AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 850760 10.1155/2013/850760 850760 Research Article Bivariate Positive Operators in Polynomial Weighted Spaces Agratini Octavian Kim Sung Guen Faculty of Mathematics and Computer Science Babeş-Bolyai University Street Kogălniceanu 1 400084 Cluj-Napoca Romania ubbcluj.ro 2013 16 4 2013 2013 25 02 2013 26 03 2013 2013 Copyright © 2013 Octavian Agratini. 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.

This paper aims to two-dimensional extension of some univariate positive approximation processes expressed by series. To be easier to use, we also modify this extension into finite sums. With respect to these two new classes designed, we investigate their approximation properties in polynomial weighted spaces. The rate of convergence is established, and special cases of our construction are highlighted.

1. Introduction

The approximation of functions by using linear positive operators is currently under research. Usually, two types of positive approximation processes are used: the discrete, respectively, continuous form. In the first case, they often are designed through a series. Since the construction of such operators requires an estimation of infinite sums, this restricts the operator usefulness from the computational point of view. In this respect, in order to approximate a function, it is interesting to consider partial sums, the number of terms considered in sum depending on the function argument. Roughly speaking, these discrete operators are truncated fading away their “tails.” Thus, they become usable for generating software approximation of functions. Among the pioneers who approached this direction we mention Gróf  and Lehnhoff . In the same direction a class of univariate linear positive operators is investigated in .

This work focuses on a general bivariate class of discrete positive linear operators expressed by infinite sums. This class acts in polynomial weighted spaces of continuous functions of two variables defined by +×+, where +=[0,). By using a certain modulus of smoothness we give theorems on the degree of approximation. Further, we replace the infinite sum by a truncated one, and we study the approximation properties of the new defined family of operators. Compared to what has been done so far, the strengths of this paper consist in using general classes of two-dimensional discrete operators, implying an arbitrary network of nodes. Finally we present some particular classes of operators that can be obtained from our family.

2. The Operators

Our linear and positive operators have the role to approximate functions defined on +×+. Therefore, on this domain we define for every (m,n)× a net of form Δm,n=Δ1,m×Δ2,n, where Δ1,m(0=xm,0<xm,1<) and Δ2,n(0=yn,0<yn,1<). Set 0={0}, and C(+) stands for the space of all real-valued continuous functions on +.

Products of parametric extensions of two univariate operators are appropriate tools to approximate functions of two variables. For this reason, the starting point is given by the following one-dimensional operators: (1)(Amf)(x)=i=0am,i(x)f(xm,i),(Bnf)(y)=j=0bn,j(y)f(yn,j), where am,i, bn,j are nonnegative functions belonging to C(+), (i,j)0×0, such that the following identities (2)i=0am,i(t)=j=0bn,j(t)=1,t0, take place. These conditions mean that the operators Am and Bn preserve the monomial e0, e0(t)=1, a property often seen at classical linear positive operators.

For each z+, define the function φz by φz(t)=t-z, t+. Also, for each r, set (3)r(Am;x)=(Amφxr)(x),r(Bn;y)=(Bnφyr)(y), representing the rth central moment of the specified operators.

For a simplified writing, we will use the common notation Ls, where Ls=As for all s or Ls=Bs for all s.

For our purposes, relative to the central moments, we require additional conditions. For each r, r(Ls;t) as a function of t is bounded by a polynomial of degree at most r. Moreover, sr/2r(Ls;·) is bounded with respect to s. These requirements can be brought together and redrafted in the following way: for each r, a polynomial Γr exists such that (4)sr/2r(Ls;t)Γr(t),t0,deg(Γr)r.

Apparently is a tough condition, but the examples that we give in the last section show that it is carried out by different classes of operators.

In what follows we specify the function spaces in which the operators act.

For univariate operators As, Bs we consider the space Cp(+), p0 fixed, consisting of all real-valued functions f continuous on + such that wpf is uniformly continuous and bounded on +, where the weight wp is defined as follows: (5)w0(t)=1,wp(t)=(1+tp)-1forp.

The space is endowed with the norm ·p, fp=supx0wp(x)|f(x)|.

