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volume 7, issue 3, article 92, 2006.

Received 13 September, 2005;

accepted 03 June, 2006.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

APPROXIMATION OF B-CONTINUOUS ANDB-DIFFERENTIABLE FUNCTIONS BY GBS OPERATORS OF BERNSTEIN

BIVARIATE POLYNOMIALS

OVIDIU T. POP AND MIRCEA FARCA ¸S

Vest University "Vasile Goldi¸s" of Arad Branch of Satu Mare 26 Mihai Viteazu Street

Satu Mare 440030, Romania.

EMail:ovidiutiberiu@yahoo.com National College "Mihai Eminescu"

5 Mihai Eminescu Street Satu Mare 440014, Romania.

EMail:mirceafarcas2005@yahoo.com

c

2000Victoria University ISSN (electronic): 1443-5756 272-05

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Approximation ofB-Continuous andB-Differentiable Functions by GBS Operators of Bernstein

Bivariate Polynomials Ovidiu T. Pop and Mircea Farca¸s

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J. Ineq. Pure and Appl. Math. 7(3) Art. 92, 2006

Abstract

In this paper we give an approximation ofB-continuous andB-differentiable functions by GBS operators of Bernstein bivariate polynomials.

2000 Mathematics Subject Classification:41A10, 41A25, 41A35, 41A36, 41A63.

Key words: Linear positive operators, Bernstein bivariate polynomials, GBS opera- tors,B-differentiable functions, approximation ofB-differentiable functions by GBS operators, mixed modulus of smoothness.

1. Preliminaries

In this section, we recall some results which we will use in this article.

In the following, let X and Y be compact real intervals. A function f : X×Y →Ris called aB-continuous (Bögel-continuous) function in(x₀, y₀)∈ X×Y if

(x,y)→(xlim₀,y0)∆f((x, y),(x₀, y₀)) = 0.

Here

∆f((x, y),(x₀, y₀)) =f(x, y)−f(x₀, y)−f(x, y₀) +f(x₀, y₀) denotes a so-called mixed difference off.

A functionf :X×Y →Ris called aB-differentiable (Bögel-differentiable) function in(x₀, y₀)∈X×Y if it exists and if the limit is finite:

(x,y)→(xlim₀,y0)

∆f((x, y),(x₀, y₀)) (x−x₀)(y−y₀) .

The limit is named the B-differential off in the point(x₀, y₀) and is denoted byD_Bf(x₀, y₀).

The definitions ofB-continuity and B-differentiability were introduced by K. Bögel in the papers [5] and [6].

The functionf :X×Y →RisB-bounded onX×Y if there existsK >0 such that

|∆f((x, y),(s, t))| ≤K for any(x, y),(s, t)∈X×Y.

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We shall use the function setsB(X×Y) =

f : X×Y → R|f bounded onX×Y with the usual sup-normk · k∞,B_b(X×Y) =

f :X×Y →R|f B-bounded onX ×Y and we setkfkB = sup

(x,y),(s,t)∈X×Y

|∆f((x, y),(s, t))|, where

f ∈Bb(X×Y), C_b(X×Y) =

f :X×Y →R|f B−continuous onX×Y , and D_b(X×Y) =

f :X×Y →R|f B−differentiable onX×Y . Letf ∈ B_b(X ×Y). The functionω_mixed(f;·,·) : [0,∞)×[0,∞) → R, defined by

(1.1) ω_mixed(f;δ₁, δ₂) = sup{|∆f((x, y),(s, t))|:|x−s| ≤δ₁,|y−t| ≤δ₂} for any(δ₁, δ₂)∈[0,∞)×[0,∞)is called the mixed modulus of smoothness.

For related topics, see [1], [2], [3] and [10].

LetL:Cb(X×Y)→B(X×Y)be a linear positive operator. The operator U L :C_b(X ×Y)→ B(X×Y)defined for any functionf ∈C_b(X×Y)and any(x, y)∈X×Y by

(1.2) (U Lf)(x, y) = (L(f(·, y) +f(x,∗)−f(·,∗))) (x, y)

is called the GBS operator ("Generalized Boolean Sum" operator) associated to the operatorL, where "·" and "∗" stand for the first and second variable.

Let the functions e_ij : X ×Y → R, (e_ij)(x, y) = xⁱy^j for any (x, y) ∈ X×Y, wherei, j ∈N. The following theorem is proved in [1].

