Some approximation properties by a class of bivariate operators

DOI: 10.1002/mma.5515

S P E C I A L I S S U E P A P E R

Some approximation properties by a class of bivariate

operators

Ana Maria Acu

Tuncer Acar

Carmen-Violeta Muraru

Voichi¸ta Adriana Radu

1_{Department of Mathematics and}

Informatics, Lucian Blaga University of Sibiu, RO-550012 Sibiu, Romania

2_{Department of Mathematics, Selcuk}

University, 42003, Konya, Turkey

3_{Department of Mathematics, Informatics,}

“Vasile Alecsandri” University of Bac˘au, 600115 Bac˘au, Romania

4_{Department of}

Statistics-Forecasts-Mathematics, Babe¸s-Bolyai University, 400591, Cluj-Napoca, Romania

Correspondence

Voichi¸ta Adriana Radu, Babe¸s-Bolyai University, Department of

Statistics-Forecasts-Mathematics, 400591, Cluj-Napoca, Romania.

Email: voichita.radu@econ.ubbcluj.ro

Communicated by: W. Sprößig

Funding information

Romanian Ministery of Research and Innovation, CNCS-UEFISCDI, Grant/Award Number: PN-III-P1-1.1-MC-2018-0792, PN-III-P1-1.1-MC-2018-1041 and PN-III-P1-1.1-MC-2018-1179

MSC Classification: 41A25; 41A36

Starting with the well- known Bernstein operators, in the present paper, we give a new generalization of the bivariate type. The approximation properties of this new class of bivariate operators are studied. Also, the extension of the proposed operators, namely, the generalized Boolean sum (GBS) in the Bögel space of con-tinuous functions is given. In order to underline the fact that in this particular case, GBS operator has better order of convergence than the original ones, some numerical examples are provided with the aid of Maple soft. Also, the error of approximation for the modified Bernstein operators and its GBS-type operator are compared.

K E Y WO R D S

Bernstein operators, GBS-type operators, modulus of continuity

1

I N T RO D U C T I O N A N D P R E L I M I NA RY R E S U LT S

The Bernstein operators of degree n are defined by

Bn(𝑓, x) = n ∑ k=0 b_n,k(x) 𝑓 ( k n ) , x ∈ [0, 1], 𝑓 ∈ C[0, 1], where bn,k(x) = ( n k ) xk_{(1 − x)}n−k_{, k = 0, n, and b} n,k(x) = 0, if k < 0 or k > n.

Over time, many mathematicians start to generalize and modify classical Bernstein operators. For example, the reader can consult papers in previous studies1-6_{and the book by Gupta et al,}7_{in which Gupta et al proposed a collection of several}

results concerning the Bernstein operator. The authors provide approximation properties, Voronovskaya-type theorems, or direct and inverse results for this modified or generalized operators. Usually, the rate of convergence for this modifica-tion is not very fast, so, a new direcmodifica-tion of study appear: how to improve the order of approximamodifica-tion of different kind of

generalization for Bernstein operators. In order to improve the order of approximation, two well-known approaches have

been established; one was introduced by Butzer.8_{Here, the author use a linear combination of the Bernstein}

polynomi-als for improving the order of approximation. The second one used the iterative combinations of the Bernstein operators,

which was introduced by Micchelli.9_{These methods were applied to obtain high-order approximations for certain positive}

linear operators.

In 2018, in their paper,10_{in order to improve the degree of approximation, Khosravian-Arab et al introduced a modified}

Bernstein operator as follows:

BM n(𝑓, x) = n ∑ k=0 bM n,k(x) 𝑓 ( k n ) , x ∈ [0, 1], (1) bM_n,k(x) = a(x, n) bn−1,k(x) + a(1 − x, n) bn−1,k−1(x), 1 ≤ k ≤ n − 1, (2) bM_n,0(x) = a(x, n)(1 − x)n−1, bnM,n(x) = a(1 − x, n)xn−1, and a(x, n) = a1(n) x + a0(n), n = 0, 1, … ,

where the two unknown sequences a0(n)and a1(n)are determined in appropriate way. Very recently, the Kantorovich

variant of the modified Bernstein operators (1) was studied in Acu et al.11

Note that throughout the paper we will assume that the sequences ai(n), i = 0, 1 verify the conditions

2a0(n) + a1(n) =1, 0 ≤ a0(n)≤ 1. (3)

Under this assumption, the operator BM

n defined in (1) is positive. Also, in case that a1(n) = −1 and a0(n) = 1 the BMn reduces to the well-known Bernstein operator.

Khosravian-Arab et al10_{obtained the moments up to order two of the operators B}M

n.