For bivariate operators we consider the space (6)Cp,q(+2)={fC(+2):wp,qfis  uniformly  continuouscand  boundedon  +2} associated to the weighted function wp,q(x,y)=wp(x)wq(y), (p,q)0×0. The norm of this space is denoted by ·p,q and is defined by (7)fp,q=sup(x,y)+2wp,q(x,y)|f(x,y)|.

These spaces are ordered with respect to inclusion as follows: if (p,q)(p,q), then Cp,q(+2)Cp,q(+2). Moreover, fp,qfp,q for any fCp,q(+2). Examining the weight wp,q, it is easy to verify the inequality (8)|xtduwp,q(u,z)||t-x|(1wp,q(x,z)+1wp,q(t,z)), for any x0 and t0.

Indeed, based on the first mean value theorem for integration, between x and t, a point ξx,t exists such that (9)|xtduwp,q(u,z)|=|t-x|(1+ξx,tp)(1+zq)|t-x|(1+xp+1+tp)(1+zq)=|t-x|(1wp,q(x,z)+1wp,q(t,z)).

Starting from (1), for each (m,n)× we introduce a linear positive operator in polynomial weighted space Cp,q(+2) as follows: (10)(Lm,nf)(x,y)=i=0j=0am,i(x)bn,j(y)f(xm,i,yn,j),(x,y)+2.

If the function fCp,q(+2) can be decomposed in the following manner f(x,y)=f1(x)f2(y), (x,y)+2, then one has (11)(Lm,nf)(x,y)=(Amf1)(x)(Bnf2)(y).

For example, we get Lm,nwp,q=(Amwp)(Bnwq).

In order to present the rate of convergence for our bivariate operators, we use a modulus of smoothness associated to any function f belonging to Cp,q(+2). It is given by the formula (12)ωf(h,δ)=sup0uh0vδΔu,vfp,q,(h,δ)+2, where (13)Δu,vf(x,y)=f(x+u,y+v)-f(x,y) for (x,y) and (u,v) belonging to Cp,q(+2). Alternative notation is ω(f;x,y). More information about moduli of smoothness can be found in the monograph .

Further, we indicate a truncated variant of operators defined at (10). Let (us)s1, (vs)s1 be strictly increasing sequences of positive numbers such that (14)limssus=limssvs=.

Taking in view the net Δ1,m, we partitioned the set 0 into two parts (15)I(x,um)={i0:xm,ix+um},I¯(x,um)=0I(x,um).

Similarly, via the network Δ2,n, we introduce J(y,vn) and J¯(y,vn).

For each (m,n)× and any fCp,q(+2) we define the linear positive operators (16)(Lm,num,vnf)(x,y)=iI(x,um)jJ(y,vn)am,i(x)bn,j(y)f(xm,i,yn,j),(x,y)+2.

3. Auxiliary Results

Throughout the paper, by c(·) we denote different real constants, in the brackets specifying the parameter(s) that the indicated constant depends.

At first we collect some useful results relative to the one-dimensional operators Ls where Ls=As(s) or Ls=Bs(s).

Lemma 1.

Let p0, and let the weight wp be given by (5). The operator Ls satisfies (17)(i)wp(t)Ls(1wp;t)c(p),(18)(ii)wp(t)|(Lsf)(t)|c(p)fp,Lsfpc(p)fp, where t0 and s.

Proof.

Let p0 and s be fixed.

(i) By using (5), (2), (3), and (4) we can write (19)wp(t)Ls(1wp;t)=wp(t)Ls(1+((x-t)+t)p;t)=wp(t)(1+tp+k=1p(pk)Ls((x-t)ktp-k;t))=1+k=1p(pk)tp-kk(Ls;t)1+tp1+k=1p(pk)tp-kΓk(t)1+tp.

Since deg(tp-kΓk(t))p, the previous expression is bounded with respect to t+, and inequality (17) follows.

(ii) To be more explicit we consider Ls=As(20)wp(t)|(Asf)(t)|wp(t)k=0as,k(t)(wp|f|)(xs,k)1wp(xs,k)fpwp(t)As(1wp;t).

By using (17) we obtain the first inequality of the relation (18). Further, applying supt0, the second inequality is proved.

Lemma 2.

