Volume 8 (2007), Issue 2, Article 39, 6 pp.
ON A FAMILY OF LINEAR AND POSITIVE OPERATORS IN WEIGHTED
SPACES
AYŞEGÜL ERENÇÝN AND FATMA TAŞDELEN
A BANT Ý ZZET BAYSAL U NIVERSITY
FACULTY OF A RTS AND S CIENCES
D EPARTMENT OF M ATHEMATICS
14280, B OLU , T URKEY
erencina@hotmail.com
A NKARA U NIVERSITY
FACULTY OF S CIENCE
D EPARTMENT OF M ATHEMATICS
06100 A NKARA , T URKEY
tasdelen@science.ankara.edu.tr
Received 06 March, 2007; accepted 30 May, 2007
Communicated by T.M. Mills
A BSTRACT. In this paper, we present a modification of the sequence of linear operators proposed by Lupaş [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. I NTRODUCTION
Lupaş in [6] studied the identity
∞
X (α)k
1
=
ak ,
(1 − a)α
k!
k=0
|a| < 1
and letting α = nx and x ≥ 0 considered the linear positive operators
∞
X
(nx)k k
k
∗
nx
Ln (f ; x) = (1 − a)
a f
k!
n
k=0
076-07
2
AY ŞEGÜL E RENÇÝN AND FATMA TA ŞDELEN
with f : [0, ∞) → R. Imposing the condition Ln (1; x) = 1 he found that a = 1/2. Therefore
Lupaş proposed the positive linear operators
∞
X
k
(nx)k
∗
−nx
(1.1)
Ln (f ; x) = 2
f
.
k
2 k!
n
k=0
Agratini [1] gave some quantitative estimates for the rate of convergence on the finite interval
[0, b] for any b > 0 and also established a Voronovskaja-type formula for these operators.
We consider the generalization of the operators (1.1)
∞
X
k
(an x)k
−an x
(1.2)
Ln (f ; x) = 2
f
, x ∈ R0 , n ∈ N,
k k!
2
b
n
k=0
where R0 = [0, ∞), N := {1, 2, . . . } and {an }, {bn } are increasing and unbounded sequences
of positive numbers such that
1
an
1
(1.3)
lim
= 0,
=1+O
.
n→∞ bn
bn
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].
Bρ : The set of all functions f defined on the real axis satisfying the condition
|f (x)| ≤ Mf ρ(x),
where Mf is a constant depending only on f and ρ(x) = 1 + x2 , −∞ < x < ∞.
The space Bρ is normed by
kf kρ = sup
x∈R
|f (x)|
,
ρ(x)
f ∈ Bρ .
Cρ : The subspace of all continuous functions belonging to Bρ .
Cρ∗ : The subspace of all functions f ∈ Cρ for which
f (x)
= k,
|x|→∞ ρ(x)
lim
where k is a constant depending on f .
Theorem A ([2, 3]). Let {Tn } be the sequence of linear positive operators which are mappings
from Cρ into Bρ satisfying the conditions
lim kTn (tν , x) − xν kρ = 0
n→∞
ν = 0, 1, 2.
Then, for any function f ∈ Cρ∗ ,
lim kTn f − f kρ = 0,
n→∞
and there exists a function f ∗ ∈ Cρ \Cρ∗ such that
lim kTn f ∗ − f ∗ kρ ≥ 1.
n→∞
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 39, 6 pp.
http://jipam.vu.edu.au/
O N A FAMILY OF L INEAR AND P OSITIVE O PERATORS IN W EIGHTED S PACES
3
2. AUXILIARY R ESULTS
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 Ln are defined by (1.2), then for all x ∈ R0 and n ∈ N the
following identities are valid
Ln (1; x) = 1,
(2.1)
Ln (t; x) =
(2.2)
(2.3)
(2.4)
Ln (t2 ; x) =
Ln (t3 ; x) =
an
x,
bn
a2n 2
an
x + 2 2 x,
2
bn
bn
a3n 3
a2n 2
an
x
+
6
x +6 3x
3
3
bn
bn
bn
and
(2.5)
Ln (t4 ; x) =
a4n 4
a3n 3
a2n 2
an
x
+
12
x
+
36
x + 26 4 x.
4
4
4
bn
bn
bn
bn
Proof. It is clear that (2.1) holds.
By using the recurrence relation (α)k = α(α + 1)k−1 , k ≥ 1 for the function f (t) = t we
have
∞
1 −an x X (an x)k
Ln (t; x) = 2
bn
2k (k − 1)!
k=1
∞
an −an x X (an x + 1)k−1
= x2
bn
2k (k − 1)!
k=1
∞
an −(an x+1) X (an x + 1)k
= x2
bn
2k k!
k=0
an
= x.
bn
In a similar way to that of (2.2), we can prove (2.3) – (2.5).
Lemma 2.2. If the operators Ln are defined by (1.2), then for all x ∈ R0 and n ∈ N
4
an
a3n
a2n
an
4
4
(2.6)
Ln (t − x) ; x =
− 1 x + 12 4 − 24 3 + 12 2 x3
bn
bn
bn
bn
2
an
an
an
+ 36 4 − 24 3 x2 + 26 4 x.
bn
bn
bn
Lemma 2.3. If the operators Ln are defined by (1.2), then for all x ∈ R0 and sufficiently large
n
1
4
(2.7)
Ln (t − x) ; x = O
x4 + x3 + x2 + x .
bn
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 39, 6 pp.
http://jipam.vu.edu.au/
4
AY ŞEGÜL E RENÇÝN AND FATMA TA ŞDELEN
3. M AIN R ESULT
In this part, we firstly prove the following theorem related to the weighted approximation of
the operators in (1.2).
Theorem 3.1. Let Ln be the sequence of linear positive operators (1.2) acting from Cρ to Bρ .
Then for each function f ∈ Cρ∗ ,
lim kLn (f ; x) − f (x)kρ = 0.
n→∞
Proof. It is sufficient to verify the conditions of Theorem A which are
lim kLn (tν , x) − xν kρ = 0
n→∞
ν = 0, 1, 2.
From (2.1) clearly we have
lim kLn (1, x) − 1kρ = 0.
n→∞
By using (1.3) and (2.2) we can write
|Ln (t, x) − x|
kLn (t, x) − xkρ = sup
1 + x2
x∈R
0
an
x
= − 1 sup
2
bn
x∈R0 1 + x
1
x
=O
sup
.
bn x∈R0 1 + x2
This implies that
lim kLn (t; x) − xkρ = 0.
n→∞
Similarly, by the equalities (1.3) and (2.3) we find that
(3.1)
|Ln (t2 , x) − x2 |
kLn (t , x) − x kρ = sup
1 + x2
x∈R0
an2
x2
an
x
≤ 2 − 1 sup
+ 2 2 sup
,
2
bn
bn x∈R0 1 + x2
x∈R0 1 + x
2
2
which gives
lim Ln t2 ; x − x2 ρ = 0.
n→∞
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 of Ω(f, δ).
Let f ∈ Cρ∗ . The weighted modulus of continuity of f 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].
Let f ∈ Cρ∗ , then
(i) Ω(f, δ) is a monotonically increasing function with respect to δ, δ ≥ 0.
(ii) For every f ∈ Cρ∗ , limδ→0 Ω(f, δ) = 0.
(iii) For each positive value of λ
Ω(f, λδ) ≤ 2(1 + λ)(1 + δ 2 )Ω(f, δ).
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 39, 6 pp.
http://jipam.vu.edu.au/
O N A FAMILY OF L INEAR AND P OSITIVE O PERATORS IN W EIGHTED S PACES
5
(iv) For every f ∈ Cρ∗ and x, t ∈ R0 :
|t − x|
|f (t) − f (x)| ≤ 2 1 +
(1 + δ 2 )Ω(f, δ)(1 + x2 )(1 + (t − x)2 ).
δ
Theorem 3.2. Let f ∈ Cρ∗ . Then the inequality
sup
x∈R0
|Ln (f, x) − f (x)|
−1/4
≤
M
Ω
f,
b
n
(1 + x2 )3
is valid for sufficiently large n, where M is a constant independent of an and bn .
Proof. By the definition of Ln and the property (iv), we get
|Ln (f, x) − f (x)| ≤ 2(1 +
δn2 )Ω(f, δn )(1
2
−an x
+ x )2
∞
X
(an x)k
k=0
where
2k k!
A1 (x),

