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Proc. Indian Acad. Sci. (Math. Sci.) Vol. 117, No. 3, August 2003, pp. 387–399. Printed in India

Approximation of functions of two variables by certain linear positive operators

arXiv:0709.3345v1 [math.CA] 21 Sep 2007

FATMA TAS ? DELEN? , ALI OLGUN? and ¨ GULEN BAS ? CANBAZ-TUNCA?

of Mathematics, Faculty of Science, Ankara University, Tandogan 06100, Ankara, Turkey ? Department of Mathematics, Faculty of Science and Arts, Kirikkale University, Yahs ?ihan 71450, Kirikkale, Turkey E-mail: tasdelen@science.ankara.edu.tr; aolgun@kku.edu.tr; tunca@science.ankara.edu.tr MS received 24 February 2006 Abstract. We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also ?nd the order of this approximation by using modulus of continuity. Moreover we de?ne an rth order generalization of these operators and observe its approximation properties. Furthermore, we study the convergence of the linear positive operators in a weighted space of functions of two variables and ?nd the rate of this convergence using weighted modulus of continuity. Keywords. Linear positive operator; modulus of continuity; order of approximation; polynomial weighted spaces.

? Department

1. Introduction

Let f ∈ C([0, 1]). The well-known Bernstein polynomial of degree n, denoted by Bn ( f ; x) is Bn ( f ; x) := where pn,k (x) = n k x (1 ? x)n?k k (1.1)

k =0

∑ pn,k (x) f

n

k , n

n ∈ N ={1, 2, . . . },

and x ∈ [0, 1] [4]. Let x ∈ [0, ∞) and f ∈ C([0, ∞)). Szazs–Mirakyan operators, denoted by Sn ( f ; x) are Sn ( f ; x) := where qn,k (x) = e?nx (nx)k . k! (1.2) 387

k =0

∑ qn,k (x) f

∞

k , n

n ∈ N,

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Fatma Tas ?delen et al

In [3] approximation properties of Sn ( f ; x) in weighted spaces were studied. Some works by Szazs–Mirakyan or modi?ed Szazs–Mirakyan operators may be found in [12,7,18] and references therein. Stancu [16] introduced the following generalization of the Bernstein polynomials. Let (α ,β ) f ∈ C([0, 1]). Stancu operators, denoted by (Pn f ) are (Pn

(α ,β )

f ) :=

k =0

∑ pn,k (x) f

n

k+α n+β

,

n ∈ N,

where pn,k (x) are the polynomials given by (1.1), α , β are positive real numbers satisfying 0 ≤ α ≤ β. Taking the operators, given above, into account we now introduce certain linear positive operators of functions of two variables as follows: α ,β Let f ∈ C(R ), R := [0, 1] × [0, ∞) and let the linear positive operators Lmi,n j , j = 1, 2, be de?ned as follows: Lmi,n j :=

α ,β

k =0 ν =0

∑ ∑ pm,ν (x)qn,k (y) f

∞

m

ν + α1 k + α2 , m + β1 n + β2

α ,β

(1.3)

and (1.2), respectively. In the sequel, whenever we mention the operators Lmi,n j , j = 1, 2, it will be mentioned that these are the operators given in (1.3). We use the notation RA to denote the following closed and bounded region in R2 , RA := [0, 1] × [0, A], A > 0.

for (x, y) ∈ R , m, n ∈ N, 0 ≤ α j ≤ β j , j = 1, 2, where pm,ν (x) and qn,k (y) are given in (1.1)

(1.4)

In this paper we ?rst study some approximation properties of the sequence of linear positive operators given by (1.3) in the space of functions, continuous on RA , and ?nd the order of this approximation using modulus of continuity. Moreover we de?ne an rth order α ,β generalization of Lmi,n j , j = 1, 2, on RA extending the results of Kirov [14] and Kirov– Popova [15] to the linear positive operators Lmi,n j , j = 1, 2, of functions of two variables and study its approximation properties. The rth order generalization of some kind of linear positive operators may also be found in [1,9]. We ?nally investigate the convergence of the sequence of linear positive operators α ,β Lmi,n j , j = 1, 2, de?ned on a weighted space of functions of two variables and ?nd the rate of this convergence by means of weighted modulus of continuity. α ,β If we take pn,k (y), k = 0, 1, . . . , n in place of qn,k (y) in (1.3), then the operators Lmi,n j , j = 1, 2, reduce to the generalized Bernstein polynomials of two variables which were studied in [5]. Approximation of functions of one or two variables by some positive linear operators in weighted spaces may be found in [8,9,13,17,18].

