On a Modification of Operators 1 Introduction

International Mathematical Forum, 4, 2009, no. 45, 2211 - 2215

On a Modification of Operators Cihan Aksop Gazi University, Department of Statistics 06500 Teknikokullar, Ankara, Turkey [email protected] Abstract In this paper, a modification of linear positive operators which results better error estimates are studied.

Mathematics Subject Classification: 41A36 Keywords: Linear positive operators, Korovkin-Bohman theorem

1

Introduction

P. P. Korovkin and H. Bohman [1] obtained the necessary and sufficient conditons for a positive linear operator sequence of {Ln } on C ([a, b]) tends to the identity operator with respect to the uniform norm of space C ([a, b]). According to this theorem, if the {Ln (ei ; x)}, i = 0, 1, 2 sequences convergence uniformly to ei (x), i = 0, 1, 2 respectively, then {Ln (f ; x)} convergence to f uniformly on [a, b], for each f ∈ C ([a, b]). Here ei (x) are called test functions and ei (x) = xi . There are several examples of operators which satistfy Korovkin-Bohman theorem. Though out of them Bernstein, Szasz, Baskakov, Durrmayer are classical examples. They are studied extensively in literatur for their convergence rates. The rate of the convergence is an important property for all linear positive operator sequences and gives the error that is made by using these operators. One of the measure for any {Ln } of positive linear operators on C ([a, b]) 

 1 1/2 |Ln (f ; x) − f (x)| ≤ ωf (δ) Ln (e0 ; x) + (Ln (e0 ; x)) αn (x) δ + |f (x)| |Ln (e0 ; x) − e0 (x)| where ωf is the modulas of continuty of f , f ∈ C ([a, b]), x ∈ [a, b] and αn2 (x) = Ln (φt ; x) with φt (x) = (t − x)2

(1)

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Cihan Aksop

To obtain better convergence rates, modifying the operator is a frequently used method. King [2] has presented a modification of Bernstein operators as follows:    n   k n k n−k (rn (x)) (1 − rn (x)) f Vn (f ; x) = k n k=0

for f ∈ C ([0, 1]) and 0 ≤ x ≤ 1 where {rn (x)} is a sequence contiuous functions defined on [0, 1] with 0 ≤ rn (x) ≤ 1. Using this idea, in this paper we study the modification of L∗n (f ; x) = Ln (f ; rn (x)) with a proper chose of {rn } function sequence and {Ln } operator sequence is studied.

2

Construction of the class {L∗n}

Throughout the paper, we consider a squence {Ln } of linear positive operators with satisfy following three condition: Ln (e0 ; x) = 1 Ln (e1 ; x) = an x + bn Ln (e2 ; x) = cn x2 + dn x + en

(2)

where {an }, {bn }, {cn }, {dn } and {en } sequences of real number with lim an , cn = 1, lim bn , dn , en = 0 and cn > 0.

n→∞

n→∞

(3)

We also consider the modification of {Ln } sequences as follows: L∗n (f ; x) = Ln (f ; rn (x)) for a proper chose of {rn } function sequence. Theorem 2.1 ([2]) limn→∞ L∗n (f ; x) = f (x) for each f ∈ C ([a, b]), x ∈ [a, b] if and only if limn→∞ rn (x) = x. At this moment, we want to minimize the error estimate in (1) for {L∗n } while Korovkin-Bohman theorem is fullfiled. Observe that, for the operators that fullfill (2-3), the error estimate in (1) reduce to   1 ∗ |Ln (f ; x) − f (x)| ≤ ωf (δ) 1 + αn (x) δ

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On a modification of operators

  where αn2 (x) = L∗n (t − x)2 ; x . To minimize the right hand side of this inequation, we need to minimize αn2 . By taking derivates of αn2 with respect to rn we obtain; d 2 α (x) = 2cn rn + dn − 2xan = 0 dr n −dn + 2xan ⇒ rn (x) = 2cn 2 d 2 α (x) = 2cn ≥ 0 dr 2 n

(4)

It is easy to see that rn satisfy the conditions of Theorem, so; {L∗n } satisfy Korovkin-Bohman theorem. The error estimate for these operators are: |L∗n

 (f ; x) − f (x)| ≤ ωf (δ) 1 +

 1 αn (rn (x)) δ

   2 a2n d 1 a d n n 1− x2 + = ωf (δ) 1 + x − n + en δ cn cn 4cn

Examples The linear positive operators we are refering to are Bernstein, Szasz, Baskakov and Kantorovich operators. (1) Bernstein operator For this class, the identities an = 1, bn = 0, cn = n−1 , dn = n1 and en = 0, n ∈ N hold, consequently conditions (2-3) are fullfilled. n Thus   n 1 rn (x) = 2x − 2 (n − 1) n and the corresponding modified operators are defined by  k  n−k   n  k n  n 1 1 ∗ f Ln (f ; x) = x− 1−x+ n − 1 k=0 k 2n 2n n with an error estimate of:

   1 1 n 1 1− x2 + x− ωf (δ) 1 + δ n−1 n−1 4n (n − 1) (2) Szasz operator In this case, conditions (2-3) are fullfield with an = 1, bn = 0, cn = 1, dn = n1 and en = 0, n ∈ N. We get rn (x) = x +

1 2n

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Cihan Aksop

and the modification as follows: L∗n

−nx−1/2

(f ; x) = e

  ∞  k (nx + 1/2)k f k! n k=0

and its error estimate



1 ωf (δ) 1 + δ



1 1 x− 2 n 4n



(3) Baskakov operator For Baskakov operator, it is easy to see that an = 1, bn = 0, cn = 1 + n1 , dn = n1 and en = 0, n ∈ N. Using (4) we obtain; rn (x) =

1 n x− n+1 2 (n + 1)

and modified Baskakov operator:  k ∞   n 1 n+k−1 ∗ Ln (f ; x) = x− k n+1 2 (n + 1) k=0  −(n+k) 1 n x− × 1+ n+1 2 (n + 1) and its error estimate is:

   n 1 1 1 1− ωf (δ) 1 + x2 + x− δ n+1 n+1 4 (n + 1) (4) Kantorovich operator In this case an = dn =

2n (n+1)2

and en =

1 , 3(n+1)2

n ∈ N. Then

n , n+1

bn =

1 , 2(n+1)

cn =

n(n−1) , (n+1)2

1 n+1 x− , n≥2 n−1 n−1 So the modified Kantorovic operator is  k n   n+1 1 n ∗ Ln (f ; x) = (n + 1) x− k n − 1 n−1 k=0  n−k k+1 n+1 n+1 1 × 1− f (t) dt x+ k n−1 n−1 n+1 rn (x) =

and error estimate    n 1 2n ωf (δ) 1 + 1− x2 + x δ n−1 (n + 1)2 (n − 1) 1 4n + − 2 (n + 1) (n − 1) 3 (n + 1)2

1/2 

On a modification of operators

3

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Conclution

The modification of operators plays an important role in approximation theory to obtain better approximations. In this paper, we present a new choise of King’s type modification which results best error estimates.

References [1] F. Altomore, M. Campiti, Korovkin-type approximation theory and its applications, Gruyter 1994. [2] J. P. King, Positive linear operators which preserve x2 , Acta Mathematica Hungarica 29 (2003) 203–208. Received: April, 2009