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1
Content available remote A note on the convergence of Phillips operators by the sequence of functions via q-calculus
Kiliçman Adem, Ayman-Mursaleen Mohammad, Nasiruzzaman Md.
Demonstratio Mathematica
|
2022
|
Vol. 55, nr 1
615--633
EN
The basic aim of this study is to include nonnegative real parameters to allow for approximation findings of the Stancu variant of Phillips operators. We concentrate on the uniform modulus of smoothness in a simple manner before moving on to the approximation in weighted Korovkin’s space. Our study’s goals and outcomes are to fully develop the uniformly approximated findings of Phillips operators. We determine the order of convergence in terms of Lipschitz maximal function and Peetre’s K-functional. In addition, the Voronovskaja-type theorem is also proved.
2
Content available remote Differences of operators of Baskakov type
Gupta Vijay
Fasciculi Mathematici
|
2019
|
nr 62
47--55
EN
In the present article, we study the approximation of difference of operators and find the quantitative estimates for the differences of Baskakov with Baskakov-Szasz and genuine Baskakov-Durrmeyer operators. We also estimate the result for the difference of Baskakov-Szasz and genuine Baskakov-Durrmeyer operators.
3
Deferred weighted A-statistical convergence based upon the (p,q)-Lagrange polynomials and its applications to approximation theorems
Srivastava H. M., Jena B. B., Paikray S. K., Misra U. K.
Journal of Applied Analysis
|
2018
|
Vol. 24, nr 1
1--16
EN
Recently, the notion of positive linear operators by means of basic (or q-) Lagrange polynomials and A-statistical convergence was introduced and studied in [M. Mursaleen, A. Khan, H. M. Srivastava and K. S. Nisar, Operators constructed by means of q-Lagrange polynomials and A-statistical approximation, Appl. Math. Comput. 219 2013, 12, 6911-6918]. In our present investigation, we introduce a certain deferred weighted A-statistical convergence in order to establish some Korovkin-type approximation theorems associated with the functions 1, t and t2 defined on a Banach space C[0,1] for a sequence of (presumably new) positive linear operators based upon (p,q)-Lagrange polynomials. Furthermore, we investigate the deferred weighted A-statistical rates for the same set of functions with the help of the modulus of continuity and the elements of the Lipschitz class. We also consider a number of interesting special cases and illustrative examples in support of our definitions and of the results which are presented in this paper.
4
Content available remote Approximation by Max-Product operators
Anastassiou G. A.
Fasciculi Mathematici
|
2018
|
Nr 60
5--28
EN
Here we study the approximation of functions by a great variety of Max-Product operators under differentiability. These are positive sublinear operators. Our study is based on our general results about positive sublinear operators. We produce Jackson type inequalities under initial conditions. So our approach is quantitative by producing inequalities with their right hand sides involving the modulus of continuity of a high order derivative of the function under approximation. We improve known related results which do not use smoothness of functions.
5
Content available The basis property of eigenfunctions in the problem of a nonhomogeneous damped string
Rzepnicki Ł.
Opuscula Mathematica
|
2017
|
Vol. 37, no. 1
141--165
EN
The equation which describes the small vibrations of a nonhomogeneous damped string can be rewritten as an abstract Cauchy problem for the densely defined closed operator iA. We prove that the set of root vectors of the operator A forms a basis of subspaces in a certain Hilbert space H. Furthermore, we give the rate of convergence for the decomposition with respect to this basis. In the second main result we show that with additional assumptions the set of root vectors of the operator A is a Riesz basis for H.
6
Content available remote Multivariate and abstract approximation theory for Banach space valued functions
Anastassiou G. A.
Demonstratio Mathematica
|
2017
|
Vol. 50, nr 1
208--222
EN
Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued Fréchet differentiable functions to the unit operator, as well as other basic approximations including those under convexity. These operators are bounded by real positive linear companion operators. The Banach spaces considered here are general and no positivity assumption is made on the initial linear operators for which we study their approximation properties. We derive pointwise and uniform estimates, which imply the approximation of these operators to the unit assuming Fréchet differentiability of functions, and then we continue with basic approximations. At the end we study the special case where the approximated function fulfills a convexity condition resulting into sharp estimates. We give applications to Bernstein operators.
7
Content available remote Approximation properties of Ibragimov-Gadjiev-Durrmeyer operators on Lp (R+)
Ulusoy G., Aral A.
Demonstratio Mathematica
|
2017
|
Vol. 50, nr 1
156--174
EN
We deal with the approximation properties of a new class of positive linear Durrmeyer type operators which offer a reconstruction of integral type operators including well known Durrmeyer operators. This reconstruction allows us to investigate approximation properties of the Durrmeyer operators at the same time. It is first shown that these operators are a positive approximation process in Lp (R+). While we are showing this property of the operators we consider the Ditzian-Totik modulus of smoothness and corresponding K-functional. Then, weighted norm convergence, whose proof is based on Korovkin type theorem on Lp (R+), is given. At the end of the paper we show several examples of classical sequences that can be obtained from the Ibragimov-Gadjiev-Durrmeyer operators.
8
Content available remote Approximation by q-analogue of modified Jakimovski-Leviatan-Stancu type operators
Mursaleen M., Nasiruzzaman M., Wafi A.
