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1
Content available remote

On a q-analogue of Stancu operators

100%
Agratini O.
Open Mathematics
|
2010
|
tom 8
|
nr 1
191-198
EN
This paper is concerned with a generalization in q-Calculus of Stancu operators. Involving modulus of continuity and Lipschitz type maximal function, we give estimates for the rate of convergence. A probabilistic approach is presented and approximation properties are established.
2
Content available remote

Bernstein type operators having 1 and x j as fixed points

81%
Finta Z.
Open Mathematics
|
2013
|
tom 11
|
nr 12
2257-2261
EN
For certain generalized Bernstein operators {L n} we show that there exist no i, j ∈ {1, 2, 3,…}, i < j, such that the functions e i(x) = x i and e j (x) = x j are preserved by L n for each n = 1, 2,… But there exist infinitely many e i such that e 0(x) = 1 and e j (x) = x j are its fixed points.
3
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Korovkin-type theorems and applications

81%
Mahmudov N.
Open Mathematics
|
2009
|
tom 7
|
nr 2
348-356
EN
Let {T n} be a sequence of linear operators on C[0,1], satisfying that {T n (e i)} converge in C[0,1] (not necessarily to e i) for i = 0,1,2, where e i = t i. We prove Korovkin-type theorem and give quantitative results on C 2[0,1] and C[0,1] for such sequences. Furthermore, we define King’s type q-Bernstein operator and give quantitative results for the approximation properties of such operators.
4
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Convergence and Voronovskaja-type theorems for derivatives of generalized Baskakov operators

81%
Wafi A., Khatoon S.
Open Mathematics
|
2008
|
tom 6
|
nr 2
325-334
EN
In the present paper our aim is to establish convergence and Voronovskaja-type theorems for first derivatives of generalized Baskakov operators for functions of one and two variables in exponential and polynomial weight spaces.
5
Content available remote

On q-Szász-Durrmeyer operators

81%
Mahmudov N., Kaffaoğlu H.
Open Mathematics
|
2010
|
tom 8
|
nr 2
399-409
EN
In the present paper, we introduce the q-Szász-Durrmeyer operators and justify a local approximation result for continuous functions in terms of moduli of continuity. We also discuss a Voronovskaya type result for the q-Szász-Durrmeyer operators.
6
Content available remote

On an approximation processes in the space of analytical functions

81%
Gadjiev A., Ghorbanalizadeh A.
Open Mathematics
|
2010
|
tom 8
|
nr 2
389-398
EN
In this paper we obtain various approximation theorems by means of k-positive linear operators defined on the space of all analytic functions on a bounded domain of the complex plane.
7
Content available remote

Statistical approximation to Bögel-type continuous and periodic functions

81%
Dirik F., Duman O., Demirci K.
Open Mathematics
|
2009
|
tom 7
|
nr 3
539-549
EN
In this paper, considering A-statistical convergence instead of Pringsheim’s sense for double sequences, we prove a Korovkin-type approximation theorem for sequences of positive linear operators defined on the space of all real valued Bögel-type continuous and periodic functions on the whole real two-dimensional space. A strong application is also presented. Furthermore, we obtain some rates of A-statistical convergence in our approximation.
8
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Lacunary equi-statistical convergence of positive linear operators

62%
Aktuğlu H., Gezer H.
Open Mathematics
|
2009
|
tom 7
|
nr 3
558-567
EN
In this paper, the concept of lacunary equi-statistical convergence is introduced and it is shown that lacunary equi-statistical convergence lies between lacunary statistical pointwise and lacunary statistical uniform convergence. Inclusion relations between equi-statistical and lacunary equi-statistical convergence are investigated and it is proved that, under some conditions, lacunary equi-statistical convergence and equi-statistical convergence are equivalent to each other. A Korovkin type approximation theorem via lacunary equi-statistical convergence is proved. Moreover it is shown that our Korovkin type approximation theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. Finally the rates of lacunary equi-statistical convergence by the help of modulus of continuity of positive linear operators are studied.
9
Content available remote