Lemma 1.1. 10_{The Bernstein operators B}M

n have the following moments:

i) BM n(e0;x) =1; ii) BMn(e1;x) = x + (1−2x)(1−a0(n)) n ; iii) BM n(e2;x) = x2+ x n(4xa0(n) −2a0(n) −5x + 3) + (1−2x)2_(1−a 0(n)) n2 .

By direct computation, the moments BM

n(ei), i = 3, 4 and the central moments of the modified Bernstein operator BMn can be obtained.

Lemma 1.2. The Bernstein operators, BMn, verify

i) BMn(e3;x) = x3+ 3x 2 n (2xa0(n) − a0(n) −3x + 2) + x(1−2x) n2 (9xa0(n) −6a0(n) −10x + 7)+ (1−2x)(1−a0(n)) n3 (6x 2_{−6x + 1);} ii) BM n(e4;x) = x4+2x 3 n (4xa0(n)−2a0(n)−7x+5)− x2 n2(48x 2_a 0(n)−60xa0(n)−59x2+18a0(n)+78x−25)+_nx3(88x 3_a 0(n)− 152x2_a 0(n) −94x3+82xa0(n) +164x2−14a0(n) −89x + 15) + (1−2x) 2_(1−a 0(n)) n4 (12x 2_{−12x + 1).}

Lemma 1. 3. The central moments of modified Bernstein operators are

i) BM n(t − x; x) = (1−2x)(1−a0(n)) n ; ii) BMn((t − x)2;x) =x(1−x)_n + (1−a0(n))(1−2x)2 n2 ; iii) BM n((t − x)4;x) = 3x2_(1−x)2 n2 − x(1−x) n3 (40a0(n)x 2_−40a 0(n)x −46x2+10a0(n) +46x − 11) +_n14(1 − a0(n))(1 − 2x) 2_(12x2₋ 12x + 1).

Using Lemma 1. 3, we get the following result:

Lemma 1.4. The modified Bernstein operators verify

i) lim n→∞nB M n(t − x; x) = (1 − 2x)(1 − a0(n)); ii) lim n→∞nB M n((t − x)2;x) = x(1 − x); iii) lim n→∞n 2_BM n((t − x)4;x) =3x2(1 − x)2.

In the following, we propose the bivariate case of the operators BM

n. Let I = [0, 1], I2 = I × I. Denote by C(I2)the class

of all real valued continuous functions on I2, and consider||𝑓|| = sup

(x,𝑦)∈I2

|𝑓(x, 𝑦)| the norm on C(I2_).

Then, for f ∈ C(I2), the bivariate Bernstein-type operators of (1) is defined as

BM n,m(𝑓; x, 𝑦) = n ∑ k=0 m ∑ 𝑗=0 bM n,k(x)bMm,𝑗(𝑦)𝑓 ( k n, 𝑗 m ) . (4)

In this paper, we desire to investigate the local approximation properties of the bivariate Bernstein-type operators (4) by means of modulus of continuity and the Peetre's K-functional. A Voronovskaja-type theorem is obtained. In order to give

better approximation properties, we introduce the generalized Boolean sum (GBS) of the operators BM

n,m. Some numerical

examples that compare the convergence of the proposed operators with the classical Bernstein operators of bivariate type are given.

2

A P P ROX I M AT I O N P RO P E RT I E S O F B

O P E R ATO R S

For the bivariate operators BM

n,mdefined in (4), we will evaluate the test functions in the following lemma.

Lemma 2.1. Let the test function eij ∶ I2 → I2, eij = xiyj(0 ≤ i + j ≤ 2 (i and j are integers). Then the following

equalities hold i) BM n,m(e00;x, 𝑦) = 1; ii) BM n,m(e10;x, 𝑦) = x +(1−2x)(1−an 0(n)); iii) BM n,m(e01;x, 𝑦) = 𝑦 + (1−2𝑦)(1−am 0(m)); iv) BM n,m(e11;x, 𝑦) = [ x +(1−2x)(1−a0(n)) n ] [ 𝑦 +(1−2𝑦)(1−a0(m)) m ] ; v) BM n,m(e20;x, 𝑦) = x2+_nx(4xa0(n) −2a0(n) −5x + 3) +(1−2x) 2_(1−a 0(n)) n2 ;

vi) BMn,m(e02;x, 𝑦) = 𝑦2+ m𝑦(4𝑦a0(m) −2a0(m) −5𝑦 + 3) +

(1−2𝑦)2_(1−a 0(m))

m2 .

Proof. Let𝜇, 𝜈 ∈N, 0 ≤ 𝜈 + 𝜇 ≤ 2. From (4), it follows

BMn,m(e𝜇,𝜈;x, 𝑦) = n ∑ k=0 m ∑ 𝑗=0 bM_n,k(x)bM_m,𝑗(𝑦) ( k n )𝜇(_𝑗 m )𝜈 = n ∑ k=0 bM_n,k(x) ( k n )_{𝜇 m∑} 𝑗=0 bM_m,𝑗(𝑦) ( 𝑗 m )_𝜈 =BMn(e𝜇;x) · BMm(e𝜈;𝑦). Using Lemma 1.1, the proof is complete.