Demonstratio Mathematica
|
2017
|
Vol. 50, nr 1
175--189
EN
In this paper, we introduce the q-analogue of the Jakimovski-Leviatan type modied operators introduced by Atakut with the help of the q-Appell polynomials. We obtain some approximation results via the well-known Korovkin’s theorem for these operators. We also study convergence properties by using the modulus of continuity and the rate of convergence of the operators for functions belonging to the Lipschitz class. Moreover, we study the rate of convergence in terms of modulus of continuity of these operators in a weighted space.
9
Content available remote Approximation theorems for Szász-Mirakjan-Durrmeyer type operators
Herzog M.
Czasopismo Techniczne. Nauki Podstawowe
|
2016
|
Y. 113, iss. 1-NP
43--53
EN
In this paper we study an integral modification of Szász-Mirakjan type operators. The modification will be called Szász-Mirakjan-Durrmeyer type operators as in many papers examining this type of operators. We give direct approximation theorems for these operators using the modulus of continuity and the modulus of smoothness for functions belonging to exponential weighted spaces.
PL
W artykule badamy całkową modyfikację operatorów typu Szásza-Mirakjana. Tę modyfikację będziemy nazywać operatorami typu Szász-Mirakjan-Durrmeyera, jak to się czyni w wielu pracach badających tego typu operatory. Podajemy twierdzenia aproksymacyjne wykorzystujące moduł ciągłości i moduł gładkości dla funkcji z wykładniczych przestrzeni wagowych.
10
Content available remote On the rates of convergence of certain bivariate linear positive operators
Olgun A, Taşdelen F.
Fasciculi Mathematici
|
2014
|
Nr 52
117--129
EN
In this paper, we present a sequence of linear positive bivariate operators and investigate the approximation properties of them. Next we study the rates of converge of this approximation by means modulus of continuity and functions from Lipschitz class. After we give a Voronovskaya type theorem for n Morever, we give an r th order generalization of these operators. Finally, we investigate approximation properties of this generalization and observe the rates of convergence for them.
11
Content available remote The degree of approximation of functions from exponential weight spaces
Herzog M.
Czasopismo Techniczne. Nauki Podstawowe
|
2014
|
R. 111, z. 3-NP
21--29
EN
This paper presents a study of the approximation properties of modified Szász-Mirakyan operators for functions from exponential weight spaces. We present theorems giving the degree of approximation by these operators using a modulus of continuity.
PL
W artykule badamy aproksymacyjne własności zmodyfikowanych operatorów typu Szásza-Mirakyana dla funkcji z wykładniczych przestrzeni wagowych. Przedstawiamy twierdzenia podające rząd aproksymacji funkcji przez operatory tego typu, wykorzystując moduł ciągłości.
12
Content available remote The voronovskaja type theorem for an extension of Szász-Mirakjan operators
Pop O. T., Bărbosu D., Miclăus D.
Demonstratio Mathematica
|
2012
|
Vol. 45, nr 1
107-115
EN
Recently ,C.Mortici defined a class of linear and positive operators depending on a certain function (…) which generalize the well known Szasz Mirakjan operators. For these generalized operators we establish a Voronovskaja type theorem, the uniform convergence and the order of approximation, using the modulus of continuity.
13
Content available remote Approximation of functions of two variables from exponential weight spaces
Herzog M.
Czasopismo Techniczne. Nauki Podstawowe
|
2012
|
R. 109, z. 1-NP
3--10
EN
In this paper we study approximative properties of modified Szasz-Mirakyan operators for functions of two variables from exponential weight spaces. We present theorems giving a degree of approximation by these operators for exponential bounded functions.
PL
W artykule przedstawiono aproksymacyjne własności zmodyfikowanych operatorów typu Szasza-Mirakyana dla funkcji dwóch zmiennych z wykładniczych przestrzeni wagowych. Wyprowadzono twierdzenia podające rząd aproksymacji funkcji ograniczonych wykładniczo przez operatory tego typu.
14
Content available remote Approximation of functions of two variables by modified Szasz-Mirakyan operators
Herzog M.
Commentationes Mathematicae
|
2012
|
Vol. 52, [Z] 1
3-9
EN
In this paper we study approximative properties of modified Szasz-Mirakyan operators for functions of two variables from polynomial weight spaces. We present some direct theorems giving a degree of approximation for these operators.
15
Content available remote Approximation by bivariate Mazhar-Totik operators
Wachnicki E.
Commentationes Mathematicae
|
2010
|
Vol. 50, [Z] 2
141-153
EN
The aim of this paper is to study a bivariate version of the operator investigated in [2], [4]. We shall present Voronovskaya type theorem and theorems giving a rate of convergence of this operator. Some applications for the limit problem are indicated.
16
Content available remote On one-sided trigonometric approximation in modular function spaces. I
Musielak A.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
|
1999
|
[Z] 39
127-138
EN
The purpose of this paper is to obtain the direct approximation theorems on one-sided approximation of 2(pi)-periodic functions f from a modular function space of 2(pi)- periodic functions by means of trigonometric polynomials of a given degree in the sense of the Luxemburg norm || || generated by a discretely convex modular .
17
Content available remote On random Lipschitz condition and its application in approximation theory
Musielak J.
Fasciculi Mathematici
|
1998
|
Nr 28
119-125
EN
There are estimated moduli of continuity of functions satisfying Lipschitz condition with random exponents both in the sense of convergence in probability and convergence in mean. The results are applied to extend Jackson's direct approximation theorem to the case of Lipschitz condition with a random exponent.
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