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Theorem 1.1. LetL :C_b(X ×Y)→B(X×Y)be a linear positive operator andU L:Cb(X×Y)→B(X×Y)the associated GBS operator. Then for any f ∈C_b(X×Y), any(x, y)∈(X×Y)and anyδ₁, δ₂ >0, we have

(1.3) |f(x, y)−(U Lf)(x, y)| ≤ |f(x, y)| |1−(Le₀₀)(x, y)|

+

(Le₀₀)(x, y) +δ₁⁻¹p

(L(· −x)²) (x, y) +δ₂⁻¹p

(L(∗ −y)²) (x, y) +δ₁⁻¹δ⁻¹₂ p

(L(· −x)²(∗ −y)²) (x, y)

ω_mixed(f;δ₁, δ₂).

In the following, we need the following theorem for estimating the rate of the convergence of theB-differentiable functions (see [11]).

Theorem 1.2. LetL :C_b(X ×Y)→B(X×Y)be a linear positive operator and U L : C_b(X ×Y) → B(X ×Y)the associated GBS operator. Then for any f ∈ D_b(X ×Y) withD_Bf ∈ B(X ×Y), any(x, y) ∈ X ×Y and any δ₁, δ₂ >0, we have

|f(x, y)−(U Lf)(x, y)|

(1.4)

≤ |f(x, y)||1−(Le₀₀)(x, y)|+3kD_Bfk_∞p

(L(· −x)²(∗ −y)²) (x, y) +

p(L(· −x)²(∗ −y)²)(x, y) +δ⁻¹₁ p

(L(· −x)⁴(∗ −y)²)(x, y) +δ⁻¹₂ p

(L(· −x)²(∗ −y)⁴)(x, y) +δ⁻¹₁ δ₂⁻¹ L(· −x)²(∗ −y)²

(x, y)

ω_mixed(D_Bf;δ₁, δ₂).

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2. Main Results

Let the sets∆2 ={(x, y)∈R×R|x, y ≥0, x+y≤1}andF(∆2) = {f|f :

∆₂ →R}. Form a non zero natural number, let the operatorsB_m : F(∆₂)→ F(∆₂), defined for any functionf ∈ F(∆₂)by

(2.1) (B_mf)(x, y) = X

k,j=0 k+j≤m

p_m,k,j(x, y)f k

m, j m

for any(x, y)∈∆₂, where

(2.2) p_m,k,j(x, y) = m!

k!j!(m−k−j)!x^ky^j(1−x−y)^m−k−j. The operators are named Bernstein bivariate polynomials (see [8]).

Lemma 2.1. The operators(B_m)m≥1 are linear and positive onF(∆₂).

Proof. The proof follows immediately.

Forma non zero natural number, let the GBS operator of Bernstein bivariate polynomialsU B_m (see [1]),U B_m :C_b(∆₂)→B(∆₂)defined for any function f ∈C_b(∆₂)and any(x, y)∈∆₂ by

(U B_mf)(x, y) (2.3)

= (B_m(f(x,∗) +f(·, y)−f(·,∗))) (x, y)

= X

k,j=0 k+j≤m

p_m,k,j(x, y)

f

x, j m

+f

k m, y

−f k

m, j m

.

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Lemma 2.2. The operators(B_m)m≥1 verify for any(x, y)∈∆₂ the following:

(B_me₀₀)(x, y) = 1;

(2.4)

B_m(· −x)²

(x, y) = x(1−x)

m ;

(2.5)

B_m(∗ −y)²

(x, y) = y(1−y)

m ;

(2.6)

(2.7) B_m(· −x)²(∗ −y)² (x, y)

= 3(m−2)

m³ x²y²−m−2

m³ (x²y+xy²) + m−1 m³ xy;

(2.8) B_m(· −x)⁴(∗ −y)² (x, y)

=−5(3m²−26m+ 24)

m⁵ x⁴y²+6(3m²−26m+ 24) m⁵ x³y²

− 6(m²−7m+ 6)

m⁵ x³y− 3m²−41m+ 42 m⁵ x²y² + 3m²−26m+ 24

m⁵ x⁴y+3m² −17m+ 14 m⁵ x²y

− m−2

m⁵ xy²+m−1 m⁵ xy and

(2.9) B_m(· −x)²(∗ −y)⁴ (x, y)

=−5(m²−26m+ 24)

m⁵ x²y⁴+ 6(3m²−26m+ 24) m⁵ x²y³

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− 6(m²−7m+ 6)

m⁵ xy³− 3m²−41m+ 42 m⁵ x²y² + 3m²−26m+ 24

m⁵ xy⁴+3m² −17m+ 14 m⁵ xy²

− m−2

m⁵ x²y+m−1 m⁵ xy for any non zero natural numberm.

