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

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Approximation ofB-Continuous andB-Differentiable Functions

Ovidiu T. Pop vol. 10, iss. 1, art. 7, 2009

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APPROXIMATION OF B-CONTINUOUS AND B-DIFFERENTIABLE FUNCTIONS BY GBS

OPERATORS DEFINED BY INFINITE SUM

OVIDIU T. POP

National College "Mihai Eminescu"

5 Mihai Eminescu Street Satu Mare 440014, Romania EMail:ovidiutiberiu@yahoo.com

Received: 27 June, 2008

Accepted: 18 March, 2009

Communicated by: S.S Dragomir

2000 AMS Sub. Class.: 41A10, 41A25, 41A35, 41A36, 41A63

Key words: Linear positive operators, GBS operators, B-continuous and B-differentiable functions, approximation of B-continuous and B-differentiable functions by GBS operators.

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 and B-differentiable functions with these operators.

Through particular cases, we obtain statements verified by the GBS operators of Mirakjan-Favard-Szász, Baskakov and Meyer-König and Zeller.

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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, letXandY be real intervals.

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

lim

(x,y)→(x0,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 onX ×Y if and only if it isB-continuous in any point ofX×Y.

A function f : X ×Y → R is 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) .

This limit is called the B-differential of f in the point(x₀, y₀)and is noted by D_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 ofB-continuity andB-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 exists k >0so that|∆f[(x, y),(s, t)]| ≤kfor any(x, y),(s, t)∈X×Y.

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We shall use the function setsB(X×Y) = {f|f :X×Y → R, f bounded on X ×Y} with the usual sup-normk·k_∞, B_b(X ×Y) = {f|f : X ×Y → R, f is B-bounded onX×Y}, C_b(X×Y) = {f|f :X×Y → R, f isB-continuous on X×Y}andD_b(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. LetX andY be compact real intervals andf ∈Bb(X×Y).

Then lim

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

For any x ∈ X consider the functionϕ_x : X → R, defined byϕ_x(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 any x ∈ [0,∞), whereC₂([0,∞)) =

f ∈ C([0,∞)) : lim

x→∞

f(x)

1+x² exists and is finite . The operators(S_m)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.

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Lemma 1.2. For anym ∈N, we have that

(1.2) Smϕ²_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.

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

(1.6) (Vmf)(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≥1 are 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 ,

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(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 form Z_m : 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

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and

(1.13) Z_mϕ²_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. 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. 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)+δ⁻¹₁ p

(L(·−x)²)(x, y)+δ⁻¹₂ p

(L(∗−y)²)(x, y) +δ₁⁻¹δ⁻¹₂ p

(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 any

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δ₁, 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).

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2. Preliminaries

Let I, J, K ⊂ R be intervals, J ⊂ K and I ∩J 6= ∅. We consider the sequence of nodes((x_m,k)k∈N0)_m≥1 so that x_m,k ∈ I∩J, k ∈ N0,m ∈ N and the functions ϕm,k : K → Rwith the property that ϕm,k(x) ≥ 0, for any k ∈ N⁰, m ∈ N and x∈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 2.2. The operators(L^∗_m)_m≥1are linear and positive onE(I ∩J).

Proof. The proof follows immediately.

Definition 2.3. Ifm, 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 2.4. The operators L^∗_m,n

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

Proof. The proof follows immediately.

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Definition 2.5. Ifm, n∈N, the operatorU 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.

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3. Main Results

Lemma 3.1. For anym, n∈N,i, j ∈N0and(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), δm,x=p

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

Then, by taking Lemma3.1into account, Theorem1.5becomes:

Theorem 3.2.

(i) For any functionf ∈C_b(I×I), any(x, y)∈(I×I)∩(J×J), anym, n∈N, anyδ₁, 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) +δ⁻¹₁ δ_m(x) +δ₂⁻¹δ_n(y) +δ₁⁻¹δ⁻¹₂ δ_m(x)δ_n(y)

ω_mixed(f;δ₁, δ₂).

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(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) +δ₂⁻¹δm(x)δn,y+δ₁⁻¹δ⁻¹₂ δ_m²(x)δ²_n(y)

ω_mixed(DBf;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 , x_m,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 function f ∈ C([0,∞)× [0,∞)), any (x, y) ∈ [0, a] × [0, b] and m, 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 ∈

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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

D_Bf; 1

√m, 1

√n #

√1 mn. Proof. It results from Theorem3.2, by choosing δ1 = ^√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 S_m,nf)(x, y) = f(x, y) is uniform on any compact[0, a]×[0, b], wherea, b >0.

Proof. It results from Theorem1.1and Theorem3.3.

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 any m, 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

.

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(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 Theorem3.2, by choosing δ1 = ^√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.

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 functionf ∈C([0,1]×[0,1]), any(x, y)∈[0,1]×[0,1]and anym, n∈N, we have that

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

2)ω_mixed

f; 1

√m , 1

√n

.

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Proof. It results from Theorem3.2, by choosing δ₁ = ^√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].

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