ON A FAMILY OF LINEAR AND POSITIVE
OPERATORS IN WEIGHTED SPACES
Linear and Positive Operators
in Weighted Spaces
AYŞEGÜL ERENÇÝN
FATMA TAŞDELEN
Abant Ýzzet Baysal University
Faculty of Arts and Sciences
Department of Mathematics
14280, Bolu, Turkey
EMail: erencina@hotmail.com
Ankara University
Faculty of Science
Department of Mathematics
06100 Ankara, Turkey
EMail: tasdelen@science.ankara.edu.tr
Ayşegül Erençýn and Fatma Taşdelen
vol. 8, iss. 2, art. 39, 2007
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Contents
Received:
06 March, 2007
Accepted:
30 May, 2007
Communicated by:
T.M. Mills
2000 AMS Sub. Class.:
41A25, 41A36.
Key words:
Linear positive operators, Weighted approximation, Rate of convergence.
Abstract:
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].
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Contents
1
Introduction
3
2
Auxiliary Results
5
3
Main Result
7
Linear and Positive Operators
in Weighted Spaces
Ayşegül Erençýn and Fatma Taşdelen
vol. 8, iss. 2, art. 39, 2007
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1.
Introduction
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
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
(nx)k
k
∗
−nx
f
.
(1.1)
Ln (f ; x) = 2
k k!
2
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
(an x)k
k
−an x
(1.2)
Ln (f ; x) = 2
f
, x ∈ R0 , n ∈ N,
k
2
k!
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
Linear and Positive Operators
in Weighted Spaces
Ayşegül Erençýn and Fatma Taşdelen
vol. 8, iss. 2, art. 39, 2007
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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
|f (x)|
, f ∈ Bρ .
x∈R ρ(x)
Cρ : The subspace of all continuous functions belonging to Bρ .
Cρ∗ : The subspace of all functions f ∈ Cρ for which
Linear and Positive Operators
in Weighted Spaces
Ayşegül Erençýn and Fatma Taşdelen
vol. 8, iss. 2, art. 39, 2007
kf kρ = sup
f (x)
= k,
|x|→∞ ρ(x)
where k is a constant depending on f .
lim
Theorem A ([2, 3]). Let {Tn } be the sequence of linear positive operators which
are mappings from Cρ into Bρ satisfying the conditions
ν
ν
lim kTn (t , x) − x kρ = 0
n→∞
Then, for any function f ∈
ν = 0, 1, 2.
Cρ∗ ,
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lim kTn f − f kρ = 0,
n→∞
and there exists a function f ∗ ∈ Cρ \Cρ∗ such that
lim kTn f ∗ − f ∗ kρ ≥ 1.
n→∞
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 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)
Linear and Positive Operators
in Weighted Spaces
Ayşegül Erençýn and Fatma Taşdelen
vol. 8, iss. 2, art. 39, 2007
Ln (t; x) =
(2.2)
an
x,
bn
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(2.3)
(2.4)
Ln (t2 ; x) =
Ln (t3 ; x) =
a2n 2
x
b2n
+2
an
x,
b2n
a3n 3
a2n 2
an
x
+
6
x +6 3x
3
3
bn
bn
bn
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and
(2.5)
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Ln (t4 ; x) =
a4n 4
x
b4n
+
a3n 3
12 4 x
bn
+
a2n 2
36 4 x
bn
+ 26
an
x.
b4n
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
Close
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
=
Linear and Positive Operators
in Weighted Spaces
Ayşegül Erençýn and Fatma Taşdelen
vol. 8, iss. 2, art. 39, 2007
Title Page
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
(2.6) Ln (t − x)4 ; x =
4
an
a3n
a2n
an
4
− 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
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3.
Main Result
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.
Linear and Positive Operators
in Weighted Spaces
Ayşegül Erençýn and Fatma Taşdelen
vol. 8, iss. 2, art. 39, 2007
Title Page
From (2.1) clearly we have
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lim kLn (1, x) − 1kρ = 0.
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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→∞
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Similarly, by the equalities (1.3) and (2.3) we find that
(3.1)
|Ln (t2 , x) − x2 |
kLn (t2 , x) − x2 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
Linear and Positive Operators
in Weighted Spaces
which gives
lim Ln t2 ; x − x2 ρ = 0.
n→∞
Ayşegül Erençýn and Fatma Taşdelen
vol. 8, iss. 2, art. 39, 2007
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).
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(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
Linear and Positive Operators
in Weighted Spaces
|Ln (f, x) − f (x)|
−1/4
sup
≤
M
Ω
f,
b
n
(1 + x2 )3
x∈R0
Ayşegül Erençýn and Fatma Taşdelen
vol. 8, iss. 2, art. 39, 2007
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 + x2 )2−an x
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∞
X
(an x)k
k=0
where
2k k!

k
2 !
bn − x
k
 1+
A1 (x) = 1 +
−x
.
δn
bn

Then for all x,
k
bn
Contents
A1 (x),
∈ R0 , by using the following inequality (see[5, p. 578])

4 
k
−x 
bn

A1 (x) ≤ 2(1 + δn2 ) 1 +
,
δn4
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we can write
4
∞
1 −an x X (an x)k k
|Ln (f, x) − f (x)| ≤ 16Ω(f, δn )(1 + x ) 1 + 4 2
−x
δn
2k k!
bn
k=0
1
4
2
= 16Ω(f, δn )(1 + x ) 1 + 4 Ln (t − x) ; x .
δn
!
2
Thus by means of (2.7), we have
Linear and Positive Operators
in Weighted Spaces
Ayşegül Erençýn and Fatma Taşdelen
1
1
2
4
3
2
|Ln (f, x) − f (x)| ≤ 16Ω(f, δn )(1 + x ) 1 + 4 O
x +x +x +x .
δn
bn
−1/4
If we choose δn = bn
for sufficiently large n, then we find
|Ln (f, x) − f (x)|
−1/4
sup
≤
M
Ω
f,
b
,
n
(1 + x2 )3
x∈R0
which is the desired result.
vol. 8, iss. 2, art. 39, 2007
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References
[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.
Linear and Positive Operators
in Weighted Spaces
[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).
Ayşegül Erençýn and Fatma Taşdelen
[4] N. ÝSPÝR, On modified Baskakov operators on weighted spaces, Turkish J.
Math., 25 (2001), 355–365.
Title Page
vol. 8, iss. 2, art. 39, 2007
Contents
[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.
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[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.
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