+2ρ(x,y)∫0δ‍|f(x+h,y+v)-f(x,y)|dv+1hδsup(x,y)∈ℝ+2ρ(x,y)∫0δ‍|f(x,y+v)-f(x,y)|dv≤2hsup(x,y)∈ℝ+2ρ(x,y)sup0≤u≤h0≤v≤δf(x+u,y+v)-f(x,y)=2hωf(h,δ).Theorem 6.
For any(m,n)∈ℕ×ℕ and f∈C2(ℝ+2), the operator Sm,nq1,q2q1,q2∈(0,1) given by (10) satisfies
(56)ρ(x,y)|(Sm,nq1,q2f)(x,y)-f(x,y)|≤103ωf(ϕ(x,q1)[m]q1,ϕ(y,q2)[n]q2),
where (x,y)∈ℝ+2, ϕ(x,q)=q/[2]q+Γ(x,q).Proof.
For any(m,n)∈ℕ×ℕ, we can write
(57)ρ(x,y)|(Sm,nq1,q2f)(x,y)-f(x,y)|≤ρ(x,y)|Sm,nq1,q2(f-fh,δ;x,y)|+ρ(x,y)|(Sm,nq1,q2fh,δ)(x,y)-fh,δ(x,y)|+ρ(x,y)|fh,δ(x,y)-f(x,y)|.
Inequalities (48) and (53) imply that
(58)ρ(x,y)|Sm,nq1,q2(f-fh,δ;x,y)|≤∥Sm,nq1,q2(f-fh,δ;x,y)∥2≤103∥f-fh,δ∥2≤103ωf(h,δ).
Let C21(ℝ+2) be the class of all functions in which partial derivatives belong to C2(ℝ+2). Since fh,δ∈C21ℝ+2 and Sm,nq1,q2 given by (10) is linear and monotone, we get
(59)ρ(x,y)|(Sm,nq1,q2fh,δ)(x,y)-fh,δ(x,y)|=ρ(x,y)|Sm,nq1,q2(∫xt‍∂∂ufh,δ(u,s)du;x,y)|+ρ(x,y)|Sm,nq1,q2(∫ys‍∂∂vfh,δ(x,s)dv;x,y)|≤ρ(x,y)Sm,nq1,q2(|∫xt‍∂∂ufh,δ(u,s)du|;x,y)+ρ(x,y)Sm,nq1,q2(|∫ys‍∂∂vfh,δ(x,s)dv|;x,y).
Then, by definition of norm ∥·∥2 and the first mean value theorem for integration, we have
(60)|∫xt‍∂∂ufh,δ(u,s)du|≤|∫xt‍ρ(u,s)|∂∂ufh,δ(u,s)|duρ(u,s)|≤∥∂∂xfh,δ∥2|∫xt‍duρ(u,s)|=∥∂∂xfh,δ∥2|t-x|1ρ(ξ,s),ξ∈(x,t)≤∥∂∂xfh,δ∥2|t-x|(1ρ(x,s)+1ρ(t,s)).
Following the same way, one finds
(61)|∫ys‍∂∂vfh,δ(x,s)dv|≤∥∂∂yfh,δ∥2|s-y|(1ρ(x,y)+1ρ(x,s)).
Using inequalities (59), (61), and (60), since Sm,nq1,q2 is linear and monotone, we get
(62)ρ(x,y)|(Sm,nq1,q2fh,δ)(x,y)-fh,δ(x,y)|≤ρ(x,y)∥∂∂xfh,δ∥2Sm,nq1,q2(|t-x|ρ(x,s);x,y)+ρ(x,y)∥∂∂xfh,δ∥2Sm,nq1,q2(|t-x|ρ(t,s);x,y)+ρ(x,y)∥∂∂yfh,δ∥2Sm,nq1,q2(|s-y|ρ(x,y);x,y)+ρ(x,y)∥∂∂yfh,δ∥2Sm,nq1,q2(|s-y|ρ(x,s);x,y).
From Lemmas 1 and 5 and (36), we can write the following:
(63)ρ(x,y)|(Sm,nq1,q2fh,δ)(x,y)-fh,δ(x,y)|≤∥∂∂xfh,δ∥2103q1[2]q1[m]q1+∥∂∂xfh,δ∥2103Γ(x,q1)[m]q1+∥∂∂yfh,δ∥2q2[2]q2[n]q2+∥∂∂yfh,δ∥2103Γ(y,q2)[n]q2,
where
(64)Γ(x,q)=(1q+1+q[2]q)x2+(1+3q[2]q+2q2[3]q)x+q3[4]q
is a polynomial of degree 2. Then, by inequalities (54), we have
(65)ρ(x,y)|(Sm,nq1,q2fh,δ)(x,y)-fh,δ(x,y)|≤[2h103(q1[2]q1[m]q1+Γ(x,q1)[m]q1)+2δ(q2[2]q2[n]q2+103Γ(y,q2)[n]q2)]ωf(h,δ).