Let p0, and let the weight wp be given by (5). For any s the operator Ls satisfies (21)(i)wp(t)Ls(φt2wp;t)Γ~(t)s,t0,(22)(ii)wp(t)Ls(|φt|wp;t)c(p)Γ~(t)s,t0, where φt(x)=x-t and Γ~=Γ2+k=0p(pk)Γk+2. The polynomials Γν, ν2, are introduced by (4).

Proof.

Let p0 and s be fixed.

Using the identity (23)1+tp=1+k=0p(pk)(x-t)ktp-k, we can write the following: (24)wp(t)Ls(φt2wp;t)=11+tpLs(φt2;t)+k=0ptp-k1+tp(pk)Ls(φtk+2;t)2(Ls;t)+k=0p(pk)k+2(Ls;t)Γ2(t)s+k=0p(pk)Γk+2(t)s1+k/2.

During the previous relations we used notation (3) and hypothesis (4). Considering the significance of Γ~, relation (21) follows.

Taking into consideration relation (1), we apply the Cauchy-Schwarz inequality, and this allows us to write (25)wp(t)Ls(|φt|wp;t)(wp(t)Ls(1wp;t))1/2(wp(t)Ls(φt2wp;t))1/2. Relations (17) and (21) imply (22).

We mention that with the help of the same Cauchy-Schwarz inequality, relations (2) and (4) lead us to the following inequality: (26)Ls(|φt|r;t)Γ2r(t)sr,t0.

Lemma 1 leads to the following result.

Lemma 3.

Let (p,q)0×0. For any (m,n)×, the operator Lm,n given by (10) verifies (27)(i)Lm,n(1wp,q;·)p,qc(p,q),(28)(ii)Lm,nfp,qc(p,q)fp,q,fCp,q(+2).

Proof.

The first statement follows immediately from the definition of the weight wp,q and relations (11), (17).

Regarding the second statement, based on (10), we get (29)wp,q(x,y)|(Lm,nf)(x,y)|wp,q(x,y)fp,qLm,n(1wp,q;x,y).

Applying sup(x,y)+2 and taking into account (27), one obtains (28).

In our investigation we appeal to the Steklov function. This can be used to approximate continuous functions by smoother functions. The Steklov function associated with fC(+2) is given as follows: (30)fh,δ(x,y)=1hδ0hdu0δf(x+u,y+v)dv,(x,y)+2, where h>0 and δ>0. By using (13) we deduce (31)fh,δ(x,y)-f(x,y)=1hδ0hdu0δΔu,vf(x,y)dv.

In the next lemma we have gathered some known properties of Steklov function fh,δ, where fCp,q(+2). These properties establish connections between fh,δ and the modulus ωf indicated at (12). For the sake of completeness we present the proofs of these inequalities.

Lemma 4.

Let f belong to Cp,q(+2), and let fh,δ be defined by (30). The following relations take place: (32)(i)fh,δ-fp,qωf(h,δ),(33)(ii)xfh,δp,q2hωf(h,δ),yfh,δ2δωf(h,δ), where h>0,δ>0, and ωf is defined by (12).

Proof.

Let h>0 and δ>0 be arbitrarily fixed.

(i) For u[0,h] and v[0,δ] we deduce (34)|Δu,vf(x,y)|sup0τ1h0τ2δ|Δτ1,τ2f(x,y)|. On the other hand, (35)sup(x,y)+2wp,q(x,y)|Δτ1,τ2f(x,y)|=Δτ1,τ2fp,q. Keeping in mind (31), we consequently obtain (36)fh,δ-fp,q1hδsup0τ1h0τ2δΔτ1,τ2fp,q0hdu0δdv=ωf(h,δ) and (32) is completed.

(ii) We justify only the first inequality, and the second inequality can be proven in the same manner.

Occurs (37)x(0hf(x+u,y+v)du)=f(x+h,y+v)-f(x,y+v)=Δh,vf(x,y)-Δ0,vf(x,y) (see (13)). Further, we can write (38)xfh,δp,q=1hδsupx0y0wp,q(x,y)|0δ(Δh,vf(x,y)-Δ0,vf(x,y))dv|1hδsupx0y0wp,q(x,y)0δ|Δh,vf(x,y)|dv+1hδsupx0y0wp,q(x,y)0δ|Δ0,vf(x,y)|dvI1+I2.