α ,β

Linear positive operators

389

2. Preliminaries

In this section we give some basic de?nitions which we shall use. We denote by ρ the function, continuous and satisfying ρ (x, y) 1 for (x, y) ∈ R and lim|r|→∞ ρ (x, y) = ∞, r = (x, y) · ρ is called a weight function. Let Bρ denote the set of functions of two variables de?ned on R satisfying | f (x, y)| ≤ M f ρ (x, y), where M f > 0 is a constant depending on f , and Cρ denote the set of functions belonging to Bρ , and continuous on R . Clearly f (x,y)| , Cρ ? Bρ · Bρ and Cρ are called weighted spaces with norm f ρ = sup(x,y)∈R |ρ (x,y) [10,11]. The Lipschitz class LipM (γ ) of the functions of f of two variables is given by | f (x1 , y1 ) ? f (x2 , y2 )| ≤ M [(x1 ? x2 )2 + (y1 ? y2 )2 ] 2 ,

γ

(2.1)

(x1 , y1 ), (x2 , y2 ) ∈ R , where M > 0, 0 < γ ≤ 1 and f ∈ C(R ). The full modulus of continuity of f ∈ C(RA ), denoted by w( f ; δ ), is de?ned as follows: w( f ; δ ) = √ max

(x1 ?x2 )2 +(y1 ?y2 )2 ≤δ

| f (x1 , y1 ) ? f (x2 , y2 )|.

(2.2)

Partial modulus of continuity with respect to x and y are given by w(1) ( f ; δ ) = max and w(2) ( f ; δ ) = max

0≤x≤1 |y1 ?y2 |≤δ 0≤y≤A |x1 ?x2 |≤δ

max | f (x1 , y) ? f (x2 , y)|

(2.3)

max | f (x, y1 ) ? f (x, y2 )|,

(2.4)

respectively. We shall also need the following properties of the full and partial modulus of continuity w( f ; λ δ ) ≤ (1 + [λ ])w( f ; δ ) (2.5)

for any λ . Here [λ ] is the greatest integer that does not exceed λ . Moreover, it is known that when f is uniformly continuous, then limδ →0 w( f ; δ ) = 0 and | f (t , τ ) ? f (x, y)| ≤ w( f ; (t ? x)2 + (τ ? y)2 ), (2.6)

(t , τ ), (x, y) ∈ RA . The analogous properties are satis?ed by the partial modulus of continuity.

3. Lemmas and theorems on RA

In this section we give some classical approximation properties of the operators Lmi,n j , j = 1, 2, on the compact set RA . Lemma 3.1. Let α j , β j , j = 1, 2, be the ?xed positive numbers such that 0 ≤ α j ≤ β j . Then we have Lmi,n j (1; x, y) = 1,

α ,β α ,β

390

Fatma Tas ?delen et al Lmi,n j (t ; x, y) = Lmi,n j (τ ; x, y) = Lmi,n j (t 2 + τ 2 ; x, y) =

α ,β α ,β α ,β

mx + α1 , m + β1 nx + α2 , n + β2

2 (m2 ? m)x2 + (2α1 + 1)mx + α1 (m + β1)2

+ for all m, n ∈ N .

2 n2 y2 + (2α2 + 1)ny + α2 2 (n + β2)

Lmi,n j , j = 1, 2.