Lemma 2.2. The following identities hold for any (x, y) ∈ I2

i) BM n,m(t − x; x, 𝑦) = (1−2x)(1−an 0(n)); ii) BM n,m(s −𝑦; x, 𝑦) = (1−2𝑦)(1−am 0(m)); iii) BMn,m((t − x)2;x, 𝑦) = x(1−x)n + (1−a0(n))(1−2x)2 n2 ; iv) BM n,m((s −𝑦)2;x, 𝑦) = 𝑦(1−𝑦)m + (1−a0(m))(1−2𝑦)2 m2 .

Proof. Let𝜇, 𝜈 ∈N, 0 ≤ 𝜈 + 𝜇 ≤ 2. From (4), it follows

BMn,m((t − x)𝜇(s −𝑦)𝜈;x, 𝑦) = BMn((t − x)𝜇;x) · BMm((s −𝑦)𝜈;𝑦). Using Lemma 1. 3, the proof is complete.

Theorem 2.1. For any f ∈ C(I2), the sequence of the bivariate modified Bernstein operators defined in relation (4)

Proof. Now, by using Lemma 2.1 and applying the well-known result because of Volkov12_{for the bivariate functions} lim n,m→∞B M n,m(e10;x, 𝑦) = x, lim n,m→∞B M n,m(e01;x, 𝑦) = 𝑦, lim n,m→∞B M n,m(e20+e02;x, 𝑦) = x2+𝑦2, the proof of this theorem is complete.

In the following, we provide some examples in order to put in evidence, for some special functions, the convergence of the BM

n,m operators and also the error of approximation, and a comparison with the classical Bernstein operators of

bivariate type are given.

FIGURE 1 The convergence of BM

n,m(green for n = m = 10, blue for n = m = 20) to the function f (red) [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 2 Error of approximation EM

n,m(green for n = m = 10, blue for n = m = 20) [Colour figure can be viewed at wileyonlinelibrary.com]

Example 2.1. Let f(x, y) = (x − 1)2_{sin(𝜋y), a0(n) =} n−1 2n, a1(n) = 1 n, and E M n,m(𝑓; x, 𝑦) = ||𝑓(x, 𝑦) − BMn,m(𝑓; x, 𝑦)|| be

the error function of the modified Bernstein operators. The graphs of function f and operator BM

n,m, for n = m = 10

and n = m = 20, respectively, are given in Figure 1. The error of approximation is illustrated in Figure 2.

Example 2.2. Let f(x, y) = x4_y2_+2x4_{y −4xy}4_{, a}

0(n) =_2n+1n , a1(n) = _2n+11 , and EM_n,m(𝑓; x, 𝑦) = ||𝑓(x, 𝑦) − BM_n,m(𝑓; x, 𝑦)||

be the error function of the modified Bernstein operators. The convergence of the modified Bernstein operators BM

n,m, considering n = m = 20 and n = m = 50, respectively, is shown in Figure 3. The error of approximation is illustrated in Figure 4.

FIGURE 3 The convergence of BM

n,m(green for n = m = 20, blue for n = m = 50) to the function f (red) [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 4 Error of approximation EM

n,m(green for n = m = 20, blue for n = m = 50) [Colour figure can be viewed at wileyonlinelibrary.com]

In the next example, we compare the modified Bernstein operators BM

n,mwith the classical Bernstein operator

Bn,m(𝑓; x, 𝑦) = n ∑ k=0 m ∑ 𝑗=0 bn,k(x)bm,𝑗(𝑦)𝑓 ( k n, 𝑗 m ) .

Example 2.3. Let𝑓(x, 𝑦) = sin(𝜋x) cos(𝜋x), a0(n) = n

2n+1, a1(n) = 1

2n+1. Denote E

M

n,m(𝑓; x, 𝑦) = ||𝑓(x, 𝑦) − BMn,m(𝑓; x, 𝑦)|| the error function of the modified Bernstein operators and En,m(𝑓; x, 𝑦) = ||𝑓(x, 𝑦) − Bn,m(𝑓; x, 𝑦)|| the error function

of the Bernstein operators. The graphs of function f and operators BM

n,m, Bn,m, for n = m = 10 are given in Figure 5.

The errors of approximation EM

n,m, En,m are illustrated in Figure 6. In some particular cases, the modified operators

can be better than the original ones.

