accepted 18 March, 2009 Communicated by S.S Dragomir ABSTRACT

APPROXIMATION OF B-CONTINUOUS AND B-DIFFERENTIABLE FUNCTIONS BY GBS OPERATORS DEFINED BY INFINITE SUM

OVIDIU T. POP

NATIONALCOLLEGE"MIHAIEMINESCU"

5 MIHAIEMINESCUSTREET

SATUMARE440014, ROMANIA

ovidiutiberiu@yahoo.com

Received 27 June, 2008; accepted 18 March, 2009 Communicated by S.S Dragomir

ABSTRACT. In this paper we start from a class of linear and positive operators defined by infinite sum. We consider the associated GBS operators and we give an approximation ofB-continuous andB-differentiable functions with these operators. Through particular cases, we obtain state- ments verified by the GBS operators of Mirakjan-Favard-Szász, Baskakov and Meyer-König and Zeller.

Key words and phrases: Linear positive operators, GBS operators,B-continuous andB-differentiable functions, approxima- tion ofB-continuous andB-differentiable functions by GBS operators.

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

1. INTRODUCTION

In this section, we recall some notions and results which we will use in this article. LetNbe the set of positive integers andN⁰ =N∪ {0}.

In the following, letX andY be real intervals.

A functionf : X ×Y → R is called aB-continuous function in(x₀, y₀) ∈ X ×Y if and only if

(x,y)→(xlim₀,y0)∆f[(x, y),(x₀, y₀)] = 0, where

∆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-continuous function on X×Y if and only if it is B-continuous in any point ofX×Y.

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

(x,y)→(xlim₀,y0)

∆f[(x, y),(x, y₀)]

(x−x₀)(y−y₀) .

180-08

This limit is called theB-differential off in the point(x₀, y₀)and is noted byD_Bf(x₀, y₀).

A functionf :X ×Y →Ris called aB-differentiable function onX×Y if and only if it isB-differentiable in any point ofX×Y.

The definition of B-continuity and B-differentiability was introduced by K. Bögel in the papers [8] and [9].

The functionf : X ×Y → RisB-bounded onX×Y if and only if there existsk > 0so that|∆f[(x, y),(s, t)]| ≤kfor any(x, y),(s, t)∈X×Y.

We shall use the function sets B(X×Y) = {f|f : X ×Y → R, f bounded on X ×Y} with the usual sup-normk·k_∞,Bb(X×Y) = {f|f :X×Y →R,f isB-bounded onX×Y}, Cb(X×Y) = {f|f : X×Y → R,f isB-continuous onX×Y}andDb(X×Y) = {f|f : X×Y →R,f isB-differentiable onX×Y}.

Letf ∈B_b(X×Y). The functionω_mixed(f;·,·) : [0,∞)×[0,∞)→Rdefined by 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.

Theorem 1.1. LetXandY be compact real intervals andf ∈B_b(X×Y).

Then lim

δ1,δ2→0ω_mixed(f;δ₁, δ₂) = 0if and only iff ∈C_b(X×Y).

For anyx∈Xconsider the functionϕx :X →R, defined byϕx(t) = |t−x|, for anyt∈X.

For additional information, see the following papers: [1], [3], [15] and [19].

Let m ∈ N and the operator S_m : C₂([0,∞)) → C([0,∞))defined for any functionf ∈ C₂([0,∞))by

(1.1) (S_mf)(x) = e^−mx

∞

X

k=0

(mx)^k k! f

k m

,

for anyx∈[0,∞), whereC₂([0,∞)) =

f ∈ C([0,∞)) : lim

x→∞

f(x)

1+x² exists and is finite . The operators(Sm)m≥1 are called the Mirakjan-Favard-Szász operators, introduced in 1941 by G.

M. Mirakjan in the paper [13].

These operators were intensively studied by J. Favard in 1944 in the paper [11] and O. Szász in the paper [20].

From [18], the following three lemmas result.

Lemma 1.2. For anym ∈N, we have that

(1.2) S_mϕ²_x

(x) = x m,

(1.3) S_mϕ⁴_x

(x) = 3mx²+x m³ for anyx∈[0,∞)and

(1.4) S_mϕ²_x

(x)≤ a m,

(1.5) S_mϕ⁴_x

(x)≤ a(3a+ 1) m² for anyx∈[0, a], wherea >0.