Taking (3.1) into account we now give the following Baskakov type theorem (see [2] to get the approximation to f (x, y) ∈ C(RA ), satisfying | f (x, y)| ≤ M f (1 + x2 + y2 ), by

α ,β

M f is a constant depending on f . Then Lmi,n j ( f ; x, y) ? f (x, y) if and only if Lmi,n j (1; x, y) ? 1 Lmi,n j (t ; x, y) ? x Lmi,n j (τ ; x, y) ? y Lmi,n j (t 2 + τ 2 ; x, y) ? (x2 + y2) as m, n → ∞ for (x, y) ∈ RA .

α ,β α ,β α ,β α ,β

C(RA ) C(RA ) C(RA ) C(RA )

Theorem 3.2. Let f (x, y) ∈ C(RA ) and | f (x, y)| ≤ M f (1 + x2 + y2 ) for (x, y) ∈ R. Here

α ,β

C(RA )

→ 0, as m, n → ∞

→ 0, → 0, → 0, → 0, (3.1)

Proof. Since the necessity is clear, then we need only to prove the suf?ciency. Let (t , τ ), (x, y) ∈ RA . By the uniform continuity of f on RA we get that for each ε > 0 there exists a number δ > 0 such that | f (t , τ ) ? f (x, y)| < ε , whenever (t ? x)2 + (τ ? y)2 < δ . Now let (x, y) ∈ RA and (t , τ ) ∈ R and let (x1 , y1 ) be an arbitrary boundary point of RA such that 0 ≤ x1 ≤ 1, 0 ≤ y1 ≤ A. Since f is continuous on the boundary points also, then for each ε > 0 there exists a δ > 0 such that | f (t , τ ) ? f (x, y)| ≤ | f (t , τ ) ? f (x1 , y1 )| + | f (x1 , y1 ) ? f (x, y)| < ε whenever (t ? x)2 + (τ ? y)2 < δ . Finally let (x, y) ∈ RA and (t , τ ) ∈ R and let (t ? x)2 + (τ ? y)2 > δ . Then easy calculations show that | f (t , τ ) ? f (x, y)| ≤ M f ((t ? x)2 + (τ ? y)2 ) ≤C (t ? x)2 + (τ ? y)2 δ2 . 2 3 + 2 + 2 (x2 + y2 ) δ2 δ

Linear positive operators Here C > 0 is a constant. Therefore we get | f (t , τ ) ? f (x, y)| ≤ ε + C (t ? x)2 + (τ ? y)2 δ2

α ,β

391

,

(3.2)

for (t , τ ) ∈ R , (x, y) ∈ RA . Applying Lmi,n j to (3.2) we get |Lmi,n j ( f (t , τ ); x, y) ? f (x, y)| ≤ Lmi,n j (| f (t , τ ) ? f (x, y)|; x, y) + f |Lmi,n j (1; x, y) ? 1|. Using (3.2) in the last inequality and taking (3.1) into account, suf?ciency is obtained easily. 2 We note that if we take f (x, y) to be bounded on R2 in the previous theorem, then α ,β we easily obtain that Lmi,n j ( f ; x, y) ? f (x, y) C(RA ) → 0, as m, n → ∞ satis?ed from Lemma 3.1 by analogous Korovkin’s theorem proved by Volkov [19]. The following theorem gives the rate of convergence of the sequence of linear positive α ,β operators {Lmi,n j } to f , by means of partial and full modulus of continuity. Theorem 3.3. Let f ∈ C(RA ). Then the following inequalities (a) (b) Lmi,n j ( f ; x, y) ? f (x, y) Lmi,n j ( f ; x, y) ? f (x, y)

α ,β α ,β

C(RA )

α ,β

α ,β

α ,β

3 ≤ {w(1) ( f ; δm ) + w(2) ( f ; δn )}, 2 3 ≤ w( f ; δm,n ) 2

(3.3) (3.4)

C(RA )

hold, where RA is the closed and bounded region given by (1.4). w(1) , w(2) and w are given by (2.3), (2.4) and (2.2) respectively, and δm , δn , δm,n are

δm =

respectively.