Approximation properties of q-Baskakov operators

62%
Finta Z., Gupta V.
Open Mathematics
|
2010
|
tom 8
|
nr 1
199-211
EN
We establish direct estimates for the q-Baskakov operator introduced by Aral and Gupta in [2], using the second order Ditzian-Totik modulus of smoothness. Furthermore, we define and study the limit q-Baskakov operator.
10
Content available remote

Approximation by the Bézier variant of the MKZ-Kantorovich operators in the case α < 1

62%
Zeng X.-M., Gupta V.
Open Mathematics
|
2009
|
tom 7
|
nr 3
550-557
EN
The pointwise approximation properties of the Bézier variant of the MKZ-Kantorovich operators $$ \hat M_{n,\alpha } (f,x) $$ for α ≥ 1 have been studied in [Comput. Math. Appl., 39 (2000), 1-13]. The aim of this paper is to deal with the pointwise approximation of the operators $$ \hat M_{n,\alpha } (f,x) $$ for the other case 0 < α < 1. By means of some new techniques and new inequalities we establish an estimate formula on the rate of convergence of the operators $$ \hat M_{n,\alpha } (f,x) $$ for the case 0 < α < 1. In the end we propose the q-analogue of MKZK operators.
11
Content available remote

Statistical approximation of Baskakov and Baskakov-Kantorovich operators based on the q-integers

62%
Mahmudov N.
Open Mathematics
|
2010
|
tom 8
|
nr 4
816-826
EN
In the present paper we introduce and investigate weighted statistical approximation properties of a q-analogue of the Baskakov and Baskakov-Kantorovich operators. By using a weighted modulus of smoothness, we give some direct estimations for error in the case 0 < q < 1.
12
Content available remote

Korovkin-type convergence results for non-positive operators

52%
Nowak O.
Open Mathematics
|
2010
|
tom 8
|
nr 5
890-907
EN
Korovkin-type approximation theory usually deals with convergence analysis for sequences of positive operators. In this work we present qualitative Korovkin-type convergence results for a class of sequences of non-positive operators, more precisely regular operators with vanishing negative parts under a limiting process. Sequences of that type are called sequences of almost positive linear operators and have not been studied before in the context of Korovkin-type approximation theory. As an example we show that operators related to the multivariate scattered data interpolation technique moving least squares interpolation originally due to Lancaster and Šalkauskas [Surfaces generated by moving least squares methods, Math. Comp., 1981, 37, 141–158] give rise to such sequences. This work also generalizes Korovkin-type results regarding Shepard interpolation [Korovkin-type convergence results for multivariate Shepard formulae, Rev. Anal. Numér. Théor. Approx., 2009, 38, 170–176] due to the author. Moreover, this work establishes connections and differences between the concepts of sequences of almost positive linear operators and sequences of quasi-positive or convexity-monotone linear operators introduced and studied by Campiti in [Convexity-monotone operators in Korovkin theory, Rend. Circ. Mat. Palermo (2) Suppl., 1993, 33, 229–238].
13
Content available remote

Local approximation properties of certain class of linear positive operators via I-convergence

52%
Özarslan M., Aktuǧlu H.
Open Mathematics
|
2008
|
tom 6
|
nr 2
281-286
EN
In this study, we obtain a local approximation theorems for a certain family of positive linear operators via I-convergence by using the first and the second modulus of continuities and the elements of Lipschitz class functions. We also give an example to show that the classical Korovkin Theory does not work but the theory works in I-convergence sense.
14
Content available remote

I-convergence theorems for a class of k-positive linear operators

52%
Özarslan M.
Open Mathematics
|
2009
|
tom 7
|
nr 2
357-362
EN
In this paper, we obtain some approximation theorems for k- positive linear operators defined on the space of analytical functions on the unit disc, via I-convergence. Some concluding remarks which includes A-statistical convergence are also given.
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