Approximation of functions of two variables by certain linear positive

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 117, No. 3, August 2007, pp. 387–399. © Printed in India

Approximation of functions of two variables by certain linear positive operators FATMA TAS¸DELEN∗ , ALI OLGUN† and ¨ GULEN BAS¸CANBAZ-TUNCA∗ ∗ Department 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: [email protected]; [email protected]; [email protected]

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 find the order of this approximation by using modulus of continuity. Moreover we define 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 find the rate of this convergence using weighted modulus of continuity. Keywords. Linear positive operator; modulus of continuity; order of approximation; polynomial weighted spaces.

1. Introduction Let f ∈ C([0, 1]). The well-known Bernstein polynomial of degree n, denoted by Bn (f ; x) is   n  k , n ∈ N ={1, 2, . . . }, pn,k (x)f Bn (f ; x) := n k=0 where pn,k (x) =

  n k x (1 − x)n−k k

(1.1)

and x ∈ [0, 1] [4]. Let x ∈ [0, ∞) and f ∈ C([0, ∞)). Szazs–Mirakyan operators, denoted by Sn (f ; x) are   ∞  k qn,k (x)f , n ∈ N, Sn (f ; x) := n k=0 387

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where qn,k (x) = e−nx

(nx)k . k!

(1.2)

In [3] approximation properties of Sn (f ; x) in weighted spaces were studied. Some works by Szazs–Mirakyan or modified 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 ) :=

n 

 pn,k (x)f

k=0

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: αi ,βj Let f ∈ C(R), R := [0, 1] × [0, ∞) and let the linear positive operators Lm,n , j = 1, 2, be defined as follows:   m ∞   ν + α1 k + α2 αi ,βj Lm,n (1.3) := pm,ν (x)qn,k (y)f , m + β1 n + β2 k=0 ν=0 for (x, y) ∈ R, m, n ∈ N, 0 ≤ αj ≤ βj , j = 1, 2, where pm,ν (x) and qn,k (y) are given in (1.1) and (1.2), respectively. In the sequel, whenever we mention the operators αi ,βj Lm,n , 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.

(1.4)

In this paper we first study some approximation properties of the sequence of linear positive operators given by (1.3) in the space of functions, continuous on RA , and find the order of this approximation using modulus of continuity. Moreover we define an rth order αi ,βj generalization of Lm,n , j = 1, 2, on RA extending the results of Kirov [14] and Kirov– αi ,βj , j = 1, 2, of functions of two variables Popova [15] to the linear positive operators Lm,n and study its approximation properties. The rth order generalization of some kind of linear positive operators may also be found in [1, 9]. We finally investigate the convergence of the sequence of linear positive operators αi ,βj Lm,n , j = 1, 2, defined on a weighted space of functions of two variables and find the rate of this convergence by means of weighted modulus of continuity. If we take pn,k (y), k = 0, 1, . . . , n in place of qn,k (y) in (1.3), then the operators αi ,βj Lm,n , 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].

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2. Preliminaries In this section we give some basic definitions 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 defined on R satisfying |f (x, y)| ≤ Mf ρ(x, y), where Mf > 0 is a constant depending on f , and Cρ denote the set of functions belonging to Bρ , and continuous on R. Clearly Cρ ⊂ Bρ · Bρ and Cρ are called weighted spaces with norm f ρ = (x,y)| sup(x,y)∈R |fρ(x,y) , [10, 11]. The Lipschitz class LipM (γ ) of the functions of f of two variables is given by γ

|f (x1 , y1 ) − f (x2 , y2 )| ≤ M[(x1 − x2 )2 + (y1 − y2 )2 ] 2 ,

(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 defined 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

max

|f (x1 , y) − f (x2 , y)|

(2.3)

w(2) (f ; δ) = max

max

|f (x, y1 ) − f (x, y2 )|,

(2.4)

0≤y≤A |x1 −x2 |≤δ

and 0≤x≤1 |y1 −y2 |≤δ

respectively. We shall also need the following properties of the full and partial modulus of continuity w(f ; λδ) ≤ (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 satisfied by the partial modulus of continuity.