FIGURE 5 The convergence of BM

n,m(green) and Bn,m(blue) to the function f (red) [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 6 Error of approximation EM

Example 2.4. Let f(x, y) = x4_y2 _{+ 2x}4_{y − 4xy}4_{, a}

0(n) = _2n+1n , a1(n) = _2n+11 . The graphs of function f and operators BM

n,m, Bn,m, for n = m = 3 are given in Figure 7. The errors of approximation E_n,mM and En,mare illustrated in Figure 8.

In some particular cases, the modified operators can be better than the original ones.

The next results concern on the evaluation of the rate of convergence in terms of modulus of continuity. First, we recall the definition of the complete modulus of continuity.

Definition 2.1. Let f ∈ C(I2). For the bivariate case, the complete modulus of continuity is defined as

̄𝜔( 𝑓; 𝜎1, 𝜎2) =sup{|𝑓(u, v) − 𝑓(x, 𝑦)| ∶ (u, v), (x, 𝑦) ∈ I2and |u − x| ≤ 𝜎1, |v − 𝑦| ≤ 𝜎2}. Also, we have the alternative form

̄𝜔( 𝑓; 𝜎) = sup{|𝑓(u, v) − 𝑓(x, 𝑦)| ∶ (u, v), (x, 𝑦) ∈ I2_and √_{(u − x)}2_{+ (v −𝑦)}2_{≤ 𝜎}}_.

FIGURE 7 The convergence of BM

n,m(green) and Bn,m(blue) to the function f (red) [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 8 Error of approximation EM

Let I1, I2⊆Rbe the compact intervals and B(I1×I2) = {𝑓 ∶ I1×I2 →R|𝑓 a bounded function on I1 × I2}. In order to

give an estimation of the error of approximation, we first need a result from 1969 from Shisha and Mond.13

Theorem 2.2. 13_{Let P ∶ C(I}

1 ×I2)→ B(I1 ×I2)be a linear positive operator. For each f ∈ C(I1 ×I2), (x, y) ∈ I1 ×I2

and any𝜎1 > 0, 𝜎2 > 0, the next inequality holds

|P𝑓(x, 𝑦) − 𝑓(x, 𝑦)| ≤ |𝑓(x, 𝑦)| · |P(1; x, 𝑦) − 1| + { P(1; x, 𝑦) + 𝜎₁−1√P(1; x, 𝑦)P((u − x)2_;x, 𝑦) + 𝜎−1 2 √ P(1; x, 𝑦)P((v − 𝑦)2_;x, 𝑦) +𝜎₁−1𝜎₂−1P(1; x, 𝑦)√P((u − x)2_{;x, 𝑦)P((v − 𝑦)}2_{;x, 𝑦)}}_{𝜔(𝜎1, 𝜎2).}

Now we can provide an estimation for the error of approximation of BM

n,moperators, using the Shisha and Monde result.

Theorem 2.3. For any f ∈ C(I2)and (x, y) ∈ I2, the following inequality holds

|BM n,m(𝑓; x, 𝑦) − 𝑓(x, 𝑦)| ≤ 4𝜔 (𝜎n(x), 𝜎n(𝑦)) , where 𝜎2 n(x) = x(1 − x) n + (1 − a0(n))(1 − 2x)2 n2 and 𝜎2 m(𝑦) = 𝑦(1 − 𝑦) m + (1 − a0(m))(1 − 2𝑦)2 m2 .

Proof. Since BMn,mis a linear operator, and BMn,m(e00;x, 𝑦) = 1 by applying Theorem 2.2, we get for every 𝛿1, 𝛿2 > 0 the following inequality: ||BM n,m(𝑓; x, 𝑦) − 𝑓(x, 𝑦)|| ≤ { 1 +𝜎₁−1 √ BM n,m((u − x)2;x, 𝑦) + 𝜎2−1 √ BM n,m((v −𝑦)2;x, 𝑦) +𝜎₁−1𝜎₂−1 √ BMn,m((u − x)2;x, 𝑦)BMn,m((v −𝑦)2;x, 𝑦) } 𝜔(𝜎1, 𝜎2). and choosing𝜎1 = 𝜎n(x), 𝜎2 = 𝜎m(y), the proof for Theorem 2.3 is obtained.

Let C2(I2) = { 𝑓 ∈ C(I2 ) ∶𝜕 i_𝑓 𝜕xi, 𝜕i_𝑓 𝜕𝑦i ∈C(I 2 ), for i = 1, 2 } ,

with the norm

||𝑓||C2_(I2₎ =‖𝑓‖_C(I2₎+ 2 ∑ i=1 (‖ ‖‖ ‖‖𝜕 i_𝑓 𝜕xi ‖‖ ‖‖ ‖C(I2₎ +‖‖‖‖ ‖ 𝜕i_𝑓 𝜕𝑦i ‖‖ ‖‖ ‖C(I2₎ ) .