k
2 !
−
x
bn
 1+ k −x
A1 (x) = 1 +
.
δn
bn

Then for all x, bkn ∈ R0 , by using the following inequality (see[5, p. 578])

4 
k
−x 
bn

A1 (x) ≤ 2(1 + δn2 ) 1 +
,
δn4
we can write
4
∞
1 −an x X (an x)k k
2
−x
|Ln (f, x) − f (x)| ≤ 16Ω(f, δn )(1 + x ) 1 + 4 2
k k!
δn
2
b
n
k=0
1
4
2
= 16Ω(f, δn )(1 + x ) 1 + 4 Ln (t − x) ; x .
δn
!
Thus by means of (2.7), we have
1
1
4
3
2
|Ln (f, x) − f (x)| ≤ 16Ω(f, δn )(1 + x ) 1 + 4 O
x +x +x +x .
δn
bn
2
−1/4
If we choose δn = bn
for sufficiently large n, then we find
sup
x∈R0
|Ln (f, x) − f (x)|
−1/4
≤
M
Ω
f,
b
,
n
(1 + x2 )3
which is the desired result.
R EFERENCES
[1] O. AGRATINI, On a sequence of linear positive operators, Facta Universitatis (Nis), Ser. Math.
Inform., 14 (1999), 41-48.
[2] A.D. GADZHIEV, The convergence problem for a sequence of positive linear operators on unbounded 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 Russian), Math. Notes, 20(5-6) (1976), 968–998 (in English).
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 39, 6 pp.
http://jipam.vu.edu.au/
6
AY ŞEGÜL E RENÇÝN AND FATMA TA ŞDELEN
[4] N. ÝSPÝR, On modified Baskakov operators on weighted spaces, Turkish J. Math., 25 (2001), 355–
365.
[5] N. ÝSPÝR AND Ç. ATAKUT, Approximation by modified Szasz-Mirakjan operators on weighted
spaces, Proc. Indian Acad. Sci.(Math. Sci), 112(4) (2002), 571–578.
[6] A. LUPAŞ, The approximation by some positive linear operators, In: Proceedings of the International Dortmund Meeting on Approximation Theory (M.W. Müller et al.,eds.), Akademie Verlag,
Berlin (1995), 201–209.
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 39, 6 pp.
http://jipam.vu.edu.au/