Karaelmas Fen ve Müh. Derg. 7(1):250-252, 2017
Karaelmas Fen ve Mühendislik Dergisi
Journal home page: http://fbd.beun.edu.tr
Research Article
Weighted Approximation Properties of Positive Linear Operators Based on q-Integer
Doğrusal Pozitif Operatörlerin q Tamsayılarla Ağırlıklı Yaklaşım Özellikleri
Tülin Coşkun
Department of Mathematics, Bülent Ecevit University, Zonguldak, Turkey
Abstract
This paper deals with q-analogues of linear positive operators defined in weighted space of continuous functions defined on real axis.
We study, using q-calculus, the existence of Korovkin’s theorem in the spaces of continuous and unbounded functions defined on
unbounded sets.
Keywords: Korovkin theorem, Linear positive operator, Q-calculus, Weighted space
Öz
Bu çalışma reel eksen üzerinde tanımlı sürekli fonksiyonların ağırlıklı uzayında tanımlı doğrusal pozitif operatörlerin q benzerleri
ile ilgilidir. q analizi kullanarak, sınırsız kümeler üzerinde tanımlı, sınırsız ve sürekli fonksiyonların uzayında Korovkin teoreminin
varlığını araştıracağız.
Anahtar Kelimeler: Korovkin teoremi, Doğrusal pozitif operatörler, Q-analiz, Ağırlıklı uzay
1. Introduction
The history of q-calculus, dates back to the eighteenth
century. It can in fact be taken as far back as Euler.
Afterwards, many remarkable results were obtained in the
nineteenth century. In the second half of the twentieth
century there was a significant increase of activity in the
area of q-calculus due to applications of q-calculus in
mathematics and physics. During the last two decades, the
applications of q-calculus emerged as a new area in the field
of approximation theory [3]. Several generalizations of wellknown positive linear operators based on q-integers were
introduced by several authors. Approximation theory has
important applications in functional analysis, numerical
solutions of differential and integral equations [8], [9],
[10]. In the classical Korovkin theorem [7] the uniform
convergence in C([a; b]), the space of all continuous realvalued functions defined on the compact interval [a; b], is
proved for a sequence of positive linear operators, assuming
the convergence only on the test functions 1, x, x2. Now, we
*Corresponding author: tulin [email protected];
[email protected]
Received / Geliş tarihi : 25.10.2016
Accepted / Kabul tarihi : 19.12.2016
give Gadjiev’s results in weighted spaces. We use the same
notation as in [1]. We now introduce some notation and
basic definitions used in this paper.
For any fixed real number q > 0 and non-negative integer r,
the q-integers of the number r are defined by
1 - qn
6n@q = * 1 - q , if q ! 1
n,
q=1
Also we have [0]q = 0.
Gadjiev defined the weighted spaces t $ 1 (t is unbounded
function) as the spaces Bt: " f: f (x) # M f t (x): x ! R ,
and C t = ^ f ! B t: f continuous h of functions which are
defined on unbounded regions. B t is a normed space with
f (x)
the norm f t = sup
. Gadjiev gave the following
p (x)
x!R
theorem in [4], [5] and [6].
Theorem 1.1 Let {Ln} be sequence of linear positive
operators taking Cρ into Bρ and satisfying the conditions
lim
L n ^ t j, x h - x j
n"3
t
= 0, j = 0, 1, 2 .
Then, there exists a function f * ∈ Cρ such that
lim
L n ^ f *, x h - f * ^ x h t ! 0 .
n"3
Coşkun / Weighted Approximation Properties of Positive Linear Operators Based on q-Integer
Then [1] extend these results of Gadjiev for different ρ1
and ρ2, acting from Cρ1 into Bρ2. We recall some notations
defined in [1]. Let positive linear operator L satisfy following
properties.
1. Positive linear operators, defined on Cρ1, acting from Cρ1
to Bρ1 iff the inequality
L ^ t 1, x h
t2
# M1
holds.
C t1 " B t2
= L ^ t 1, x h
L ^ f, x h
t2
# L ^ t 1, x h
f
t2
t1
It is clear that An(f,x) ≥ 0 for each x ! 60, 6n@q@ and f(x) ≥ 0.
.
3. Let A n: C t " B t be positive linear operators. Suppose that
there exists M > 0 such that for all x ! R , t 1 ^ x h # M t ^ x h .
If
1
2
2
lim
A n ^ t 1, x h - t 1 ^ x h
n"3
t2
=0
then the sequence of norms A n
C t1 " B t2
is uniformly bounded.
In this work we show that a Korovkin’s theorem does not
hold for a class of positive linear operators acting from Cρ1
into Bρ2 for different ρ1 and ρ2 using q-calculus.