The Peetre's K-functional of the function f ∈ C(I2_{)is defined by}

K(𝑓; 𝜎) = inf

u∈C2_(I2₎

{

||𝑓 − u||C(I2₎+𝜎||u||_C2_(I2), 𝜎 > 0

}

,

and the second-order modulus of continuity is given as

̄𝜔2(𝑓;√𝜎) = sup {| || || | 2 ∑ 𝜈=0 (−1)2−𝜈_{𝑓(x + 𝜈h, 𝑦 + 𝜈k)}|||| ||∶ (x, 𝑦), (x + 2h, 𝑦 + 2k) ∈ I 2_{, |h| ≤ 𝜎, |k| ≤ 𝜎} } .

From Butzer and Berens work,14_{the following link between the K-functional and the second-order modulus of continuity}

is known K(𝑓; 𝜎) ≤ M { ̄𝜔2(𝑓;√𝜎) + min(1, 𝜎)||𝑓||C(I2₎, } for all𝜎 > 0,

where M is a constant independent of f and𝜎.

The next result establishes the rate of approximation of the modified Bernstein-type operators by means of Peetre's K-functional.

Theorem 2.4. For the function f ∈ C(I2), we have the following inequality

||BM n,m(𝑓; x, 𝑦) − 𝑓(x, 𝑦)|| ≤ 4K ( 𝑓;1 4An,m(x, 𝑦) ) +𝜔 ( 𝑓;√𝜈n,m(x, 𝑦) ) ,

where 𝜈n,m(x, 𝑦) = ( (1 − 2x)(1 − a0(n)) n )2 + ( (1 − 2𝑦)(1 − a0(m)) m )2 and A_n,m(x, 𝑦) = 𝜎2 n(x) +𝜎m2(𝑦) + 𝜈n,m(x, 𝑦). Proof. We define ̃BM n,m(𝑓; x, 𝑦) = BMn,m(𝑓; x, 𝑦) − 𝑓 ( x +(1 − 2x)(1 − a0(n)) n , 𝑦 + (1 − 2𝑦)(1 − a0(m)) m ) +𝑓(x, 𝑦). From Lemma 2.1, we have

̃BM

n,m(t − x; x, 𝑦) = 0, ̃BMn,m(s −𝑦; x, 𝑦) = 0. Using Taylor's theorem for g ∈ C2(I2), it follows

g(t, s) − g(x, 𝑦) = 𝜕g(x, 𝑦) 𝜕x (t − x) + ∫ t x (t − u)𝜕 2_{g(u, 𝑦)} 𝜕u2 du + 𝜕g(x, 𝑦) 𝜕𝑦 (s −𝑦) + ∫ s 𝑦 ( s − v)𝜕 2_{g(x, v)} 𝜕v2 dv. (5) Applying ̃BM

n,mon both side of (5), we get

̃BM n,m(g; x, 𝑦) − g(x, 𝑦) = ̃BMn,m ( ∫ t x (t − u)𝜕 2_{g(u, 𝑦)} 𝜕u2 du; x, 𝑦 ) + ̃BMn,m ( ∫ s 𝑦 (s − v) 𝜕2_{g(x, v)} 𝜕v2 dv; x, 𝑦 ) =BMn,m ( ∫ t x (t − u)𝜕 2_{g(u, 𝑦)} 𝜕u2 du; x, 𝑦 ) − ∫ x+(1−2x)(1−a0(n))_n x ( x +(1 − 2x)(1 − a0(n)) n −u )_{𝜕2g(u, 𝑦)} 𝜕u2 du +BM_n,m ( ∫ s 𝑦 (s − v) 𝜕2_{g(x, v)} 𝜕v2 dv; x, 𝑦 ) − ∫ 𝑦+(1−2𝑦)(1−a0(m)) m 𝑦 ( 𝑦 +(1 − 2𝑦)(1 − a0(m)) m −v ) 𝜕2_{g(x, v)} 𝜕v2 dv. Therefore, ||̃BM n,m(g; x, 𝑦) − g(x, 𝑦)|| ≤ BMn,m (| || ||∫ t x |t − u|||_|| | 𝜕2_{g(u, 𝑦)} 𝜕u2 || || |du || || |;x, 𝑦 ) +|||| ||∫ x+(1−2x)(1−a0(n)) n x || ||x +(1 − 2x)(1 − a0(n)) n −u||||· || || | 𝜕2_{g(u, 𝑦)} 𝜕u2 || || |du || || || +BMn,m (| || ||∫ s 𝑦 |s − v| || || | 𝜕2_{g(x, v)} 𝜕v2 || || |dv || || |;x, 𝑦 ) +|||| ||∫ 𝑦+(1−2_{𝑦)(1−a0(m))} m 𝑦 || ||𝑦+(1 − 2𝑦)(1 − a0(m)) m −v||||· || || | 𝜕2_{g(x, v)} 𝜕v2 || || |dv || || || ≤ { BMn,m((t − x)2;x, 𝑦)+ ( (1 − 2x)(1 − a0(n)) n )2} ||g||C2_(I2₎ + { BM n,m((s −𝑦)2;x, 𝑦)+ ( (1 − 2𝑦)(1 − a0(m)) m )2} ||g||C2_(I2₎ ≤ { 𝜎2 n(x) + ( (1 − 2x)(1 − a0(n)) n )2 +𝜎m2(𝑦) + ( (1 − 2𝑦)(1 − a0(m)) m )2} ||g||C2_(I2). But, ||̃BM n,m(𝑓; x, 𝑦)|| ≤ 3||𝑓||C(I2).