Proof. Let(x, y)∈∆₂andmbe a non zero natural number. We have (B_me₀₀)(x, y) = X

k,j=0 k+j≤m

m!

k!j!(m−k−j)!x^ky^j(1−x−y)^m−k−j

= (x+y+ 1−x−y)^m = 1, so (2.4) holds,

(B_me₁₀)(x, y) = X

k,j=0 k+j≤m

m!

k!j!(m−k−j)!x^ky^j(1−x−y)^m−k−j k m

=x X

k=1,j=0 k+j≤m

(m−1)!

(k−1)!j!(m−k−j)!x^k−1y^j(1−x−y)^m−k−j

=x, it results that

(2.10) (Bme10)(x, y) =x

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and similarly

(2.11) (B_me₀₁)(x, y) =y.

In the same way, using the formulas k² =k(k−1) +k,

k³ =k(k−1)(k−2) + 3k(k−1) +k,

k⁴ =k(k−1)(k−2)(k−3) + 6k(k−1)(k−2) + 7k(k−1) +k, we obtain

(Bme20)(x, y) = m−1

m x² + 1 mx, (2.12)

(B_me₃₀)(x, y) = (m−1)(m−2)

m² x³+3(m−1)

m² x²+ 1 m² x, (2.13)

(2.14) (B_me₄₀)(x, y) = (m−1)(m−2)(m−3)

m³ x⁴

+6(m−1)(m−2)

m³ x³+7(m−1)

m³ x²+ 1 m³x and similarly the relations(B_me₀₂)(x, y),(B_me₀₃)(x, y),(B_me₀₄)(x, y).

We have (B_me₁₁)(x, y)

= m−1

m y X

k=0,j=1 k+j≤m

(m−1)!

k!(j−1)!(m−k−j)!x^ky^j−1(1−x−y)^m−k−j k m−1

= m−1

m y(Bm−1e₁₀)(x, y),

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(B_me₂₁)(x, y)

=

m−1 m

2

y X

k=0,j=1 k+j≤m

(m−1)!

k!(j−1)!(m−k−j)!x^ky^j−1(1−x−y)^m⁻^k⁻^j k

m−1 2

=

m−1 m

2

y(B_m−1e₂₀)(x, y),

and in the same way, we write (B_me₃₁)(x, y), (B_me₄₁)(x, y), (B_me₃₂)(x, y), (B_me₄₂)(x, y). Taking (2.12) - (2.14) into account, we obtain

(B_me₁₁)(x, y) = m−1 m xy, (2.15)

(B_me₂₁)(x, y) = (m−1)(m−2)

m² x²y+ m−1 m² xy, (2.16)

(2.17) (B_me₃₁)(x, y) = (m−1)(m−2)(m−3) m³ x³y +3(m−1)(m−2)

m³ x²y+ m−1 m³ xy,

(2.18) (Bme41)(x, y) = (m−1)(m−2)(m−3)(m−4)

m⁴ x⁴y

+6(m−1)(m−2)(m−3)

m⁴ x³y

+ 7(m−1)(m−2)

m⁴ x²y+ m−1 m⁴ xy,

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(2.19) (Bme22)(x, y) = (m−1)(m−2)(m−3) m³ x²y² +(m−1)(m−2)

m³ (x²y+xy²) + m−1 m³ xy,

(2.20) (B_me₃₂)(x, y) = (m−1)(m−2)(m−3)(m−4)

m⁴ x³y²

+ (m−1)(m−2)(m−3)

m⁴ x³y+3(m−1)(m−2)(m−3) m⁴ x²y² +3(m−1)(m−2)

m⁴ x²y+ (m−1)(m−2)

m⁴ xy²+ m−1 m⁴ xy,

(B_me₄₂)(x, y) (2.21)

= (m−1)(m−2)(m−3)(m−4)(m−5)

m⁵ x⁴y²

+ (m−1)(m−2)(m−3)(m−4)

m⁵ x⁴y

+ 6(m−1)(m−2)(m−3)(m−4)

m⁵ x³y²

+ 6(m−1)(m−2)(m−3)

m⁵ x³y

+ 7(m−1)(m−2)(m−3) m⁵ x²y² + 7(m−1)(m−2)

m⁵ x²y+(m−1)(m−2)

m⁵ xy²+m−1 m⁵ xy

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and similarly the relations (B_me₁₂)(x, y), (B_me₁₃)(x, y), (B_me₁₄)(x, y), (Bme23)(x, y),(Bme24)(x, y).