2+m 4β1

m + β1

,

δn =

2 A2 + nA β2

n + β2

,

δm,n =

2 + 4δ 2 , δm n

(3.5)

Proof. From (1.3) we have |Lmi,n j ( f (t , τ ); x, y) ? f (x, y)| ≤ e?ny ∑ × f

α ,β

∞

k =0 ν =0

∑

m

(ny)k m ν x (1 ? x)m?ν ν k! ? f (x, y) . (3.6)

ν + α1 k + α2 , m + β1 n + β2

ν +α1 Let us ?rst add and drop the function f m +β1 , y inside the absolute value sign on the right-hand side of (3.6). Using the analogous property of (2.6) for the partial modulus of continuity and ?nally applying the Cauchy–Schwartz inequality to the resulting term, then we arrive at (3.3) on RA , which proves (a). Using (2.6) directly in (3.6) and applying Cauchy–Schwartz inequality to the resulting term we then reach to (3.4) on RA , which gives (b). 2

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COROLLARY 3.4. Let f ∈ LipM (γ ). Then the inequality

′ γ δm,n |Lmi,n j ( f ; x, y) ? f (x, y)| ≤ M1 ′ holds, where M1 =3 2 M1 , M1 > 0 and δm,n is given in (3.5).

α ,β

COROLLARY 3.5. If f satis?es the following Lipschitz conditions | f (x1 , y) ? f (x2 , y)| ≤ M2 |x1 ? x2|α and | f (x, y1 ) ? f (x, y2 )| ≤ M3 |y1 ? y2 |β , 0 < α , β ≤ 1, M j > 0, j = 2, 3, then the inequality

′ α ′ |Lmi,n j ( f ; x, y) ? f (x, y)| ≤ M2 δm + M3 (2δn )β ′ = 3 M and M ′ = 3 M , and δ , δ are given in (3.5). holds, where M2 m n 3 2 2 2 3

α ,β

4. A generalization of order r of Lmi,n j

Let Cr (RA ), r ∈ N ∪ {0}, denote the set of all functions f having all continuous partial derivatives up to order r at (x, y) ∈ RA . α ,β α ,β By (Lmi,n j )[r] , j = 1, 2, we denote the following generalization of Lmi,n j : (Lmi,n j )[r] ( f ; x, y) := e?ny ∑ ×P 0 ≤ α j ≤ β j , j = 1, 2, where P

r, m+β1 , n+β2 1 2

ν +α

k +α

α ,β

α ,β

∞

k =0 ν =0

ν +α

k +α

∑

m

(ny)k m ν x (1 ? x)m?ν ν k! x? k + α2 ν + α1 ,y ? , m + β1 n + β2 (4.1)

r, m+β1 , n+β2 1 2

x?

k + α2 ν + α1 ,y ? m + β1 n + β2

=

1 h fi j j xy h h=0 i+ j =h !

∑ ∑

r

ν + α1 k + α2 , m + β1 n + β2

k + α2 n + β2

j

× x?

ν + α1 m + β1

i

y?

,

?r ? xi ? y j

(4.2) f (x, y).

fxi y j denotes the partial derivatives of f , i.e.: fxi y j :=

Linear positive operators

α ,β α ,β

393

(Lmi,n j )[r] reduce to Lmi,n j , when r = 0. Now let us write x?

α ,β

(Lmi,n j )[r] are called the rth order of Lmi,n j (see [14] for one variable). Obviously

α ,β

ν + α1 k + α2 ,y ? m + β1 n + β2

= u(α , β ),

(4.3)

where (α , β ) is a unit vector, u > 0 and let F (u) = f =f

ν + α1 k + α2 + uα , + uβ m + β1 n + β2

k + α2 k + α2 ν + α1 ν + α1 + x? + y? , m + β1 m + β1 n + β2 n + β2 . (4.4)

It is clear that Taylor’s formula for F (u) at u = 0 turns into Taylor’s formula for f (x, y) at ν +α1 k +α2 m+β , n+β . Morever rth derivative takes the form