Now, we have ||BM n,m(𝑓; x, 𝑦) − 𝑓(x, 𝑦)|| ≤ ||̃BMn,m(𝑓 − g; x, 𝑦)|| + ||̃BMn,m(g; x, 𝑦) − g(x, 𝑦)|| + |g(x, 𝑦) − 𝑓(x, 𝑦)| +|||| |𝑓 ( x +(1 − 2x)(1 − a0(n)) n , 𝑦 + (1 − 2𝑦)(1 − a0(m)) m ) −𝑓(x, 𝑦)|||| | ≤ 4||𝑓 − g||C(I2₎+A_n,m(x, 𝑦)||g||_C2_(I2₎+𝜔 ( 𝑓;√𝜈n,m(x, 𝑦) ) =4 { ||𝑓 − g||C(I2₎+1 4An,m(x, 𝑦)||g||C2(I2) } +𝜔 ( 𝑓;√𝜈n,m(x, 𝑦) ) .

Taking the infimum on the right hand side over all g ∈ C2(I2_{), it follows}

||BM n,m(𝑓; x, 𝑦) − 𝑓(x, 𝑦)|| ≤ 4K ( 𝑓;1 4An,m(x, 𝑦) ) +𝜔 ( 𝑓;√𝜈n,m(x, 𝑦) ) .

2.1

Voronovskaya-type theorem

Some asymptotic properties of the modified Bernstein operators introduced in Section 1 are studied in this subsection.

Theorem 2.5. If𝑓 ∈ C2(_I2)_{, then} lim n→∞n { BM_n,n(𝑓; x, 𝑦) − 𝑓(x, 𝑦)} = (1 − a0(n)) [ (1 − 2x)𝑓′ x(x, 𝑦) + (1 − 2𝑦)𝑓𝑦′(x, 𝑦)]+ 1₂ { x(1 − x)𝑓′′ x2(x, 𝑦) + 𝑦(1 − 𝑦)𝑓_𝑦′′2(x, 𝑦) } . Proof. Let (x0, y₀) ∈ I2be a fixed point. Using the Taylor formula, we get

𝑓(u, v) = 𝑓(x0, 𝑦0) +𝑓x′(x0, 𝑦0)(u − x0) +𝑓𝑦′(x0, 𝑦0)(v −𝑦0) +1 2 { 𝑓′′ x2(x0, 𝑦0)(u − x0) 2_+2𝑓′′ x𝑦(x0, 𝑦0)(u − x0)(v −𝑦0) +𝑓_𝑦′′2(x0, 𝑦0)(v −𝑦0) 2} +𝜃(u, v)((u − x0)2+ (v −𝑦0)2 ) ,

where (u, v) ∈ I2_{and lim}

(u,v)→(x0,𝑦0)

𝜃(u, v) = 0.

From the linearity of BM

n,n, we obtain BMn,n(𝑓(u, v); x0, 𝑦0) =𝑓(x0, 𝑦0) +𝑓x′(x0, 𝑦0)BMn,n(u − x0;x0, 𝑦0) +𝑓_𝑦′(x0, 𝑦0)BnM,n(v −𝑦0;x0, 𝑦0) +1₂ { 𝑓′′ x2(x0, 𝑦0)B M n,n((u − x0)2;x0, 𝑦0) +2𝑓x′′𝑦(x0, 𝑦0)BMn,n((u − x0)(v −𝑦0);x0, 𝑦0) +𝑓𝑦′′2(x0, 𝑦0)B M n,n((v −𝑦0)2;x0, 𝑦0) } +BM_n,n(𝜃(u, v)((u − x0)2+ (v −𝑦0)2);x0, 𝑦0) =𝑓(x0, 𝑦0) +𝑓x′(x0, 𝑦0)BMn(u − x0;x0) +𝑓𝑦′(x0, 𝑦0)BMn(v −𝑦0;𝑦0) +1 2 { 𝑓′′ x2(x0, 𝑦0)B M n((u − x0)2;x0) +𝑓_𝑦′′2(x0, 𝑦0)B M n((v −𝑦0)2;𝑦0) +2𝑓′′ x𝑦(x0, 𝑦0)BMn((u − x0);x0)BMn((v −𝑦0);𝑦0) } +BM_n,n(𝜃(u, v)((u − x0)2+ (v −𝑦0)2);x0, 𝑦0). Applying the Hölder inequality, we get