3. Lemmas and theorems on RA α ,β

i j In this section we give some classical approximation properties of the operators Lm,n ,j = 1, 2, on the compact set RA .

Lemma 3.1. Let αj , βj , j = 1, 2, be the fixed positive numbers such that 0 ≤ αj ≤ βj . Then we have α ,β

i j (1; x, y) = 1, Lm,n

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mx + α1 , m + β1

α ,β

nx + α2 , n + β2

i j Lm,n (t; x, y) =

i j Lm,n (τ ; x, y) =

α ,β

i j (t 2 + τ 2 ; x, y) = Lm,n

(m2 − m)x 2 + (2α1 + 1)mx + α12 (m + β1 )2 +

n2 y 2 + (2α2 + 1)ny + α22 (n + β2 )2

for all m, n ∈ N. 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)| ≤ Mf (1 + x 2 + y 2 ), by α ,β

i j , j = 1, 2. Lm,n

Theorem 3.2. Let f (x, y) ∈ C(RA ) and |f (x, y)| ≤ Mf (1 + x 2 + y 2 ) for (x, y) ∈ R. α ,β

i j (f ; x, y) − f (x, y)C(RA ) → 0, as Here Mf is a constant depending on f . Then Lm,n m, n → ∞ if and only if

α ,β

i j Lm,n (1; x, y) − 1C(RA ) → 0,

α ,β

i j Lm,n (t; x, y) − xC(RA ) → 0,

α ,β

i j Lm,n (τ ; x, y) − yC(RA ) → 0,

α ,β

i j Lm,n (t 2 + τ 2 ; x, y) − (x 2 + y 2 )C(RA ) → 0,

(3.1)

as m, n → ∞ for (x, y) ∈ RA . Proof. Since the necessity is clear, then we need only to prove the sufficiency. Let that for each ε > 0 there (t, τ ), (x, y) ∈ RA . By the uniform continuity of f on RA we get 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   2 3 |f (t, τ ) − f (x, y)| ≤ Mf ((t − x)2 + (τ − y)2 ) 2 + 2 + 2 (x 2 + y 2 ) δ δ   (t − x)2 + (τ − y)2 ≤C . δ2

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

(3.2)

α ,β

i j to (3.2) we get for (t, τ ) ∈ R, (x, y) ∈ RA . Applying Lm,n

α ,β

α ,β

i j i j |Lm,n (f (t, τ ); x, y) − f (x, y)| ≤ Lm,n (|f (t, τ ) − f (x, y)|; x, y)

α ,β

i j + f |Lm,n (1; x, y) − 1|.

Using (3.2) in the last inequality and taking (3.1) into account, sufficiency is obtained easily. 2 We note that if we take f (x, y) to be bounded on R2 in the previous theorem, then we αi ,βj easily obtain that Lm,n (f ; x, y) − f (x, y)C(RA ) → 0, as m, n → ∞ satisfied 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 αi ,βj operators {Lm,n } to f, by means of partial and full modulus of continuity. Theorem 3.3. Let f ∈ C(RA ). Then the following inequalities α ,β

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

(3.3)

α ,β

3 w(f ; δm,n ) 2

(3.4)

(a)

i j (f ; x, y) − f (x, y)C(RA ) ≤ Lm,n

(b)

i j Lm,n (f ; x, y) − f (x, y)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    4β12 + m β22 A2 + nA 2 + 4δ 2 , δm = , δn = , δm,n = δm (3.5) n m + β1 n + β2 respectively. Proof. From (1.3) we have m   ∞   m

(ny)k ν k! k=0 ν=0       ν + α 1 k + α2 − f (x, y) . , × f m + β1 n + β2 (3.6)  ν+α1 Let us first 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 finally 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 α ,β

i j (f (t, τ ); x, y) − f (x, y)| ≤ e−ny |Lm,n

x ν (1 − x)m−ν

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

γ

i j |Lm,n (f ; x, y) − f (x, y)| ≤ M1 δm,n

holds, where M1 = 23 M1 , M1 > 0 and δm,n is given in (3.5). COROLLARY 3.5 If f satisfies 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, Mj > 0, j = 2, 3, then the inequality α ,β