1 2

F (r) (u) =

i+ j =r

∑

r j

fxi y j

ν + α1 k + α2 + uα , + uβ α i β j , m + β1 n + β2

(4.5)

r ∈ N (see Chapter 3 of [6]). By means of the modi?cation stated above ((4.3)–(4.5)), we get the following result. Theorem 4.1. Let f ∈ Cr (RA ) and F (r) (u) ∈ LipM (γ ). Then the inequality (Lmi,n j )[r] ( f ; x, y) ? f (x, y) ≤

α ,β

C(RA )

γ M B(γ , r) αi ,β j Lm,n (|(x, y) ? (t , τ )|r+γ ; x, y) γ + r (r ? 1)!

C(RA )

(4.6)

holds, where F (r) (u) are given by (4.5), B(γ , r) is the well-known beta function, r, m, n ∈ N , 0 < γ ≤ 1 and M > 0. Proof. From (4.1) and (4.2) we have f (x, y) ? (Lmi,n j )[r] ( f ; x, y) =

ν =0

∞

α ,β

∑

m

(ny)k m ν x (1 ? x)m?ν e?ny k! ν

×∑ ×

k =0

f (x, y) ? ∑

1 h fi j ∑ j xy h=0 h! i+ j =h x?

r

ν + α1 k + α2 , m + β1 n + β2

ν + α1 m + β1

i

y?

k + α2 n + β2

j

.

(4.7)

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Fatma Tas ?delen et al

We now consider Taylor’s formula with the remainder for the functions of two variables. Using the integral form of the remainder term that appeared in (4.7), we arrive at f (x, y) ? P =

r, m+β1 , n+β2 1 2 1

ν +α

k +α

x? h j

k + α2 ν + α1 ,y ? m + β1 n + β2 x?

1 (r ? 1)! × fxi y j

0 i+ j =h

∑

ν + α1 m + β1

i

y?

k + α2 n + β2

j

ν + α1 ν + α1 k + α2 k + α2 +t x? , +t y? m + β1 m + β1 n + β2 n + β2

(4.8)

1 0

Taking (4.3)–(4.5) into account, (4.8) turns into the following form: F (u) ? ∑ ur 1 (h ) F (0)uh = (r ? 1)! h =0 h !

r

× (1 ? t )r?1dt .

[F (r) (tu) ? F (r) (0)](1 ? t )r?1dt . (4.9)

From (4.3), (4.8), (4.9) and the fact that F (r) ∈ LipM (γ ) it follows that f (x, y) ? P

r, m+β1 , n+β2 1 2 r

ν +α

k +α

x?

ν + α1 k + α2 ,y ? m + β1 n + β2

= F (u) ? ∑ ≤ ≤ ≤ ≤ |u|r (r ? 1)!

1 (h ) F (0)uh h ! h =0

1 0

[F (r) (tu) ? F (r) (0)](1 ? t )r?1dt

|u|r+γ MB(γ + 1, r) (r ? 1)! M γ B(γ , r)|u|r+γ (r ? 1)! γ + r

Hence combining (4.7) and (4.10), we obtain (4.6), which completes the proof. Now we take a function g ∈ C(RA ) which is given by Obviously g(x, y) = 0. From Theorem 3.2 it follows that Lm,n (g; x, y)

α ,β

C(RA )

γ ν + α1 k + α2 M ,y ? B(γ , r) x ? (r ? 1)! γ + r m + β1 n + β2

r +γ

.

(4.10) 2 (4.11)

g(t , τ ) = |(x, y) ? (t , τ )|r+γ . →0

From (4.6) we arrive at the following result: (Lmi,n j )[r] ( f ; x, y) ? f (x, y)

as m, n → ∞.

C(RA )

Using (3.3) and Corollary 3.4 we get to the following results by means of Theorem 4.1.

→ 0 as m, n → ∞.