Letm ∈ N and the operatorV_m : C₂([0,∞)) → C([0,∞)), defined for any function f ∈ C₂([0,∞))by

(1.6) (V_mf)(x) = (1 +x)^−m

∞

X

k=0

m+k−1 k

x 1 +x

k

f k

m

for anyx∈[0,∞).

The operators(V_m)m≥1are called Baskakov operators, introduced in 1957 by V. A. Baskakov in the paper [5].

Lemma 1.3. For anym ∈N, we have that

(1.7) V_mϕ²_x

(x) = x(1 +x)

m ,

(1.8) V_mϕ⁴_x

(x) = 3(m+ 2)x⁴+ 6(m+ 2)x³+ (3m+ 7)x²+x m³

for anyx∈[0,∞)and

(1.9) V_mϕ²_x

(x)≤ a(1 +a)

m ,

(1.10) V_mϕ⁴_x

(x)≤ a(9a³+ 18a²+ 10a+ 1) m²

for anyx∈[0, a], wherea >0.

W. Meyer-König and K. Zeller have introduced a sequence of linear positive operators in paper [12]. After a slight adjustment, given by E. W. Cheney and A. Sharma in [10], these operators take the formZm :B([0,1))→C([0,1)), defined for any functionf ∈B([0,1))by

(1.11) (Z_mf)(x) =

∞

X

k=0

m+k k

(1−x)^m+1x^kf k

m+k

, for anym∈Nand for anyx∈[0,1).

These operators are called the Meyer-König and Zeller operators.

In the following we considerZ_m :C([0,1]) →C([0,1]), for anym∈N. Lemma 1.4. For anym ∈Nand anyx∈[0,1], we have that

(1.12) Z_mϕ²_x

(x)≤ x(1−x)² m+ 1

1 + 2x m+ 1

and

(1.13) Zmϕ²_x

(x)≤ 2 m.

The inequality of Corollary 5 from [4], in the condition (1.14) becomes inequality (1.15).

Inequality (1.16) is demonstrated in [16].

Theorem 1.5. Let L : 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. Supposing that the operatorLhas the property

(1.14) L(· −x)²ⁱ(∗ −y)^2j

(x, y) = L(· −x)²ⁱ

(x, y) L(∗ −y)^2j (x, y)

for any(x, y) ∈X×Y and anyi, j ∈ {1,2}, where "·" and "∗" stand for the first and second variable. Then:

(i) For any functionf ∈C_b(X×Y), any(x, y)∈X×Y and anyδ₁, δ₂ >0, we have that (1.15) |f(x, y)−(U Lf)(x, y)| ≤ |f(x, y)||1−(Le₀₀)(x, y)|

+h

(Le₀₀)(x, y)+δ₁⁻¹p

(L(·−x)²)(x, y)+δ₂⁻¹p

(L(∗−y)²)(x, y) +δ₁⁻¹δ₂⁻¹p

(L(· −x)²)(x, y)(L(∗ −y)²)(x, y)i

ω_mixed(f;δ₁, δ₂).

(ii) For anyf ∈D_b(X×Y)withD_Bf ∈B(X×Y), any(x, y)∈X×Y and anyδ₁, δ₂ >0, we have that

|f(x, y)−(U Lf)(x, y)|

(1.16)

≤ |f(x, y)||1−(Le00)(x, y)|+ 3kDBfk∞

p(L(· −x)²)(x, y)(L(∗ −y)²)(x, y) +hp

(L(· −x)²)(x, y)(L(∗ −y)²)(x, y) +δ⁻¹₁ p

(L(· −x)⁴)(x, y)(L(∗ −y)²)(x, y) +δ⁻¹₂ p

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

ω_mixed(D_Bf;δ₁, δ₂).

2. PRELIMINARIES

LetI, J, K ⊂ R be intervals, J ⊂ K andI ∩J 6= ∅. We consider the sequence of nodes ((x_m,k)k∈N0)_m≥1 so thatx_m,k ∈ I ∩J, k ∈N0, m ∈ Nand the functionsϕ_m,k :K →Rwith the property thatϕ_m,k(x)≥0, for anyk∈N0,m ∈Nandx∈J.

Definition 2.1. Ifm∈N, we define the operatorL^∗_m :E(I)→F(K)by

(2.1) (L^∗_mf) (x) =

∞

X

k=0

ϕ_m,k(x)f(x_m,k)

for any functionf ∈E(I)and anyx∈K, whereE(I)andF(K)are subsets of the set of real functions defined onI, respectively onK.