Since 0vδ, it is clear that |Δh,vf(x,y)|sup0τ1h0τ2δ|Δτ1,τ2f(x,y)|, and we obtain (39)I11hδsup0τ1h0τ2δ0δsupx0y0wp,q(x,y)|Δτ1,τ2f(x,y)|dv1hsup0τ1h0τ2δΔτ1,τ2p,q=1hωf(h,δ).

In the same manner we show I2(1/h)ωf(h,δ). Returning at (38), the proof is ended.

In the following we denote by Cp,q1(+2) the space of all functions g:+2 having the first order partial derivatives such that the functions g/x,g/y, and g belong to Cp,q(+2).

Lemma 5.

Let (p,q)0×0. If gCp,q1(+2), then for any (m,n)× the operator Lm,n given by (10) verifies (40)wp,q(x,y)|(Lm,ng)(x,y)-g(x,y)|c(p,q)(gxp,qΦ(x)m+gyp,qΦ(y)n),(gxp,qΦ(x)m+gyp,q)(x,y)+2, where (41)Φ(t)=Γ2(t)+Γ2(t)+k=0p(pk)Γk+2(t), the polynomials Γν, ν=2,p+2¯, being indicated at (4).

Proof.

Let (x,y)+2 and (m,n)2 be arbitrarily fixed. Since gCp,q1(+2), for any (t,z)+2 we can write (42)g(t,z)-g(x,y)=xtug(u,z)du+yzvg(x,v)dv.

Since Lm,n is linear monotone and reproduces the constants, from the previous identity we obtain (43)|(Lm,ng)(x,y)-g(x,y)|=|(yzvg(x,v)dv;x,y)Lm,n(xtug(u,z)du;x,y)+Lm,n(yzvg(x,v)dv;x,y)|Lm,n(|xtug(u,z)du|;x,y)+Lm,n(|yzvg(x,v)dv|;x,y)J1+J2.

Further, one has (44)|xtug(u,z)du||xtwp,q(u,z)|ug(u,z)|duwp,q(u,z)|gxp,q|xtduwp,q(u,z)|gxp,q(1wp,q(x,z)+1wp,q(t,z))|t-x|, see (8). Applying Lm,n and by using successively (11), (26), (17), and (22) we have (45)J1gxp,q(Lm,n(|t-x|wp,q(x,z);x,y)gxp,qcc+Lm,n(|t-x|wp,q(t,z);x,y))=gxp,q(1wp(x)Am(|φx|;x)Bn(1wq;y)gxp,qcc+Am(|φx|wp;x)Bn(1wq;y))gxp,q(1wp(x)Γ2(x)m+c(p)wp(x)Γ~(x)m)c(q)wq(y)c(p,q)wp,q(x,y)gxp,qΓ2(x)+Γ~(x)m, where c(p,q) is a suitable constant. Following the same pathway, we find (46)J1c(p,q)wp,q(x,y)gyp,qΓ2(y)+Γ~(y)n.

Considering the increases established for J1,J2 and returning to the relation (43), the inequality (40) is completely proven.

4. Main Results

The rate of convergence for Lm,n operator will be read as follows.

Theorem 6.

Let (p,q)0×0. For any (m,n)×, the operator Lm,n given by (10) satisfies (47)wp,q(x,y)|(Lm,nf)(x,y)-f(x,y)|c(p,q)ωf(Φ(x)m,Φ(y)n),(x,y)+2, where Φ is given by (41) and c(p,q) is a suitable constant.

Proof.

Setting (48)T1wp,q(x,y)|Lm,n(f-fn,δ;x,y)|,T2wp,q(x,y)|(Lm,nfh,δ)(x,y)-fh,δ(x,y)|,T3wp,q(x,y)|fh,δ(x,y)-f(x,y)|, we can write (49)wp,q(x,y)|(Lm,nf)(x,y)-f(x,y)|T1+T2+T3.

We establish upper bounds for these three quantities. Relations (28) and (32) imply that (50)T1Lm,n(f-fh,δ;x,y)p,qc(p,q)f-fh,δp,qc(p,q)ωf(h,δ).

For T2 we use Lemma 5 choosing g=fh,δCp,q1(+2); see definition (31).