Linear positive operators COROLLARY 4.2. Let f ∈ Cr (RA ) and F (r) ∈ LipM (γ ). Then the inequality (Lmi,n j )[r] ( f ; x, y) ? f (x, y)

α ,β

C(RA )

395

≤

MB(γ , r) γ 3 w(g; δm,n ) (r ? 1)! γ + r 2

holds, where F (r) , δm,n and g are given by (4.5), (3.5) and (4.11), respectively. COROLLARY 4.3.

r (γ ) in Corollary 3.4. Let f ∈ Cr (RA ) and F (r) ∈ LipM (γ ), and assume that g ∈ Lip (1 + A 2 ) 2 Then we arrive at

(Lmi,n j )[r] ( f ; x, y) ? f (x, y) where δm,n is given by (3.5).

α ,β

C(RA ) ≤

M (1 + A2 ) 2 γ γ B(γ , r)δm ,n , (r ? 1)! γ + r

r

5. Weighted approximation of functions of two variables by Lmi,n j

In this section we investigate the convergence of the sequence {Lmi,n j } mapping the weighted space Cρ into Bρ1 . We also study the rates of convergence of the sequence {Lmi,n j } de?ned on weighted spaces. In the rest of the article ρ will be given by ρ (x, y) = 1 + x2 + y2 . We ?rst give the following important Korovkin type theorem (in weighted spaces) proved by Gadjiev in [11]. Theorem of Gadjiev. Let {An } be the sequence of linear positive operators mapping from Cρ (Rm ) into Bρ (Rm ), m 1, and satisfying the conditions An (1; x) ? 1 An (|t |2 ; x) ? |x|2

ρ ρ α ,β α ,β

α i ,β j

→ 0, →0

An (t j ; x) ? x j

ρ

→ 0 , j = 1 , . . . , m,

By taking the result of the last theorem into account we conclude that verifying the α ,β α ,β conditions of the above theorem by the operators Lmi,n j , j = 1, 2, is not suf?cient for Lmi,n j to be convergent to any function f in ρ norm. Hence we need to show the convergence in another norm for any function in Cρ (R ). For this purpose, we now give the following lemma, which we shall use.

as n → ∞ for ρ (x) = 1 + |x|2 , x ∈ Rm . Then there exists a function f ? ∈ Cρ (Rm ) such that An ( f ? ; x) ? f?(x) ρ 1.

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Fatma Tas ?delen et al

α ,β

Lemma 5.1. The operators Lmi,n j possess the following: (a) {Lmi,n j }, m, n ∈ N , is the sequence of linear positive operators from the weighted space Cρ (R ) into the weighted space Bρ (R ). (b) The norms Lmi,n j

α ,β Lmi,n j Cρ →Bρ α ,β α ,β

≤ M ).

Cρ →Bρ

are uniformly bounded (i.e. there exists an M > 0 such that

Proof. From Lemma 3.1 we easily obtain that |Lmi,n j (ρ ; x, y)| ≤ M (1 + x2 + y2) Lmi,n j

α ,β

Cρ →Bρ

α ,β

which proves (a). Taking Lemma 3.1 into account we get the following inequality:

≤ Lmi,n j (ρ ; x, y) ? ρ (x, y) ≤ Lmi,n j (1; x, y) ? 1 = sup

(x,y)∈R

α ,β α ,β

ρ

+1

ρ

ρ

+ Lmi,n j (t 2 + τ 2 ; x, y) ? x2 ? y2

α ,β

+1

2 (m2 ? m)x2 + (2α1 + 1)mx + α1 (m + β1 )2

+

2 n2 y2 + (2α2 + 1)ny + α2 1 ? x2 ? y2 +1 (n + β2)2 1 + x2 + y2

(5.1) 2

so (b) is obtained from (5.1), which completes the proof.

Now the following theorem shows the convergence of the sequence of linear positive α ,β operators {Lmi,n j }, mapping from Cρ into Bρ1 , in ρ1 norm. Theorem 5.2. Let ρ1 (x, y) be a weight function satisfying

|x|→∞ ρ1 (x, y)

lim

ρ (x, y)

= 0.

ρ1

(5.2) → 0, m, n → ∞ for all f ∈ Cρ (R ), where x = (x, y) ∈ R.

