7_Lucia Cabulea, Mihaela Aldea. 163KB Jun 04 2011 12:08:17 AM

ON SOME PROPERTIES OF KANTOROVICH BIVARIATE

OPERATORS

by

Lucia A. C

ă

bulea and Mihaela Aldea

Abstract. In this paper we continue our earlier investigations concerning the use of probabilistic methods for constructing linear positive operators useful in approximation theory of functions. The main result of this paper consists in introducing and investigating the approximation properties of Kantorovich bivariate operator, which is an integral linear positive operator reproducing the linear functions.

Keywords:

bivariate operator

1. THE OPERATORS OF KANTOROVICH

Let m∈Nbe fixed. The operators K_m:L₁([0,1])→C([0,1]), defined by

(

)

∑

∫

=

+ +

+ −

−     +

= m

k

m k

m k k m k

m x x f t dt

k m m

x f K

0

1 1

1

) ( )

1 ( )

1 ( )

( (1.1)

are the operators of Kantorovich.

If we noticed χ_m the characteristic function of

  

 

+1 1 , 0

m then the operators

(1.1) can be define

(

)

∑

∫

= 

 

 

+ − +

= m

k

m k

m

m dt

m k t t f x p m

x f K

0

1 0 ,

1 )

( ) ( )

1 ( )

( χ , (1.2)

where _mk xk x m k

k m

p − −

   

= (1 )

, , k =0,m , x∈[0,1] are the Bernstein polynomials.

Lemma[1]. The operators of Kantorovich satisfy the relations

ii)

(

)

) 1 ( 2

1 1

) (

1

+ + + =

m x m

m x e

K_m ,

iii)

(

)

2 2

2 2 2

) 1 ( 3

1 )

1 (

2 )

1 (

) 1 ( ) (

+ + + + +− =

m x m

m x

m m m x e

K_m .

iv) For all f∈L₁

( )

[ ]

0,1 ,

(

)

( )

(

B ₁F

)

(x)

dx d x f

K_m = _m+ , where F x =

∫

xf t dt

0 ( )

)

( and

( )

∑

=

− _

     −

    = m

k

k m k

m

m k f x x k m x

f B

0

) 1 ( )

( , x∈

[ ]

0,1 are the operators of Bernstein.

Theorem [8]. The operators of Kantorovich have the properties

i) K_mf f

mlim→∞ = , uniformly on [0,1], (∀)f ∈C

[ ]

0,1 . ii) K_mf f

mlim→∞ = , (∀)f ∈Lp

[ ]

0,1 , p≥1.

2. THE OPERATORS OF STANCU-KANTOROVICH

The Stancu polynomials [7] defined by

( )

∑

( )

= 

    = n

k k n n

n k f x w x

f S

0 ,

; α

α _{, x∈I=}

[ ]

_{0,1 , α ≥0}

where

( )

( ) ( )

( α) α α

α

− − − − ₋

   

= , _, ,

,

1 ) 1 (

n k n k

k n

x x

k n x

w , x(k,−α) =x(x+α)...(x+(k−1)α) , can be used for constructing the operators of Stancu-Kantorovich[5]

( ) ( )

∑

( )

∫

( )

=

++

+

+

=

n

k

n k

n k k n

n

f

x

n

w

x

f

t

dt

K

0

1 1

1 ,

1

;

α

α _{. (2.1)}

It is clear that the operators defined by (2.1) are linear and positive too. In the particular case

α

=0, the operator reduces obviously to the

n

th classical Kantorovich operator defined by (1.1).

i) K_nα

( )

1;x =1 ,

ii)

(

)

) 1 ( 2

2 1 ;

+ − = −

n x x

x t

K_nα ,

iii)

(

( )

)

( ) ( )

( )

2 2

2

1 3

1 1

1 1

1 1

;

+ + +

− ++ −

= −

n n

n n x x x x t

K_n α

α

α _,

iv) If

n C ≤ ≤α

0 , with C a positive constant, it follows

( )

(

)

( )

( ) ( )

_

 

+− + + ≤

− x

n x x n

C x x t

K_nα ψ_n

1 1 1

1 ;

2 2 2 2

where

( )

  

∈∈ =

n n n

E I x

E x x

\ ,

0 , 1

ψ , with

  

 ₋

=

n A n A

E_n ,1 and A > 0 fixed.

Theorem. Let

( )

K

_n( )α _n≥1 be defined by (2.1) and

( ) ( ) ( )

(

1/

)

( )

2 1

1 3

1 1

1 ,

+ +

+ +

+

= −

n p

n n p n

p

α

β .

If r≥3 is an integer number and

( )

r p n

r

2 1 ,

/

1 _≤

β , n∈N, the for every

function f ∈L_p

[ ]

0,1 we have

( )

(

( )

(

( )

)

)

p r

r p r

p

n f f C n p f r n p f

K α − ≤ _, β , +ω 2 β1/ , ,

with C_p,r a positive constant independent of f and n.