i j α (f ; x, y) − f (x, y)| ≤ M2 δm + M3 (2δn )β |Lm,n

holds, where M2 = 23 M2 and M3 = 23 M3 , and δm , δn are given in (3.5). α i ,β j

i j 4. A generalization of order r of Lm,n

Let C r (RA ), r ∈ N ∪ {0}, denote the set of all functions f having all continuous partial derivatives up to order r at (x, y) ∈ RA . αi ,βj [r] αi ,βj By (Lm,n ) , j = 1, 2, we denote the following generalization of Lm,n : α ,β

i j [r] ) (f ; x, y) := e−ny (Lm,n

m   ∞   m k=0 ν=0

ν

x ν (1 − x)m−ν

(ny)k k!

  k + α2 ν + α1 , (4.1) ,y − × Pr, ν+α1 , k+α2 x − n + β2 m + β1 m+β1 n+β2 0 ≤ αj ≤ βj , j = 1, 2, where   ν + α1 k + α2 ,y − Pr, ν+α1 , k+α2 x − m + β1 n + β2 m+β1 n+β2 =

  r   1 h ν + α 1 k + α2 , fx i y j h! j m + β1 n + β2 h=0 i+j =h

ν + α1 × x− m + β1

i

k + α2 y− n + β2

j ,

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

(4.2) ∂r ∂x i ∂y j

f (x, y).

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α ,β

α ,β

i j [r] i j (Lm,n ) are called the rth order of Lm,n (see [14] for one variable). Obviously αi ,βj [r] αi ,βj ) reduce to Lm,n , when r = 0. (Lm,n Now let us write   ν + α1 k + α2 x− = u(α, β), (4.3) ,y − m + β1 n + β2

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

 ν + α1 k + α2 F (u) = f + uα, + uβ m + β1 n + β2   

 k + α2 k + α2 ν + α1 ν + α1 , . + y− =f + x− m + β1 n + β2 n + β2 m + β1

(4.4)

It is clear that Taylor’s formula for F (u) at u = 0 turns into Taylor’s formula for f (x, y)  ν+α 1 k+α2 . Morever rth derivative takes the form , at m+β 1 n+β2 F (r) (u) =

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

(4.5)

r ∈ N (see Chapter 3 of [6]). By means of the modification stated above ((4.3)–(4.5)), we get the following result. Theorem 4.1. Let f ∈ C r (RA ) and F (r) (u) ∈ LipM (γ ). Then the inequality α ,β

i j [r] ) (f ; x, y) − f (x, y)C(RA ) (Lm,n

≤

γ M B(γ , r) αi ,βj Lm,n (|(x, y) − (t, τ )|r+γ ; x, y)C(RA ) γ + r (r − 1)!

(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 α ,β

i j [r] f (x, y) − (Lm,n ) (f ; x, y)

=

m    m ν=0

×

ν

∞  k=0

 ×

x ν (1 − x)m−ν e−ny

(ny)k k!



  r  1  h f (x, y) − f i j h! i+j =h j x y h=0

ν + α 1 k + α2 , m + β1 n + β2



ν + α1 x− m + β1

i

k + α2 y− n + β2

j .

(4.7)

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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   k + α2 ν + α1 ,y − f (x, y) − Pr, ν+α1 , k+α2 x − m + β1 n + β2 m+β1 n+β2 =

1 (r − 1)!

 0

× fx i y j

1



  h

ν + α1 i k + α2 j x− y− j m + β1 n + β2 i+j =h

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

× (1 − t)r−1 dt.

(4.8)

Taking (4.3)–(4.5) into account, (4.8) turns into the following form: F (u) −

 1 r  1 (h) ur [F (r) (tu) − F (r) (0)](1 − t)r−1 dt. F (0)uh = (r − 1)! h! 0 h=0 (4.9)

From (4.3), (4.8), (4.9) and the fact that F (r) ∈ LipM (γ ) it follows that   ν + α1 k + α2 ,y − f (x, y) − Pr, ν+α1 , k+α2 x − m + β1 n + β2 m+β1 n+β2   r    1 (h)   = F (u) − F (0)uh    h! h=0 |u|r ≤ (r − 1)! ≤



1

[F (r) (tu) − F (r) (0)](1 − t)r−1 dt

0

|u|r+γ MB(γ + 1, r) (r − 1)!