Proposition 2.1. The operators(L^∗_m)_m≥1 are linear and positive onE(I∩J).

Proof. The proof follows immediately.

Definition 2.2. If m, n ∈ N, the operator L^∗_m,n : E(I ×I) → F(K ×K) defined for any functionf ∈E(I×I)and any(x, y)∈K×Kby

(2.2) L^∗_m,nf

(x, y) =

∞

X

k=0

∞

X

j=0

ϕ_m,k(x)ϕ_n,j(y)f(x_m,k, x_n,j) is called the bivariate operator ofL^∗- type.

Proposition 2.2. The operators L^∗_m,n

m,n≥1 are linear and positive onE[(I×I)∩(J×J)].

Proof. The proof follows immediately.

Definition 2.3. If m, n ∈ N, the operator U L^∗_m,n : E(I ×I) → F(K ×K)defined for any functionf ∈E(I×I)and any(x, y)∈K×Kby

(2.3) U L^∗_m,nf

(x, y) =

∞

X

k=0

∞

X

j=0

ϕ_m,k(x)ϕ_n,j(y)

f(x_m,k, y) +f(x, x_n,j)−f(x_m,k, x_n,j) is called a GBS operator ofL^∗- type.

3. MAINRESULTS

Lemma 3.1. For anym, n∈N,i, j ∈N⁰ and(x, y)∈K×K, the identity (3.1) L^∗_m,n(· −x)²ⁱ(∗ −y)^2j

(x, y) = L^∗_m(· −x)²ⁱ

(x) L^∗_n(∗ −y)^2j (y) holds.

Proof. We have that

L^∗_m,n(· −x)²ⁱ(∗ −y)^2j

(x, y) =

∞

X

k=0

∞

X

j=0

ϕ_m,k(x)ϕ_n,j(y)(x_m,k−x)²ⁱ(x_n,j −y)^2j

=

∞

X

k=0

ϕ_m,k(x)(x_m,k−x)²ⁱ

∞

X

j=0

ϕ_n,j(y)(x_n,j −y)^2j

= L^∗_m(· −x)²ⁱ

(x) L^∗_n(∗ −y)^2j (y),

so (3.1) holds.

For the operators constructed in this section, we note thatδ_m(x) = p

(L^∗_mϕ²_x) (x), δ_m,x = p(L^∗_mϕ⁴_x) (x), wherex∈I∩J,m∈N,m 6= 0.

Then, by taking Lemma 3.1 into account, Theorem 1.5 becomes:

Theorem 3.2.

(i) For any function f ∈ C_b(I×I), any(x, y) ∈ (I ×I)∩(J ×J), anym, n ∈ N, any δ₁, δ₂ >0, we have that

(3.2) |f(x, y)−(U L^∗_m,nf)(x, y)|

≤ |f(x, y)||1−(Le₀₀)(x, y)|+ (Le₀₀)(x, y) +δ₁⁻¹δ_m(x) +δ₂⁻¹δ_n(y) +δ⁻¹₁ δ₂⁻¹δ_m(x)δ_n(y)

ω_mixed(f;δ₁, δ₂).

(ii) For any functionf ∈D_b(I×I)withD_Bf ∈B(I×I), any(x, y)∈(I×I)∩(J×J), anym, n∈N, anyδ₁, δ₂ >0, we have that

(3.3) |f(x, y)−(U L^∗f)(x, y)| ≤ |f(x, y)||1−(Le₀₀)(x, y)|

+ 3kD_Bfk∞δ_m(x)δ_n(y) +

δ_m(x)δ_n(y) +δ₁⁻¹δ_m,xδ_n(y) +δ₂⁻¹δ_m(x)δ_n,y+δ₁⁻¹δ₂⁻¹δ_m²(x)δ_n²(y)

ω_mixed(D_Bf;δ₁, δ₂).

In the following, we give examples of operators and of the associated GBS operators.

Application 1. IfI = J =K = [0,∞), E(I) = C₂([0,∞)), F(K) = C([0,∞)),ϕ_m,k(x) = e^−mx(mx)_k!^k , xm,k = _m^k , x ∈ [0,∞), m, k ∈ N⁰, m 6= 0, then we obtain the Mirakjan-Favard- Szász operators.