|| |BMn,n(𝜃(u, v)((u − x0)2+ (v −𝑦0)2 ) ;x0, 𝑦0)||| ≤{BM_n,n(𝜃2(u, v); x0, 𝑦0)}1∕2 { BM_n,n(((u − x0)2+ (v −𝑦0)2)2;x0, 𝑦0 )}1∕2 ≤√2{BMn,n(𝜃2(u, v); x0, 𝑦0)}1∕2·{BMn,n((u − x0)4;x0, 𝑦0)+BMn,n((v −𝑦0)4;x0, 𝑦0)}1∕2.

From Theorem 2.1, we get

lim n→∞B

M

n,n(𝜃2(u, v); x0, 𝑦0)=𝜃2(x0, 𝑦0) =0, and using Lemma 2.2, we obtain

lim n→∞nB

M

n,n(𝜃(u, v)((u − x0)2+ (v −𝑦0)2);x0, 𝑦0)=0. Applying Lemma 2.2, the proof is completed.

3

A P P ROX I M AT I O N BY A S S O C I AT E D G B S O P E R ATO R S

In 1934 and 1935, Bögel15,16_{introduced a new concept, namely, the Bögel-continuous and Bögel-differentiable functions.}

Consequently, using these concepts, he established some important theorems.

Let A, B ⊂Rbe compact sets. Denote the mixed difference of f as follows:

Δ𝑓[(x, 𝑦); (𝛼0, 𝛽0)] =𝑓(x, 𝑦) − 𝑓(𝛼0, 𝑦) − 𝑓(x, 𝛽0) +𝑓(𝛼0, 𝛽0),

where𝑓 ∶ A × B →Rand (𝛼0, 𝛽0) ∈ A × B. The function f is the Bögel continuous (B-continuous) at (𝛼0, 𝛽0) ∈ A × Bif

lim

(x,𝑦)→(𝛼0,𝛽0)

Δ𝑓[(x, 𝑦); (𝛼0, 𝛽0)] =0.

A function𝑓 ∶ A × B →Ris the Bögel-differentiable (B-differentiable) at (𝛼0, 𝛽0) ∈ A × Bif the limit

Db𝑓(𝛼0, 𝛽0) ∶= lim

(x,𝑦)→(𝛼0,𝛽0)

Δ𝑓[(x, 𝑦); (𝛼0, 𝛽0)] (x −𝛼0)(𝑦 − 𝛽0) exists and is finite. The limit Dbf(𝛼0, 𝛽0)is named B-differential of f at the point (𝛼0, 𝛽0).

The function𝑓 ∶ X ⊂ A × B →Ris B-bounded on X if there exists M > 0, such that

|Δ𝑓 [(u, v); (x, 𝑦)]| ≤ M,

for every (u, v), (x, y) ∈ X. Denote by Bb(X)all B-bounded functions defined on X endowed with the norm

‖𝑓‖B= sup

(x,𝑦),(u,v)∈X

|Δ𝑓 [(u, v); (x, 𝑦)]| .

Denote by Cb(X), Db(X), B(X), and C(X) the space of B-continuous functions, B-differentiable functions, bounded

func-tions, continuous functions on X, respectively. As we can see in the study of Bögel,17page 52 _{it is known that C(X) ⊂}

Cb(X).

Let𝑓 ∈RX_{= {𝑓 ∶ X →R}. The GBS operator associated to a linear operator P ∶R}X_→RX_{is defined as}

G(𝑓; x, 𝑦) = G[𝑓(u, v); x, 𝑦]

=P[𝑓(u, 𝑦) + 𝑓(x, v) − 𝑓(u, v); x, 𝑦].

During time some researchers constructed and studied different type of GBS operators as we can see for example papers in B˘arbosu et al,18_{Kajla and Micl˘au¸s,}19_{and Ruchi et al.}20

Let f ∈ Cb(I2). The GBS operator GMn,massociated to BMn,mcan be introduced as follows

GMn,m(𝑓; x, 𝑦) = BMn,m[𝑓(u, 𝑦) + 𝑓(x, v) − 𝑓(u, v); x, 𝑦]. (6)

In order to determine the degree of approximation of B-continuous functions by using GBS operators, we define the

mixed modulus of smoothness for𝑓 ∈ Cb

(

I2)_{as follows:}

𝜔mixed(𝑓; 𝜎1, 𝜎2) ∶=sup {|Δ𝑓 [(u, v); (x, 𝑦)]| ∶ |x − u| < 𝜎1, |𝑦 − v| < 𝜎2}, for all (x, y), (u, v) ∈ I2_{and for any (𝜎1,𝜎2) ∈ (0, ∞) × (0, ∞).}

Badea et al21,22_{obtained the basic properties of}𝜔

mixed. These properties are similar to the ones of the usual modulus of continuity.