One has (51)T2c(p,q)(fh,δxp,qΦ(x)m+fh,δyp,qΦ(y)n)c(p,q)(2Φ(x)hm+2Φ(y)δn)ωf(h,δ) (see also (33)). Finally, inequality (32) implies (52)T3fh,δ-fp,qωf(h,δ).

Setting h=Φ(x)/m,δ=Φ(y)/n and coming back to (49), we can affirm that a certain constant c(p,q) exists such that (47) holds.

Knowing that the modulus ωf enjoys the property lim(h,δ)(0+,0+)ωf(h,δ)=0, from Theorem 6 we deduce the following result.

Theorem 7.

Let (p,q)0×0, and let the operators Lm,n, (m,n)×, be defined by (10).

For any (x,y)+2 the pointwise convergence takes place (53)limm,n(Lm,nf)(x,y)=f(x,y),fCp,q(+2).

If K1,K2 are compact intervals included in +, then (53) holds uniformly on the domain K1×K2.

Theorem 8.

Let (p,q)0×0, and let the operators Lm,num,vn, (m,n)×, be defined by (16). For any (x,y)+2 the pointwise convergence takes place (54)limm,n(Lm,num,vnf)(x,y)=f(x,y),fCp,q(+2).

If K1,K2 are compact intervals included in +, then (54) holds uniformly on the domain K1×K2.

Proof.

Let (x,y)+2 be arbitrarily fixed. Taking in view the partitions of 0 (see (15)), we use the following decomposition: (55)0×0=(I×J)(I¯×J¯)(I¯×J)(I×J¯).

Setting (56)(1Rm,num,vnf)(x,y)=iI¯(x,um)jJ¯(y,vn)am,i(x)bn,j(y)f(xm,i,yn,j),(2Rm,num,vnf)(x,y)=iI¯(x,um)jJ(y,vn)am,i(x)bn,j(y)f(xm,i,yn,j),(3Rm,num,vnf)(x,y)=iI(x,um)jJ¯(y,vn)am,i(x)bn,j(y)f(xm,i,yn,j), we can write (57)|(Lm,num,vnf)(x,y)-f(x,y)||(Lm,nf)(x,y)-f(x)|+k=13|(kRm,num,vnf)(x,y)|.

If we show limm,n(kRm,num,vnf)(x,y)=0,  k{1,2,3}, then, based on (53), our statement (54) follows, and the proof is ended.

Further, we prove the previous limit only for k=1, other two following similar routes.

Since fCp,q(+2), a constant Mf exists such that (58)|f(x,y)|Mf(1+xp)(1+yq),(x,y)+2.

Based on the classical inequality (a+b)s2s-1(as+bs), a0,  b0, s0, we deduce (59)ts(|t-z|+z)s2s-1(|t-z|s+zs),t0,z0,s0.

Consequently, (58) implies that (60)|f(xm,i,yn,j)|Mf(1+2p-1(|xm,i-x|p+xp))×(1+2q-1(|yn,j-y|q+yq)), where (i,j)I¯(x,um)×J¯(y,vn).

Using this relation we have (61)|(1Rm,num,vnf)(x,y)|MfiI¯(x,um)am,i(x)(1+2p-1xp+2p-1|xm,i-x|p)×jJ¯(y,vn)bn,j(y)(1+2q-1yq+2q-1|yn,j-y|q)MfS1S2.

We establish an upper bound for S1. Since iI¯(x,um), clearly (62)1<um-p|xm,i-x|p,p1, and we can write (63)S1(1+2p-1xp)1umpiI¯(x,um)am,i(x)|xm,i-x|p+2p-1iI¯(x,um)am,i(x)|xm,i-x|p(1+2p-1xpump+2p-1)2p1/2(Am;x)(1+2p-1xpump+2p-1)Γ2p(x)mp (see (26) and (4)). Arguing similarly we get (64)S2(1+2q-1yqvnq+2q-1)Γ2q(y)nq.

Considering (14), relation (61) leads to the claimed result.

5. Particular Cases

In presenting these cases, we are looking for one-dimensional linear operators that verify conditions (2) and (4).