ρ1

Proof. Let us denote the region [0, 1] × [0, s], s > 0, by Rs . Therefore we have Lmi,n j ( f ; x, y) ? f (x, y)

α ,β α ,β

Then Lmi,n j ( f ; x, y) ? f (x, y)

α ,β

= sup

(x,y)∈R

Lmi,n j ( f ; x, y) ? f (x, y) ρ1 (x, y)

α ,β

|Lmi,n j ( f ; x, y) ? f (x, y)| ρ (x, y) = sup ρ (x, y) ρ1 (x, y) (x,y)∈Rs + sup

(x,y)∈R

|Lmi,n j ( f ; x, y) ? f (x, y)| ρ (x, y) . ρ (x, y) ρ1 (x, y) Rs

α ,β

(5.3)

Linear positive operators

397

Since ρ /ρ1 is bounded on Rs , the ?rst term on the right-hand side of (5.3) approaches zero when m, n → ∞ by Theorem 3.2. The second term also approaches zero when m, n → ∞ by Lemma 5.1(b) and the condition (5.2). So proof is completed. 2 As a result we give the approximation order of Lmi,n j , j = 1, 2, m, n ∈ N, by means of the weighted modulus of continuity. Theorem 5.3. For any s > 0 and all m, n ∈ N the inequality ? ? ? ? αi ,β j sup sup | L ( f ; x , y ) ? f ( x , y ) | ≤ c sup [wρ ( f , δ )] m , n √ ? f ρ =1 ? f ρ =1 2 2

x +y ≤s

α ,β

(5.4)

δ=

holds for the linear positive operators {Lmi,n j }, j = 1, 2, de?ned on Cρ , where Lmi,n j [(t ? x)2 + (τ ? y)2 ] and c > 0 is a constant depending on s.

α ,β α ,β α ,β

α ,β

Proof. Since Lmi,n j , j = 1, 2, m, n ∈ N, are linear positive operators, we have |Lmi,n j ( f ; x, y) ? f (x, y)| ≤ Lmi,n j (| f (t , τ ) ? f (x, y)|; x, y) + | f (x, y)|(Lmi,n j (1; x, y) ? 1) ≤ Lmi,n j

α ,β α ,β α ,β

ρ (x, y)wρ f ;

(t ? x)2 + (τ ? y)2 δ ; x, y , δ

by Lemma 3.1. Using (2.5) we get |Lmi,n j ( f ; x, y) ? f (x, y)| ≤ ρ (x, y)wρ ( f ; δ )Lmi,n j

α ,β α ,β α ,β

1+

(t ? x)2 + (τ ? y)2 ; x, y δ (t ? x)2 + (τ ? y)2 ; x, y δ2 1 αi ,β j Lm,n ([(t ?x)2 + (τ ?y)2 ]; x, y). (5.5) δ2

α ,β

≤ ρ (x, y)wρ ( f ; δ )Lmi,n j 1 +

α ,β

≤ ρ (x, y)wρ ( f ; δ )Lmi,n j (ρ ; x, y)+ Since (t ? x)2 + (τ ? y)2 ∈ Cρ , (5.5) gives |Lmi,n j ( f ; x, y) ? f (x, y)| ≤ √ sup +

α ,β

c2 wρ ( f ; δ ) Lmi,n j (ρ ; x, y)

ρ

x2 +y2 ≤ s

1 αi ,β j Lm,n ((t ? x)2 + (τ ? y)2 ; x, y) δ2

ρ

ρ,

α ,β where c = sup√x2 +y2 ≤ s ρ (x, y). Lmi,n j (ρ ; x, y)

is bounded since

Lmi,n j (ρ ; x, y)

α ,β

ρ

= Lmi,n j (ρ ; x, y)

α ,β

Cρ →Bρ

398

Fatma Tas ?delen et al

which is uniformly bounded, for all m, n ∈ N, by Lemma 5.1. From (5.5) and (5.6) we arrive at √ sup which implies that sup ? ? √ sup ? ? α ,β |Lmi,n j ( f ; x, y) ? f (x, y)| ? |Lmi,n j ( f ; x, y) ? f (x, y)| ≤ c2 (1 + M )wρ ( f ; δ ),

α ,β

x2 +y2 ≤ s

f ρ =1 ?

x2 +y2 ≤s

≤ c2 (1 + M ) sup [wρ ( f , δ )],

f ρ =1

where δ = Lmi,n j ([(t ? x)2 + (τ ? y)2 ]; x, y) pletes the proof.