In the next, we provide two examples in which we compare the convergence and also the error of approximation for the modified Bernstein operators with its GBS type operators. We note that for this particular function, GBS has better order of approximation than the original operators.

FIGURE 9 The convergence of GM

n,m(green) and BMn,m(blue) to f (red) [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 10 The errors of approximation ̃EM

n,m(green) and EMn,n(blue) [Colour figure can be viewed at wileyonlinelibrary.com]

Example 3.1. Let𝑓(x, 𝑦) = 𝑦2 ₋₂√_{2(1 − x −𝑦)}2 _{−8x𝑦, n = m = 20, a0 =} n−1

2n, and a1 =

1

n. In Figure 9 we

compare the modified Bernstein operators and its GBS-type operators. Denote EM

n,m(𝑓; x, 𝑦) = ||BMn,m(𝑓; x, 𝑦) − 𝑓(x, 𝑦)|| and ̃EM

n,m(𝑓; x, 𝑦) = ||GMn,m(𝑓; x, 𝑦) − 𝑓(x, 𝑦)||. The error of approximation for the modified Bernstein operators and its GBS-type operator are compared in Figure 10. For this particular case, GBS operator has better order of convergence than the original ones.

Example 3.2. Let𝑓(x, 𝑦) = sin(𝜋x) cos(𝜋x), n = m = 10, a0 = n−1

2n , a1 =

1

n. In Figure 11 we compare the modified

Bernstein operators and its GBS type operator. The error of approximation for the modified Bernstein operators (EnM,n)

and its GBS type operator ( ̃EM

n,m) are compared in Figure 12. For this particular case, GBS operator has better order of

FIGURE 11 The convergence of GM

n,m(green) and BMn,m(blue) to f (red) [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 12 The errors of approximation ̃EM

n,m(green) and EMn,n(blue) [Colour figure can be viewed at wileyonlinelibrary.com]

In order to give the rate of convergence for B-continuous functions using GBS operators, Badea et al22 _{proved the}

following Shisha and Mond type result:

Theorem 3.1. If G ∶ Cb(X)→ Cb(X) is the GBS operator associated to the linear positive operator P ∶ Cb(X)→ Cb(X),

then |G( 𝑓; x, 𝑦) − 𝑓(x, 𝑦)| ≤ |𝑓(x, 𝑦)||P(1; x, 𝑦) − 1| + {P(1; x, 𝑦) +𝜎₁−1√P((u − x)2_{;x, 𝑦) + 𝜎}−1 2 √ P((v −𝑦)2_{;x, 𝑦)} +𝜎₁−1√P((u − x)2_{;x, 𝑦)𝜎}−1 2 √ P((v −𝑦)2_{;x, 𝑦)}𝜔} mixed(𝑓; 𝜎1, 𝜎2),

Conclusion1. Let f ∈ Cb(I2)and (x, y) ∈ I2. Applying Theorem 3.1, we have |GM n,m(𝑓; x, 𝑦) − 𝑓(x, 𝑦)| ≤ |𝑓(x, 𝑦)||BMn,m(1; x, 𝑦) − 1| + {BMn,m(1; x, 𝑦) +𝜎₁−1 √ BM n,m((u − x)2;x, 𝑦) + 𝜎2−1 √ BM n,m((v −𝑦)2;x, 𝑦) +𝜎₁−1𝜎₂−1 √ BM n,m((u − x)2;x, 𝑦) √ BM n,m((v −𝑦)2;x, 𝑦)}𝜔mixed(𝑓; 𝜎1, 𝜎2).

Now, choosing𝜎1 = 𝜎n(x)and𝜎2 = 𝜎m(y), we obtain for the GBS operator associated to the modified Bernstein

operator defined in (4) the following result: |GM

n,m(𝑓; x, 𝑦) − 𝑓(x, 𝑦)| ≤ 4𝜔mixed(𝑓, 𝜎n(x), 𝜎m(𝑦)).

AC K N OW L E D G E M E N T

The work was supported by a mobility grant of the Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, project numbers PN-III-P1-1.1-MC-2018-0792, PN-III-P1-1.1-MC-2018-1041 and PN-III-P1-1.1-MC-2018-1179 within PNCDI III.

O RC I D

Ana Maria Acu https://orcid.org/0000-0002-0488-1058

Voichi¸ta Adriana Radu https://orcid.org/0000-0003-0581-8381

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How to cite this article: Acu AM, Acar T, Muraru C-V, Radu VA. Some approximation properties by a class of