In this paper, we present a modification of the sequence of linear operators pro- posed by Lupa¸s [6] and studied by Agratini [1]

ON A FAMILY OF LINEAR AND POSITIVE OPERATORS IN WEIGHTED SPACES

AY ¸SEGÜL ERENÇÝN AND FATMA TA ¸SDELEN

ABANTÝZZETBAYSALUNIVERSITY

FACULTY OFARTS ANDSCIENCES

DEPARTMENT OFMATHEMATICS

14280, BOLU, TURKEY

erencina@hotmail.com

ANKARAUNIVERSITY

FACULTY OFSCIENCE

DEPARTMENT OFMATHEMATICS

06100 ANKARA, TURKEY

tasdelen@science.ankara.edu.tr

Received 06 March, 2007; accepted 30 May, 2007 Communicated by T.M. Mills

ABSTRACT. In this paper, we present a modification of the sequence of linear operators pro- posed by Lupa¸s [6] and studied by Agratini [1]. Some convergence properties of these operators are given in weighted spaces of continuous functions on positive semi-axis by using the same approach as in [4] and [5].

Key words and phrases: Linear positive operators, Weighted approximation, Rate of convergence.

2000 Mathematics Subject Classification. 41A25, 41A36.

1. INTRODUCTION

Lupa¸s in [6] studied the identity 1 (1−a)^α =

∞

X

k=0

(α)_k

k! a^k, |a|<1 and lettingα=nxandx≥0considered the linear positive operators

L^∗_n(f;x) = (1−a)^nx

∞

X

k=0

(nx)_k k! a^kf

k n

076-07

withf : [0,∞)→ R. Imposing the conditionL_n(1;x) = 1he found thata = 1/2. Therefore Lupa¸s proposed the positive linear operators

(1.1) L^∗_n(f;x) = 2^−nx

∞

X

k=0

(nx)_k 2^kk! f

k n

.

Agratini [1] gave some quantitative estimates for the rate of convergence on the finite interval [0, b]for anyb >0and also established a Voronovskaja-type formula for these operators.

We consider the generalization of the operators (1.1) (1.2) L_n(f;x) = 2^−anx

∞

X

k=0

(a_nx)_k 2^kk! f

k bn

, x∈R0, n ∈N,

whereR⁰ = [0,∞), N:= {1,2, . . .}and{an},{bn}are increasing and unbounded sequences of positive numbers such that

(1.3) lim

n→∞

1 bn

= 0, a_n bn

= 1 +O 1

bn

.

In this work, we study the convergence properties of these operators in the weighted spaces of continuous functions on positive semi-axis with the help of a weighted Korovkin type theorem, proved by Gadzhiev in [2, 3]. For this purpose, we now recall the results of [2, 3].

B_ρ: The set of all functionsf defined on the real axis satisfying the condition

|f(x)| ≤M_fρ(x),

whereMf is a constant depending only onf andρ(x) = 1 +x²,−∞< x < ∞.

The spaceBρis normed by

kfk_ρ= sup

x∈R

|f(x)|

ρ(x) , f ∈B_ρ. C_ρ: The subspace of all continuous functions belonging toB_ρ. C_ρ^∗: The subspace of all functionsf ∈C_ρfor which

|x|→∞lim f(x) ρ(x) =k, wherekis a constant depending onf.

Theorem A ([2, 3]). Let{T_n}be the sequence of linear positive operators which are mappings fromC_ρintoB_ρsatisfying the conditions

n→∞lim kT_n(t^ν, x)−x^νk_ρ= 0 ν = 0,1,2.

Then, for any functionf ∈C_ρ^∗,

n→∞lim kT_nf−fk_ρ= 0, and there exists a functionf^∗ ∈C_ρ\C_ρ^∗ such that

n→∞lim kT_nf^∗−f^∗k_ρ≥1.

2. AUXILIARY RESULTS

In this section we shall give some properties of the operators (1.2), which we shall use in the proofs of the main theorems.

Lemma 2.1. If the operators L_n are defined by (1.2), then for all x ∈ R0 and n ∈ N the following identities are valid

(2.1) L_n(1;x) = 1,

(2.2) L_n(t;x) = a_n

b_nx,

(2.3) L_n(t²;x) = a²_n

b²_nx²+ 2a_n b²_nx,

(2.4) L_n(t³;x) = a³_n

b³_nx³+ 6a²_n

b³_nx²+ 6a_n b³_nx and

(2.5) Ln(t⁴;x) = a⁴_n

b⁴_nx⁴+ 12a³_n

b⁴_nx³+ 36a²_n

b⁴_nx²+ 26a_n b⁴_nx.

Proof. It is clear that (2.1) holds.

By using the recurrence relation(α)_k = α(α+ 1)k−1, k ≥ 1for the function f(t) = twe have

Ln(t;x) = 1 b_n2^−anx

∞

X

k=1

(anx)k

2^k(k−1)!

= an

b_nx2^−anx

∞

X

k=1

(anx+ 1)k−1

2^k(k−1)!

= an

b_nx2^−(anx+1)

∞

X

k=0

(anx+ 1)k

2^kk!

= a_n bn

x.

In a similar way to that of (2.2), we can prove (2.3) – (2.5).