2. q-Analogues in Different Weighted Spaces
Theorem 2.1 Let q > 0, n ! N and { 1, { 2 be two monotone
increasing continuous functions, on real axis such that
lim { 1 ^ x h = xlim
{ 2 ^ x h = !3 and that t 1 ^ x h # Mt 2 ^ x h
x "!3
"!3
(M>0 is arbitrary constant) for all x ! R where
t k ^ x h = 1 + { 2k ^ x h, k = 1, 2
and
lim
x"3
t 1 ^ xh
= a ! 0.
t 2 ^ xh
1
t2
2
= 0, v = 0, 1, 2 .
Then there exist f * ! C t such that
1
lim
sup A n ^ f *, x h - f * ^ x h
n"3
t2
! 0.
Proof. { 1, { 2 are two continuous functions that provide
given conditions, An be a sequence of operators for every
n ! N defined as follows
A n ^ f, x h: = *
f^ xh +
f^ xh
Now we show that this sequence of operators
A n: C t " B t . Since { 1 is a monotonic function satisfying
{ 21 ^ x h # { 21 ^ x + 1 h and using properties 1 we get the
following inequality
J { 21 (x)
N
^ 1 + { 21 (x + 1 h) -O
t 2 (x) K 2
A n (t 1, x) = t 1 (x) +
{ 1 (x + 1)
OO
2t 2 ([n] q) KK
2
1
{
(
+
1 (x))
L
P
# t 1 (x) # Mt 2 (x)
1
2
So A n (t 1, x) ! B t . In this way are positive linear operators
A n; C t " B t . Now we show that our operators satisfy three
conditions. Since for each x ! 60, 6n@q@
2
1
2
t 2 (x)
{ 2 (x)
< 2 1
- 1F
2t 2 ([n] q) { 1 (x + 1)
A n (1, x) - 1
{ 21 (x)
1
1
=
-1 #
2
t 2 (x)
2t 2 ([n] q) { 1 (x + 1)
2t 2 ([n])
A n (1, x) = 1 +
1
= 1 - q . For this reason, we
For q > 0 one has lim
n"3 6 @
nq
q n = 1.
consider a sequence q = qn such that lim
n"3
So we get
Then there exist a sequence A n: C p " B t of positive linear
operators satisfying the following three conditions
lim
A n ^ { v1, x h - { v1 ^ x h
n"3
Since [n]q = 1 + q + q2 +...+ qn-1, for all 0 ≤ x ≤ [n]q, ρ2(x) ≤
t 2 (x)
1
ρ2([n]q) and so 1 $ 2 2 0 then we get
2t 2 ([n] q)
A n (f, x) = f ^ x h +
t2
and therefore for all f ∈ Cρ1;
It is obvious that for every n ! N ; An are linear. Now we
show that this operator is positive.
t 2 (x)
{ 2 (x)
d 2 1
f (x + 1) - f (x) n
2t 2 ([n] q) { 1 (x + 1)
t 2 (x)
t 2 (x)
{ 2 (x)
d 2 1
n+
= f (x) d 1 f (x + 1) - f (x) n
2t 2 ([n] q
2t 2 ([n] q) { 1 (x + 1)
2. Let L: Cρ1Õ Bρ1 be positive linear operator. Then
L
We can assume that { 1 (0) = 0 and { 2 ^ 0 h = 0 since
{ 1 (x): = { 1 (x) - { 1 (0) implies { 1 (0) = 0 whenever
{ 1 (0) ! 0 .
t 2 (x)
{ 21 ^ x h
d
f ^ x + 1 h - f ^ x hn; 0 # x # 6n@q
2t 2 ^6n@q h { 21 ^ x + 1 h
; if others
Karaelmas Fen Müh. Derg., 2017; 7(1):250-252
lim
A n (1, x) - 1
n"3
t2
= 0.
Similary since
A n ({ 1, x) - { 1 (x)
{ 1 (x)
{ 1 (x)
=
-1
t 2 (x)
2t 2 ([n] q) { 1 (x + 1)
{ 1 (x)
- 1 1 1. Then
and by using monotonicity of { 1,
{ 1 (x + 1)
we get
{ 1 ^6n@q h
1
A n ({ 1, x) - { 1 (x) t # sup
{ 1 (x) #
2t 2 ^6n@q h
70,6n@ A 2t 2 ([n] q)
Therefore
2
q
lim
A n ^ { 1, x h - { 1 (x)
n"3
t2
= 0.
Lastly, we have
251
Coşkun / Weighted Approximation Properties of Positive Linear Operators Based on q-Integer
A n ^ { 21, x h - { 21 (x) =
So
2
t 2 (x)
{ 2 (x) 1 + { 1 (x + 1)) e 2 1
e
oo = 0
2t 2 ^6n@q h { 1 (x + 1) (1 + { 21 (x)
lim
A n ^ { 21, x h - { 21 (x)
n"3
t2
lim
sup A n ^ f *, x h - f * (x)
n"3
t2
t2
2
= 0 , v = 0, 1, 2 .