3.THE BIVARIATE OPERATORS OF KANTOROVICH

(

)( ) (

)( )

∑∑

( )

( )

∫ ∫

( )

= =

+ +

+ + +

+ −

− ₋

− +

+

= m

k n

h

m k

m k n

h

n h h n h k m k

mnf x y m n x x y y f u v dudv

K

0 0

1 1

1 1 1

1

, 1

1 1

1

,

where f belongs to the spaces L₁

(

[ ] [ ]

0,1× 0,1

)

.

The operators defined by (3.1) are the bivariate operators of Kantorovich. Lemma . The bivariate operators of Kntorovich satisfy the relations:

i)

(

K_mne₀₀

)( )

x,y =1,

ii)

(

)( )

) 1 ( 2

1 1

,

10 = ₊ + ₊

m x m

m y x e

K_mn ,

iii)

(

)

) 1 ( 2

1 1

) , (

01 = ₊ + ₊

n y n

n y x e

K_mn ,

iv)

(

)( )

  



+ + +   



+ + + =

) 1 ( 2

1 1

) 1 ( 2

1 1

,

11

n y n

n m

x m

m y

x e

K_mn ,

v)

(

)( )

2 2

2 2 20

) 1 ( 3

1 )

1 (

2 )

1 (

) 1 ( ,

+ + + + +− =

m x m

m x

m m m y x e

K_mn ,

vi)

(

₀₂

)( )

₂ 2 _{2 2}

) 1 ( 3

1 )

1 (

2 )

1 (

) 1 ( ,

+ + + + +

− =

n y n

n y

n n n y x e

K_mn .

Theorem . The bivariate operators of Kantorovich have the properties

i) K_mnf f

nlim→∞ = , uniformly on

[ ] [ ]

0,1× 0,1 ,

( )

∀ f ∈C

(

[ ] [ ]

0,1× 0,1

)

, ii) K_mnf f

n→∞ =

lim ,

( )

∀ f ∈L_p

(

[ ] [ ]

0,1 × 0,1

)

, p≥1.

4.THE BIVARIATE OPERATORS OF STANCU–KANTOROVICH

(

) ( )(

)

∑∑

( ) ( )

∫ ∫

++

( )

+ + + + = = + + = 1 1 1 1 1 1 0 0 , , , 1 1 , ; n k n k m h m h n k m h k m k n

nm f x y n m w x w y f u v dudv

Kα α α (4.1)

where

( )

( ) ( ) ( α) α α α − − − − ₋     = , , , , 1 ) 1 ( n k n k k n x x k n x

w , x(k,−α) =x(x+α)...(x+(k−1)α) and

( )

( ) ₍ α₎( ) α α α − − − − ₋     = , , , , 1 ) 1 ( m h m h h m y y h m x

w , y(h,−α) = y(y+α)...(y+(h−1)α) .

The operators defined by (4.1) are the bivariate operators of Stancu-Kantorovich.

Lemma . The operators defined by (4.1) satisfy the relations

i) K_nmα

(

1;x,y

)

=1,

ii)

(

) ( )

1 2 2 1 , ; + − = − n x y x x u

K_nmα ,

iii)

(

)

) 1 ( 2 2 1 , ; + − = − m y y x y v

K_nmα ,

iv)

(

(

)(

)

)

) 1 ( 2 2 1 ) 1 ( 2 2 1 , ; + − ⋅ + − = − − m y n x y x y v x u K_nmα

v)

(

(

)

)

( ) ( )

2 2 2 ) 1 ( 3 1 1 1 1 1 1 , ; + + + − ++ − = − n n n n x x y x x u

K_nm α

α

α _,

vi)

(

(

)

)

( ) ( )

2 2 2 ) 1 ( 3 1 1 1 1 1 1 , ; + + + − ++ − = − m m m m y y y x y v

K_nm α

α

α _.

REFERENCES

[1]. Agratini, O., Aproximare prin operatori liniari, Presa Universitară Clujeană, 2000. [2]. Căbulea, L., Aldea, M., Generalizations of Kantorovich type, Analele Universităţii „Aurel Vlaicu” Arad, Seria Matematica, Arad 2002, 27-32.

D. Stancu, Studia Univ. Babeş-Bolyai, Math. XXXVII (1992), 3-36.

[4]. Della Vechia, B., Mache, D. H., On approximation properties of Stancu-Kantorovich operators, Rev. D'Analyse Num. Et de Théorie de L'Approx., 27(1998), no. 1-2, 1-10. [5]. Kantorovich, L. V., Surcertains dévelopments suivant Ies polynômes de la forme de S. Bemstein, I, II, C. R. Acad. URSS (1930), 563-568, 595-600.

[6]. Razi, Q., Approximation of a function by Kantorovich type operators, Mat. Veşnic. 41(1989), 183-192.

[7]. Sendov, B., Popov, V. A., The Averaged Moduli of Smoothness, Pure and Applied Mathematics, John Wiley & Soons, 1988.

[8]. Stancu D. D., Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl. 8(1969), 1173-1194.

[9]. Stancu, D. D., Coman, Gh., Agratini, O., Trîmbiţaş, R., Analiză numericăşi teoria aproximării, voi. l, Presa Universitară Clujeană, 2001.

Authors:

Lucia A. Căbulea,"1 Decembrie 1918" University of Alba lulia, România. Departament of Mathematics, e-mail: lcabulea@uab.ro