Theorem 3.3. Leta, b∈R,a >0andb >0. Then:

(i) For any functionf ∈C([0,∞)×[0,∞)), any(x, y)∈[0, a]×[0, b]andm, n∈N, we have that

(3.4) |f(x, y)−(U S_m,nf)(x, y)| ≤ 1+√ a

1+√ b

ω_mixed

f; 1

√m , 1

√n

.

(ii) For any functionf ∈D_b([0,∞)×[0,∞))∩C([0,∞)×[0,∞))withD_Bf ∈B([0, a]× [0, b]), any(x, y)∈[0, a]×[0, b], anym, n∈N, we have that

(3.5) |f(x, y)−(U S_m,nf)(x, y)| ≤√ ab

"

3kD_Bfk∞+ 1 +√

3a+ 1

+√

3b+ 1 +

√ ab

ω_mixed

DBf; 1

√m , 1

√n #

√1 mn. Proof. It results from Theorem 3.2, by choosingδ₁ = ^√1_m,δ₂ = ^√1_n and Lemma 1.2.

Theorem 3.4. Iff ∈C([0,∞)×[0,∞)), then the convergence

(3.6) lim

m,n→∞(U Sm,nf)(x, y) = f(x, y) is uniform on any compact[0, a]×[0, b], wherea, b >0.

Proof. It results from Theorem 1.1 and Theorem 3.3.

Application 2. IfI = J =K = [0,∞), E(I) = C₂([0,∞)), F(K) = C([0,∞)),ϕ_m,k(x) = (1 + x)^{−m m+k−1}_{k x}

1+x

k

, x_m,k = _m^k , x ∈ [0,∞), m, k ∈ N0, m 6= 0, then we obtain the Baskakov operators.

Theorem 3.5. Leta, b∈R,a >0andb >0. Then:

(i) For any functionf ∈C([0,∞)×[0,∞)), any(x, y)∈[0, a]×[0, b]and anym, n∈N, we have that

(3.7) |f(x, y)−(U V_m,nf)(x, y)|

≤ 1 +p

a(1 +a) 1 +p

b(1 +b) ω_mixed

f; 1

√m , 1

√n

. (ii) For any functionf ∈D_b([0,∞)×[0,∞))∩C([0,∞)×[0,∞))withD_Bf ∈B([0, a]×

[0, b]), any(x, y)∈[0, a]×[0, b], anym, n∈N, we have that (3.8) |f(x, y)−(U V_m,nf)(x, y)| ≤p

ab(1 +a)(1 +b) (

3kD_Bk∞

+h 1 +√

9a³+ 18a²+ 10a+ 1 +√

9b³+ 18b²+ 10b+ 1 +p

ab(1 +a)(1 +b)i ω_mixed

D_Bf; 1

√m , 1

√n )

√1 mn. Proof. It results from Theorem 3.2, by choosingδ₁ = ^√1_m,δ₂ = ^√1_n and Lemma 1.3.

Theorem 3.6. Iff ∈C([0,∞)×[0,∞)), then the convergence

(3.9) lim

m,n→∞(U V_m,nf)(x, y) =f(x, y) is uniform on any compact[0, a]×[0, b], wherea, b >0.

Proof. It results from Theorem 1.1 and Theorem 3.5.

Application 3. If I = J = K = [0,1], E(I) = F(K) = C([0,1]), ϕ_m,k(x) = ^m+k_k (1− x)^m+1x^k,x_m,k = _m^k ,x∈[0,1],m, k ∈N0,m 6= 0, then we obtain the Meyer-König and Zeller operators.

Theorem 3.7. For any function f ∈ C([0,1]×[0,1]), any (x, y) ∈ [0,1] ×[0,1] and any m, n∈N, we have that

(3.10) |f(x, y)−(U Z_m,nf)(x, y)| ≤(3 + 2√

2)ω_mixed

f; 1

√m, 1

√n

.

Proof. It results from Theorem 3.2, by choosingδ₁ = ^√1_m,δ₂ = ^√1_n and Lemma 1.4.

Theorem 3.8. Iff ∈C([0,1]×[0,1]), then the convergence

(3.11) lim

m,n→∞(U Z_m,nf)(x, y) =f(x, y) is uniform on[0,1]×[0,1].

Proof. It results from Theorem 1.1 and Theorem 3.7.

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