Then there exist f * ! C t such that
1
1
lim
sup A n ^ f *, x h - f * ^ x h
n"3
! 0.
Let g(x) be a function defined on the interval [-1,1] given
as follows
g (x): = (
1
lim
A n ^ { v1, x h - { v1 ^ x h
n"3
= 0.
Finally we show that for a f * ! C t
Theorem 2.2 Let q > 0, ρ1 and ρ2 be as in Theorem 2.1.
Then there exist a sequence A n: C t " B t of positive linear
operators such that
2 (1 + x); - 1 # x # 0
2 (1 - x); 0 1 x 1 1
and let us extend g(x) to a function h(x) on R with period
2. If f * is defined by
t2
! 0.
The positive statements of Korovkin type theorems for
linear positive operators A n: C t " B t may be proved as in
[2],[4] and [5] in some subspace of C t .
1
2
2
We study using the concepts of q-analysis the existence
of Korovkin’s theorem in the spaces of continuous and
unbounded functions defined on unbounded sets.
f * (x): = { 21 (x) .h ^ x h
4. Acknowledgements
for all x ! R ; then we have the following equality for
x ! 60, 6n@q@ and n ! N ,
The author would like to thank the referees for their careful
reading of this paper.
t 2 (x)
{ 2 (x) *
< 2 1
f (x + 1) - f * (x)F
2t 2 (6n@q { 1 (x + 1)
t 2 (x)
{ 2 (x)
< 2 1
= f * (x) +
{ 21 (x + 1) h (x + 1) - { 21 (x) h (x)F
2t 2 (6n@q { 1 (x + 1)
t 2 (x) 2
= f * (x) +
{ 1 (x) (h (x + 1) - h (x)) .
2t 2 (6n@q
t 2 (x) 2
{ 1 (x) 6h (x + 1) - h (x)@ .
A n ^ f *, x h - f * (x) =
2t 2 (6n@q
5. References
A n ^ f *, x h = f * (x) +
Therefore we have the following inequality,
{ 21 ^6n@q h
A n ^ f *, x h - f * (x)
h ^6n@q + 1 h - h ^6n@q h
$
t 2 (x)
2t 2 ^6n@q h
x ! 70,6n@ A
{ 21 ^6n@q h
{ 21 ^6n@q h
.
2=
=
2t 2 ^6n@q h
1 + { 22 ^6n@q h
sup
q
Then we get lim
n"3
1
= 0 and
1 + { 22 ^6n@q h
Coskun, T., 1997. Some Properties of Linear Positive Operators
in the Weighted Spaces of Unbounded Functions., Commun.
Fac. Sci. Univ. Ank. Series, 47: 175-191.
Coskun, T., 2012. On the Order of Weighted Approximation of
by Sequences of Positive Linear Operators., Turk J Math., 36:
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Ernst, T., A., 2012. Comprehensive Treatment of q-Calculus,
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Gadjiev, A. D., 1976. On P. P. Korovkin Type Theorems., Math.
Zametki, 20(5):995-998.
Gadjiev, A. D., 1974. The Convergence Problem for a Sequence of
Positive Linear Operators on Unbounded Sets and Theorems
Analogous To That of P. P. Korovkin , Engl. Translated. Sov.
Math. Dokl., 15: 1433-1436.
1 + { 21 ^6n@q h
lim
=a!0,
n"3
1 + { 22 ^6n@q h
Gadjiev and Hacisalihoglu, 1995. On convergence of the
sequences of linear positive operators, Ankara University, in
Turkish.
3. Results
Lupas, A., 1987. A q-analogue of the Bernstein operator, Seminar
on Numerical and Statistical Calculus, University of ClujNapoca, 9, 85-92.
Korovkin, P. P. 1960. Linear Operators and Approximation
Theory, Hindustan Publishing Corp., 219: 1-20.
which proves the theorem.
Theorem 2.1 shows that there exists no theorem of Korovkin
type for the class of positive linear operators from C t to
B t means of q-calculus. Note that a more general statement
can be proved for a class of positive linear operators acting
from C t to C t2 . We get following theorems.
1
2
1
252
Phillips, G.M., 1997. Bernstein polynomials based on the
q-integers, The heritage of P.L.Chebyshev, Ann. Numer.Math.,
4, 511-518.
Videnskii, V S., 2005. On some classes of q-parametric positive
operators, Operator Theory Adv. Appl., 158, 213-222.
Karaelmas Fen Müh. Derg., 2017; 7(1):250-252