α ,β

ρ.

Last inequality gives (5.4), which com2

References

[1] Alt?n A, Do? gru O and Tas ? delen F, The generalization of Meyer–K¨ onig and Zeller operators by generating functions, J. Math. Anal. Appl. (in print) [2] Baskakov V A, On a construction of converging sequences of linear positive operators, Studies of Modern Problems of Constructive Theory of Functions (1961) 314–318 [3] Becker M, Global approximation theorems for Szazs–Mirakyan and Baskakov operators in polynomial weight spaces, Indiana Univ. Math. J. 27(1) (1978) 127–142 [4] Bernstein S N, D? emonstration du th? eor? eme de Weierstrass fond? ee sur la calcules des probabilites, Comm. Soc. Math. Charkow S? er. 13(2) (1912) 1–2 [5] B¨ uy¨ ukyaz?c? ˙ I and ˙ Ibikli E, The approximation properties of generalized Bernstein polynomials of two variables, Appl. Math. Comput. 156(2) (2004) 367–380 [6] Callahan J, Advanced Calculus, Lecture Notes (USA: Smith College) [7] Zhou D X, Weighted approximation by Sz? asz–Mirakjan operators, J. Approx. Theory 76(3) (1994) 393–402 [8] Do? gru O, Weighted approximation of continuous functions on the all positive axis by modi?ed linear positive operators, Int. J. Comput. Numerical Anal. Appl. 1(2) (2002) 135–147 ¨ [9] Do? gru O, Ozarslan M A and Tas ?delen F, On positive operators involving a certain class of generating functions, Studia Scientiarum Mathematicarum Hungarica 41(4) (2004) 415–429 [10] Gadjiev A D, Weighted approximation of continuous functions by positive linear operators on the whole real axis, (Russian) Izv. Akad. Nauk Azerba? ?dˇ zan. SSR Ser. Fiz.-Tehn. Mat. Nauk 5 (1975) 41–45 [11] Gadˇ ziev A D, Positive linear operators in weighted spaces of functions of several variables, Izv. Akad. Nauk Azerba? ?dzhan. SSR Ser. Fiz.-Tekhn. Mat. Nauk 1 4 (1980) 32–37 [12] Hermann T, On the Sz? asz-Mirakian operator, Acta Math. Acad. Sci. Hungar. 32(1–2) (1978) 163–173 [13] Ispir N and Atakut C ? , Approximation by modi?ed Szasz-Mirakjan operators on weighted spaces, Proc. Indian Acad. Sci. (Math. Sci.) 112(4) (2002) 571–578 [14] Kirov G H, A generalization of the Bernstein polynomials, Math. Balkanica (N.S.) 6(2) (1992) 147–153

Linear positive operators

399

[15] Kirov G H and Popova L, A generalization of the linear positive operators, Math. Balkanica (N.S.) 7(2) (1993) 149–162 [16] Stancu D D, Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl. 13 (1968) 1173–1194 [17] Stancu D D, A new class of uniform approximating polynomial operators in two and several variables, Proceedings of the Conference on the Constructive Theory of Functions (Approximation Theory) (Budapest, 1969) pp. 443–455 (Budapest: Akad? emiai Kiad? o) (1972) [18] Walczak Z, Approximation of functions of two variables by some linear positive operators, Acta. Math. Univ. Comenianae LXXIV(1) (2005) 37–48 [19] Volkov V I, On the convergence of sequences of linear positive operators in the space of continuous functions of two variables, (Russian) Dokl. Akad. Nauk SSSR (N.S.) 115 (1957) 17–19

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