Finally, we write from (53)
(66)ρ(x,y)|fh,δ(x,y)-f(x,y)|≤∥fh,δ-f∥2≤ωf(h,δ).
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 replaceq1 and q2 in (10) by sequences (q1,m), (q2,n) so that
(67)limm→∞q1,m=1,limn→∞q2,n=1,limm→∞1[n]q1,m=0,limn→∞1[n]q2,n=0.
So, (h,δ)→(0+,0+) as n,m→∞. Knowing that modulus of smoothness function ωf satisfies the property lim(h,δ)→(0+,0+)ωf(h,δ)=0, from Theorem 6, we deduce the following result.Theorem 7.
Let(m,n)∈ℕ×ℕ and let (q1,m), (q2,n) be sequences in the interval (0,1) satisfying (67). Let the operator Sm,nq1,q2 given by (10) and f∈C2(ℝ+2)qR-integrable function. For any (x,y)∈ℝ+2 the pointwise convergence takes place
(68)limm,n→∞(Sm,nq1,m,q2,nf)(x,y)=f(x,y).
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(q1,n), (q2,n) be sequences in the interval (0,1) satisfying (67). Suppose that f∈C22(ℝ+2) is the class of all functions in which the second partial derivatives belong to C2(ℝ+2) and qR-integrable function. Then, for every (x,y)∈ℝ+2, one has
(69)limn→∞n{Sn,nq1,n,q2,n(f;x,y)-f(x,y)}=x2fxx(x,y)+y2fyy(x,y)+12fx(x,y)+12fy(x,y).Proof.
Letf∈C22(ℝ+2) and qR-integrable function and let (x0,y0)∈ℝ+2 be fixed point. Then, by the Taylor formula, we can write
(70)f(t,s)=f(x0,y0)+fx(x0,y0)(t-x0)+fy(x0,y0)(s-y0)+12{fxx(x0,y0)(t-x0)2+2fxy(x0,y0)×(t-x0)(s-y0)+fyy(x0,y0)(s-y0)2}+φ(t,s;x0,y0)(t-x0)4+(s-y0)4,
where (t,s)∈ℝ+2, φ(t,s)=φ(t,s;x0,y0) belongs to C2(ℝ+2), and lim(t,s)→(x0,y0)φ(t,s)=0 for n∈ℕ. From the linearity of Sn,nq1,q2, we have
(71)Sn,nq1,q2(f(t,s);x0,y0)=f(x0,y0)+fx(x0,y0)Sn,nq1,q2(t-x0;x0)+fy(x0,y0)Sn,nq1,q2(s-y0;y0)+12{fxx(x0,y0)Sn,nq1,q2((t-x0)2;x0)+2fxy(x0,y0)Sn,nq1,q2(t-x0;x0)Sn,nq1,q2(s-y0;y0)+fyy(x0,y0)Sn,nq1,q2((s-y0)2;y0)}+Sn,nq1,q2(φ(t,s)(t-x0)4+(s-y0)4;x0,y0).
By Lemma 1 and since the sequences q1,n, q2,n satisfy (67), we obtain
(72)limn→∞nSn,nq1,n,q2,n(t-x0;x0)=12=limn→∞nSn,nq1,n,q2,n(s-y0;y0),limn→∞nSn,nq1,n,q2,n((t-x0)2;x0)=x0,limn→∞nSn,nq1,n,q2,n((s-y0)2;y0)=y0.
By the Hölder inequality, we have
(73)|Sn,nq1,q2(φ(t,s)(t-x0)4+(s-y0)4;x0,y0)|≤{Sn,nq1,q2(φ2(t,s);x0,y0)}1/2×{Sn,nq1,q2((t-x0)4;x0)+Sn,nq1,q2((s-y0)4;y0)}1/2.
By properties of φ and Theorem 7, we get
(74)limn→∞Sn,nq1,n,q2,n(φ2(t,s);x0,y0)=φ2(x0,y0)=0.
From the foregoing facts and using (12) and (38), we obtain
(75)limn→∞nSn,nq1,n,q2,n(φ(t,s)(t-x0)4+(s-y0)4;x0,y0)=0.
Then, using (72) and (75), we reproduce from (71)
(76)limn→∞n{Sn,nq1,n,q2,n(f(t,s);x0,y0)-f(x0,y0)}=x02fxx(x0,y0)+y02fyy(x0,y0)+12fx(x,y)+12fy(x,y).
Thus, the proof is completed for f∈C22(ℝ+2).



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*Source: 823803-2013-12-03.xml*