Lemma 2.2. If the operatorsLnare defined by (1.2), then for allx∈R⁰andn∈N (2.6) Ln (t−x)⁴;x

= a_n

b_n −1 4

x⁴+

12a³_n

b⁴_n −24a²_n

b³_n + 12a_n b²_n

x³ +

36a²_n

b⁴_n −24a_n b³_n

x²+ 26a_n b⁴_nx.

Lemma 2.3. If the operatorsL_nare defined by (1.2), then for allx∈R0 and sufficiently large n

(2.7) L_n (t−x)⁴;x

=O 1

b_n

x⁴+x³+x²+x .

3. MAINRESULT

In this part, we firstly prove the following theorem related to the weighted approximation of the operators in (1.2).

Theorem 3.1. LetL_nbe the sequence of linear positive operators (1.2) acting fromC_ρtoB_ρ. Then for each functionf ∈C_ρ^∗,

n→∞lim kL_n(f;x)−f(x)k_ρ= 0.

Proof. It is sufficient to verify the conditions of Theorem A which are

n→∞lim kL_n(t^ν, x)−x^νk_ρ= 0 ν= 0,1,2.

From (2.1) clearly we have

n→∞lim kL_n(1, x)−1k_ρ= 0.

By using (1.3) and (2.2) we can write

kL_n(t, x)−xk_ρ= sup

x∈R0

|L_n(t, x)−x|

1 +x²

=

an

b_n −1

sup

x∈R0

x 1 +x²

=O 1

b_n

sup

x∈R0

x 1 +x². This implies that

n→∞lim kL_n(t;x)−xk_ρ = 0.

Similarly, by the equalities (1.3) and (2.3) we find that kLn(t², x)−x²kρ= sup

x∈R0

|Ln(t², x)−x²| 1 +x² (3.1)

≤

a_n² b²_n −1

sup

x∈R0

x²

1 +x² + 2a_n b²_n sup

x∈R0

x 1 +x², which gives

n→∞lim

L_n t²;x

−x² ρ= 0.

Thus all conditions of Theorem A hold and the proof is completed.

Now, we find the rate of convergence for the operators (1.2) in the weighted spaces by means of the weighted modulus of continuity Ω(f, δ) which tends to zero as δ → 0 on an infinite interval, defined in [5]. We now recall the definition ofΩ(f, δ).

Letf ∈C_ρ^∗. The weighted modulus of continuity off is denoted by Ω(f, δ) = sup

|h|≤δ,x∈R0

|f(x+h)−f(x)|

(1 +h²)(1 +x²) . Ω(f, δ)has the following properties [4, 5].

Letf ∈C_ρ^∗, then

(i) Ω(f, δ)is a monotonically increasing function with respect toδ,δ≥0.

(ii) For everyf ∈C_ρ^∗,lim_δ→0Ω(f, δ) = 0.

(iii) For each positive value ofλ

Ω(f, λδ)≤2(1 +λ)(1 +δ²)Ω(f, δ).

(iv) For everyf ∈C_ρ^∗ andx, t∈R0 :

|f(t)−f(x)| ≤2

1 + |t−x|

δ

(1 +δ²)Ω(f, δ)(1 +x²)(1 + (t−x)²).

Theorem 3.2. Letf ∈C_ρ^∗. Then the inequality sup

x∈R0

|L_n(f, x)−f(x)|

(1 +x²)³ ≤MΩ f, b^−1/4_n

is valid for sufficiently largen, whereM is a constant independent ofa_nandb_n. Proof. By the definition ofL_nand the property(iv), we get

|L_n(f, x)−f(x)| ≤2(1 +δ_n²)Ω(f, δ_n)(1 +x²)2^−anx

∞

X

k=0

(a_nx)_k

2^kk! A₁(x), where

A₁(x) =



1 +

k bn −x

δ_n



 1 + k

b_n −x 2!

. Then for allx,_b^k

n ∈R0,by using the following inequality (see[5, p. 578]) A₁(x)≤2(1 +δ²_n)





1 + k

bn −x 4

δ_n⁴





, we can write

|Ln(f, x)−f(x)| ≤16Ω(f, δn)(1 +x²) 1 + 1 δ_n⁴2^−anx

∞

X

k=0

(a_nx)_k 2^kk!

k b_n −x

4!

= 16Ω(f, δ_n)(1 +x²)

1 + 1

δ⁴_nL_n (t−x)⁴;x

. Thus by means of (2.7), we have

|L_n(f, x)−f(x)| ≤16Ω(f, δ_n)(1 +x²)

1 + 1 δ⁴_nO

1 b_n

x⁴+x³ +x²+x

. If we chooseδ_n=b^−1/4n for sufficiently largen,then we find

sup

x∈R0

|L_n(f, x)−f(x)|

(1 +x²)³ ≤MΩ f, b^−1/4_n ,

which is the desired result.

REFERENCES

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[2] A.D. GADZHIEV, The convergence problem for a sequence of positive linear operators on un- bounded sets, and theorems analogous to that of P.P. Korovkin, Soviet Math. Dokl., 15(5) (1974), 1433–1436.

[3] A.D. GADZHIEV, On P.P. Korovkin type theorems, Math. Zametki, 20(5) (1976), 781–786 (in Rus- sian), Math. Notes, 20(5-6) (1976), 968–998 (in English).

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