Now, we have

(Bm(· −x)²)(x, y) = (Bme20)(x, y)−2x(Bme10)(x, y) +x²(Bme02)(x, y), (B_m(·−x)²(∗−y)²)(x, y) = (B_me₂₂)(x, y)−2y(B_me₂₁)(x, y)+y²(B_me₂₀)(x, y)

−2x(Bme12)(x, y) + 4xy(Bme11)(x, y)−2xy²(Bme10)(x, y) +x²(B_me₀₂)(x, y)−2x²y(B_me₀₁)(x, y) +x²y²(B_me₀₀)(x, y), (Bm(· −x)⁴(∗ −y)²)(x, y)

= (B_me₄₀)(x, y)−2y(B_me₄₁)(x, y) +y²(B_me₄₀)(x, y)

−4x(B_me₃₂)(x, y) + 8xy(B_me₃₁)(x, y)−4xy²(B_me₃₀)(x, y) + 6x²(B_me₂₂)(x, y)−12x²y(B_me₂₁)(x, y) + 6x²y²(B_me₂₀)(x, y)

−4x³(B_me₁₂)(x, y) + 8x³y(B_me₁₁)(x, y)−4x³y²(B_me₁₀)(x, y) +x⁴(B_me₀₂)(x, y)−2x⁴y(B_me₀₁)(x, y) +x⁴y²(B_me₀₀)(x, y) and taking (2.9) – (2.21) into account, we obtain (2.5), (2.7) and (2.8). Similarly we obtain (2.9).

Lemma 2.3. The operators(B_m)_m≥1 verify for any(x, y) ∈ ∆₂ the following inequalities:

(B_m(· −x)²)(x, y)≤ 1 4m, (2.22)

(B_m(∗ −y)²)(x, y)≤ 1 4m, (2.23)

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for any non zero natural numberm,

(2.24) B_m(· −x)²(∗ −y)²

(x, y)≤ 9 4m² , for any natural numberm,m≥2,

B_m(· −x)⁴(∗ −y)²

(x, y)≤ 9 m³, (2.25)

B_m(· −x)²(∗ −y)⁴

(x, y)≤ 9 m³, (2.26)

for any natural numberm,m≥8.

Proof. Becausex(1−x)≤ ¹₄ for anyx∈[0,1], (2.22) and (2.23) results.

From (2.7), we have

B_m(· −x)²(∗ −y)² (x, y)

= 2(m−2)

m³ x²y²+ m−2

m³ x(1−x)y(1−y) + 1 m³ xy

≤ 2(m−2)

m³ + m−2 16m³ + 1

m³

= 33m−50 16m³ , from where (2.24) results.

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From (2.8), we have Bm(· −x)⁴(∗ −y)²

(x, y)

= 6(3m²−26m+ 24)

m⁵ x³y²(1−x) + 3m²−26m+ 24

m⁵ x⁴y(y+ 1)

− 6(m²−7m+ 6)

m⁵ x³y+ 3m²−17m+ 14

m⁵ x²y(1−y) +24m−28

m⁵ x²y²+m−2

m⁵ xy(1−y) +xy.

But

3m²−26m+ 24

m⁵ x⁴y(y+ 1)≤23m²−26m+ 24 m⁵ x²y

= 6m²−42m+ 36

m⁵ x²y− 10m−12 m⁵ x²y

≤ 6m²−42m+ 36

m⁵ x²y− 10m−12 m⁵ x³y² and then, from the inequalities above, we obtain

(2.27) B_m(· −x)⁴(∗ −y)² (x, y)

≤ 6(3m²−26m+ 24)

m⁵ x³y²(1−x) + 6m²−42m+ 36

m⁵ x²y(1−y) + 3m²−17m+ 14

m⁵ x²y(1−y) + 10m−12

m⁵ x²y²(1−y) +14m−16

m⁵ x²y²+m−2

m⁵ xy(1−y) +xy.

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Becausex(1−x)≤ ¹₄,y(1−y)≤ ¹₄,xy≤1for anyx, y ∈[0,1], from (2.27) we have

B_m(· −x)⁴(∗ −y)² (x, y)

≤ 6(3m²−26m+ 24)

4m⁵ + 6m²−42m+ 36 4m⁵ + 3m²−17m+ 14

4m⁵ +10m−12

4m⁵ + 14m−16

m⁵ +m−2 4m⁵ + 1

= 27m²−148m+ 170

m⁵ ,

from where (2.25) results.