M γ B(γ , r)|u|r+γ (r − 1)! γ + r    ν + α1 M k + α2 r+γ γ  B(γ , r) x − ≤ ,y − . m + β1 n + β2  (r − 1)! γ + r

≤

Hence combining (4.7) and (4.10), we obtain (4.6), which completes the proof.

(4.10) 2

Now we take a function g ∈ C(RA ) which is given by g(t, τ ) = |(x, y) − (t, τ )|r+γ .

(4.11)

Obviously g(x, y) = 0. From Theorem 3.2 it follows that Lm,n (g; x, y)C(RA ) → 0

as

m, n → ∞.

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From (4.6) we arrive at the following result: α ,β

i j [r] ) (f ; x, y) − f (x, y)C(RA ) → 0 (Lm,n

as

m, n → ∞.

Using (3.3) and Corollary 3.4 we get to the following results by means of Theorem 4.1. COROLLARY 4.2 Let f ∈ C r (RA ) and F (r) ∈ LipM (γ ). Then the inequality α ,β

i j [r] ) (f ; x, y) − f (x, y)C(RA ) ≤ (Lm,n

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 CorolLet f ∈ C r (RA ) and F (r) ∈ LipM (γ ), and assume that g ∈ Lip (1+A2 ) 2 lary 3.4. Then we arrive at

α ,β

r

i j [r] ) (f ; x, y) − f (x, y)C(RA ) ≤ (Lm,n

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

where δm,n is given by (3.5).

α i ,β j

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

α ,β

i j } mapping the In this section we investigate the convergence of the sequence {Lm,n weighted space Cρ into Bρ1 . We also study the rates of convergence of the sequence

α ,β

i j } defined on weighted spaces. In the rest of the article ρ will be given by ρ(x, y) = {Lm,n 1 + x2 + y2. We first 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) − 1ρ → 0,

An (tj ; x) − xj ρ → 0, j = 1, . . . , m,

An (|t|2 ; x) − |x|2 ρ → 0 as n → ∞ for ρ(x) = 1 + |x|2 , x ∈ R m . Then there exists a function f ∗ ∈ Cρ (Rm ) such that An (f ∗ ; x) − f ∗ (x)ρ  1. By taking the result of the last theorem into account we conclude that verifying the conαi ,βj αi ,βj ditions of the above theorem by the operators Lm,n , j = 1, 2, is not sufficient for Lm,n 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.

396

Fatma Tas¸delen et al α ,β

i j Lemma 5.1. The operators Lm,n possess the following:

α ,β

i j }, m, n ∈ N , is the sequence of linear positive operators from the weighted (a) {Lm,n space Cρ (R) into the weighted space Bρ (R).

α ,β

i j Cρ →Bρ are uniformly bounded (i.e. there exists an M > 0 such (b) The norms Lm,n

α ,β

i j Cρ →Bρ ≤ M). that Lm,n

Proof. From Lemma 3.1 we easily obtain that α ,β

i j |Lm,n (ρ; x, y)| ≤ M(1 + x 2 + y 2 )

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

i j Lm,n Cρ →Bρ

α ,β

i j (ρ; x, y) − ρ(x, y)ρ + 1 ≤ Lm,n

α ,β

α ,β

i j i j ≤ Lm,n (1; x, y) − 1ρ + Lm,n (t 2 + τ 2 ; x, y) − x 2 − y 2 ρ + 1   (m2 − m)x 2 + (2α + 1)mx + α 2 1  1 = sup  2  (m + β ) 1 (x,y)∈R

  n2 y 2 + (2α2 + 1)ny + α22 1 2 2 − x − y +1 +   1 + x2 + y2 (n + β2 )2 so (b) is obtained from (5.1), which completes the proof.

(5.1) 2

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

lim

|x|→∞ ρ1 (x, y)

= 0.

(5.2)

α ,β

i j Then Lm,n (f ; x, y) − f (x, y)ρ1 → 0, m, n → ∞ for all f ∈ Cρ (R), where x = (x, y) ∈ R.