(1) Baskakov operators  are given by the formula (65)(Vnf)(x)=k=0bn,k(x)f(kn),  bn,k(x)=(n+k-1k)xk(1+x)-n-k,x0, where Vne0=e0, hence (2) fulfilled. Since Vn reproduces the monomial e1, e1(t)=t, its first central moment 1(Vn;·) is null.

For r2, the rth central moment is given as follows [6, Lemma 4]: (66)r(Vn;x)=j=1[r/2]bn,r,j(x(1+x)n)j(1+2xn)δr, where δr=1 if r is odd, δr=0 if r is even, and bn,r,j are positive coefficients bounded with respect to n. In particular, r(Vn;x) is a polynomial of degree r without a constant term. These properties ensure that condition (4) is achieved.

The study of these operators in polynomial weighted spaces was carried out in . Choosing in (10) Am=Vm and Bn=Vn we obtain the Baskakov operator for functions of two variables. The net is Δm,n=(i/m,j/n)i,j0. Our results indicated at (47) and (53) are identified with the results established by Gurdek et al. [7, Equations (22), (28)].

The univariate truncated operators has been approached in . The truncated version specified in (16) coincides with the operators studied by Walczak [9, Equation (17)]. In this case I(x,um) from (15) becomes {i0:i[m(x+um)]}. Here [λ] indicates the largest integer not exceeding λ.

(2) Szász operators  are of the form: (67)(Snf)(x)=k=0pn,k(x)f(kn),pn,k(x)=e-nx(nx)kk!,x0. One has Sne0=e0 and 1(Sn;x)=0. The central moments have the following form: (68)2r(Sn;x)=j=0r-1qjnr+jxr-j,2r+1(Sn;x)=j=0r-1pjnr+j+1xr-j, where qj,pj are constants; see, for example, [11, Equations (9.5.10)-(9.5.11)]. For this class, conditions (2) and (4) are achieved.

The research of Sn, n, operators in polynomial weighted spaces has appeared in . The truncated univariate Szász operators and another extension to functions of two variables in weighted spaces have been considered in  and , respectively. In the latter paper instead wp,q was used the weight ρ, ρ(x,y)=1+x2+y2.

Our theorems of the previous section lead us to two-dimensional versions of genuine Szász operators and of their truncated form. In this case the net is Δm,n=(i/m,j/n)i,j0.

The next example comes from the world of Quantum Calculus which, in the past two decades, has gained popularity in the construction of linear approximation processes. We choose a q-analogue of Szász-Mirakjan operators recently introduced and studied by Mahmudov .

(3) q-Szász-Mirakjan-Mahmudov operators. Let q>1 and n. For f:+ one defines the operator (69)(Mn,qf)(x)=k=0f([k]q[n]q)[n]qkxkqk(k-1)/2[k]q!eq(-[n]qq-kx), where eq(z)=j=0(1+(q-1)(z/qj+1)). We recall the standard notations in q-calculus. For n consider (70)[n]q={1-qn1-q,ifq(0,1)(1,),n,ifq=1;[n]q!=qq[n]q.

Also, =0 and !=1. In  it was proved that Mn,qe0=e0 and Mn,q is a linear positive operator from Cp(+) to Cp(+) for any p0. Withal, for all moments Mn,qer,  er(t)=tr, explicit formulas were given as follows: (71)(Mn,qer;x)=k=1rsq(r,k)xk[n]qr-j, where (72)sq(0,0)=1,sq(r,0)=0forr>0,sq(r,k)=0forr<k,sq(r+1,k)=[k]qsq(r,k)+sq(r,k-1),k-1cforr0,k1, represent q-analogue of Stirling numbers; see [13, Lemma 2.6]. One can see that (Mn,qer)(x) is a polynomial in x of degree r without a constant term, and it is bounded with respect to [n]q. These properties are transferred to the central moments Mn,q(φxr;x), and consequently (4) takes place, provided to replace n with [n]q. To construct two-dimensional operators of the form (10) and (16) we choose Am=Mm,q,  Bn=Mn,q, and the network will be (73)Δm,n=([i]q[m]q,[j]q[n]q)i,j0.

In time were carried out q-analogues of these operators not only for q>1 but for the case q(0,1); see, for example, [14, 15]. Extensions of these classes of operators by our method also work there.