Theorem 2.4. Let the function f ∈ C_b(∆₂). Then, for any (x, y) ∈ ∆₂, any natural numberm,m≥2, we have

(2.28) |f(x, y)−(U Bmf)(x, y)|

≤

1 +δ₁⁻¹ 1 2√

m +δ⁻¹₂ 1 2√

m +δ⁻¹₁ δ₂⁻¹ 3 2m

ω_mixed(f;δ₁, δ₂) for anyδ₁,δ₂ >0and

(2.29) |f(x, y)−(U B_mf)(x, y)| ≤ 7 2ω_mixed

f; 1

√m, 1

√m

.

Proof. For the first inequality we apply Theorem 1.1and Lemma 2.3. The inequality (2.29) is obtained from (2.28) by choosingδ₁ =δ₂ = ^√¹_m .

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Corollary 2.5. Iff ∈C_b(∆₂), then

(2.30) lim

m→∞(U B_mf)(x, y) =f(x, y) uniformly on∆₂.

Proof. Because f ∈ C_b(∆₂), there results that f is uniform B-continuous on

∆₂ and then lim

m→∞ω_mixed

f;^√¹_m,^√¹_m

= 0(see [2] or [3]). From (2.29), there results the conclusion.

Theorem 2.6. Let the functionf ∈D_b(∆₂)withD_Bf ∈B(∆₂). Then for any (x, y)∈∆₂, any natural numberm,m≥8, we have

(2.31) |f(x, y)−(U B_mf)(x, y)| ≤ 9

2m kD_bfk∞

+ 3

2m+δ₁⁻¹ 3 m√

m+δ⁻¹₂ 3 m√

m+δ₁⁻¹δ₂⁻¹ 9 4m²

ω_mixed(D_Bf;δ₁, δ₂) for anyδ₁, δ₂ >0and

(2.32) |f(x, y)−(U B_mf)(x, y)|

≤ 3 4m

6kD_Bfk∞+ 13ω_mixed

D_Bf; 1

√m

√1 m

.

Proof. It results from Theorem1.2and Lemma2.3.

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References

[1] C. BADEA AND C. COTTIN, Korovkin-type theorems for generalised boolean sum operators, Colloquia Mathematica Societatis János Bolyai, 58, Approximation Theory, Kecskemét (Hunagary), 1990, 51–67.

[2] C. BADEA, Modul de continuitate în sens Bögel ¸si unele aplica¸tii în aproximarea printr-un operator Bernstein, Studia Univ. "Babe¸s-Bolyai", Ser. Math.-Mech., 18(2) (1973), 69–78 (Romanian).

[3] C. BADEA, I. BADEA, C. COTTIN AND H.H. GONSKA, Notes on the degree of approximation ofB-continuous andB-differentiable functions, J. Approx. Theory Appl., 4 (1988), 95–108.

[4] D. B ˘ARBOSU, Aproximarea func¸tiilor de mai multe variabile prin sume booleene de operatori liniari de tip interpolator, Ed. Risoprint, Cluj- Napoca, 2002 (Romanian).

[5] K. BÖGEL, Mehrdimensionale Differentiation von Funtionen mehrerer Veränderlicher, J. Reine Angew. Math., 170 (1934), 197–217.

[6] K. BÖGEL, Über die mehrdimensionale differentiation, integration und beschränkte variation, J. Reine Angew. Math., 173 (1935), 5–29.

[7] K. BÖGEL, Über die mehrdimensionale differentiation, Jber. DMV, 65 (1962), 45–71.

[8] G.G. LORENTZ, Bernstein Polynomials, University of Toronto Press, Toronto, 1953.

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[9] M. NICOLESCU, Contribu¸tii la o analiz˘a de tip hiperbolic a planului, St.

Cerc. Mat., III, 1-2, 1952, 7–51 (Romanian).

[10] M. NICOLESCU, Analiz˘a Matematic˘a, II, E. D. P. Bucure¸sti, 1980 (Ro- manian).

[11] O.T. POP, Approximation ofB-differentiable functions by GBS operators (to appear in Anal. Univ. Oradea).

[12] D.D. STANCU, Gh. COMAN, O. AGRATINIANDR. TRÎMBI ¸TA ¸S, Anal- iz˘a Numeric˘a ¸si Teoria Aproxim˘arii, I, Presa Universitar˘a Clujean˘a, Cluj- Napoca, 2001 (Romanian).