Generalized Statistically Almost Convergence Based on the Difference Operator which Includes the (p, q)-Gamma Function and Related Approximation Theorems
This paper is devoted to extend the notion of almost convergence and its statistical forms with respect to the difference operator involving ( p , q )-gamma function and an increasing sequence ( λ n ) of positive numbers. We firstly introduce some new concepts of almost Δ h , α , β [ a , b , c ] (...
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| Vydané v: | Resultate der Mathematik Ročník 73; číslo 1 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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Springer International Publishing
01.03.2018
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| ISSN: | 1422-6383, 1420-9012 |
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| Abstract | This paper is devoted to extend the notion of almost convergence and its statistical forms with respect to the difference operator involving (
p
,
q
)-gamma function and an increasing sequence
(
λ
n
)
of positive numbers. We firstly introduce some new concepts of almost
Δ
h
,
α
,
β
[
a
,
b
,
c
]
(
λ
)
-statistical convergence, statistical almost
Δ
h
,
α
,
β
[
a
,
b
,
c
]
(
λ
)
-convergence and strong almost
[
Δ
h
,
α
,
β
[
a
,
b
,
c
]
(
λ
)
]
r
-convergence. Moreover, we present some inclusion relations between these newly proposed methods and give some counterexamples to show that these are non-trivial generalizations of existing literature on this topic. We then prove a Korovkin type approximation theorem for functions of two variables through statistically almost
Δ
h
,
α
,
β
[
a
,
b
,
c
]
(
λ
)
-convergence and also present an illustrative example via bivariate non-tensor type Meyer–König and Zeller generalization of Bernstein power series. Furthermore, we estimate the rate of almost convergence of approximating linear operators by means of the modulus of continuity and derive some Voronovskaja type results by using the generalized Meyer–König and Zeller operators. Finally, some computational and geometrical interpretations for the convergence of operators to a function are presented. |
|---|---|
| AbstractList | This paper is devoted to extend the notion of almost convergence and its statistical forms with respect to the difference operator involving (
p
,
q
)-gamma function and an increasing sequence
(
λ
n
)
of positive numbers. We firstly introduce some new concepts of almost
Δ
h
,
α
,
β
[
a
,
b
,
c
]
(
λ
)
-statistical convergence, statistical almost
Δ
h
,
α
,
β
[
a
,
b
,
c
]
(
λ
)
-convergence and strong almost
[
Δ
h
,
α
,
β
[
a
,
b
,
c
]
(
λ
)
]
r
-convergence. Moreover, we present some inclusion relations between these newly proposed methods and give some counterexamples to show that these are non-trivial generalizations of existing literature on this topic. We then prove a Korovkin type approximation theorem for functions of two variables through statistically almost
Δ
h
,
α
,
β
[
a
,
b
,
c
]
(
λ
)
-convergence and also present an illustrative example via bivariate non-tensor type Meyer–König and Zeller generalization of Bernstein power series. Furthermore, we estimate the rate of almost convergence of approximating linear operators by means of the modulus of continuity and derive some Voronovskaja type results by using the generalized Meyer–König and Zeller operators. Finally, some computational and geometrical interpretations for the convergence of operators to a function are presented. |
| ArticleNumber | 9 |
| Author | Mohiuddine, S. A. Kadak, Uğur |
| Author_xml | – sequence: 1 givenname: Uğur surname: Kadak fullname: Kadak, Uğur email: ugurkadak@gmail.com organization: Department of Mathematics, Gazi University – sequence: 2 givenname: S. A. surname: Mohiuddine fullname: Mohiuddine, S. A. organization: Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University |
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| Keywords | bivariate nontensor type Meyer–Konig and Zeller operators 40A30 41A25 41A36 Korovkin and Voronovskaja type approximation theorems 41A10 rate of convergence Banach limit Gamma function 40G15 Statistically almost convergence |
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| Title | Generalized Statistically Almost Convergence Based on the Difference Operator which Includes the (p, q)-Gamma Function and Related Approximation Theorems |
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