Proof. Let us denote the region [0, 1] × [0, s], s > 0, by Rs . Therefore we have α ,β

i j Lm,n (f ; x, y) − f (x, y)ρ1  α ,β   L i j (f ; x, y) − f (x, y)   m,n  = sup     ρ (x, y) 1 (x,y)∈R

α ,β

=

i j |Lm,n (f ; x, y) − f (x, y)| ρ(x, y) ρ(x, y) ρ1 (x, y) (x,y)∈Rs

sup

α ,β

+

i j |Lm,n (f ; x, y) − f (x, y)| ρ(x, y) . ρ(x, y) ρ1 (x, y) (x,y)∈RRs

sup

(5.3)

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Since ρ/ρ1 is bounded on Rs , the first 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 α ,β

i j As a result we give the approximation order of Lm,n , 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 ,βj sup sup |L (f ; x, y) − f (x, y)| ≤ c sup [wρ (f, δ)] (5.4) m,n √ ⎭ f ρ =1 ⎩ f ρ =1 2 2 x +y ≤s

α ,β

i j holds for the linear positive operators {Lm,n }, j = 1, 2, defined on Cρ , where δ =  αi ,βj Lm,n [(t − x)2 + (τ − y)2 ] and c > 0 is a constant depending on s.

α ,β

i j , j = 1, 2, m, n ∈ N, are linear positive operators, we have Proof. Since Lm,n

α ,β

i j |Lm,n (f ; x, y) − f (x, y)|

α ,β

α ,β

i j i j ≤ Lm,n (|f (t, τ ) − f (x, y)|; x, y) + |f (x, y)|(Lm,n (1; x, y) − 1)

 ≤

αi ,βj Lm,n

 ρ(x, y)wρ

   (t − x)2 + (τ − y)2 f; δ ; x, y , δ

by Lemma 3.1. Using (2.5) we get α ,β

i j |Lm,n (f ; x, y) − f (x, y)|

 ≤

αi ,βj ρ(x, y)wρ (f ; δ)Lm,n

α ,β

i j ≤ ρ(x, y)wρ (f ; δ)Lm,n

 1+



1+

  (t − x)2 + (τ − y)2 ; x, y δ

  (t − x)2 + (τ − y)2 ; x, y δ2

α ,β

i j ≤ ρ(x, y)wρ (f ; δ)Lm,n (ρ; x, y)+

1 αi ,βj Lm,n ([(t −x)2 + (τ −y)2 ]; x, y). δ2 (5.5)

Since (t − x)2 + (τ − y)2 ∈ Cρ , (5.5) gives α ,β

i j (f ; x, y) − f (x, y)| ≤ √ sup |Lm,n

α ,β

i j c2 wρ (f ; δ)Lm,n (ρ; x, y)ρ

x 2 +y 2 ≤ s

+

1 αi ,βj Lm,n ((t − x)2 + (τ − y)2 ; x, y)ρ , δ2

398

Fatma Tas¸delen et al α ,β

i j where c = sup√x 2 +y 2 ≤ s ρ(x, y). Lm,n (ρ; x, y)ρ is bounded since

α ,β

α ,β

i j i j (ρ; x, y)ρ = Lm,n (ρ; x, y)Cρ →Bρ Lm,n

which is uniformly bounded, for all m, n ∈ N, by Lemma 5.1. From (5.5) and (5.6) we arrive at α ,β

i j |Lm,n (f ; x, y) − f (x, y)| ≤ c2 (1 + M)wρ (f ; δ),

√ sup

x 2 +y 2 ≤ s

which implies that ⎧ ⎨ sup

⎫ ⎬

√ sup

f ρ =1 ⎩

αi ,βj |Lm,n (f ; x, y) − f (x, y)|

x 2 +y 2 ≤s

⎭

≤ c2 (1 + M) sup [wρ (f, δ)], f ρ =1

 αi ,βj ([(t − x)2 + (τ − y)2 ]; x, y)ρ . Last inequality gives (5.4), which where δ = Lm,n completes the proof. 2

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