scholarly journals On a Durrmeyer-type modification of the Exponential sampling series

2020 ◽  
Author(s):  
Carlo Bardaro ◽  
Ilaria Mantellini
Keyword(s):  
Asymptotic Formula ◽  
Uniform Convergence ◽  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Convergence Properties ◽  
Sampling Series ◽  
Quantitative Results ◽  
Uniformly Continuous

Abstract In this paper we introduce the exponential sampling Durrmeyer series. We discuss pointwise and uniform convergence properties and an asymptotic formula of Voronovskaja type. Quantitative results are given, using the usual modulus of continuity for uniformly continuous functions. Some examples are also described.

scholarly journals Approximation by pseudo-linear discrete operators

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
İsmail Aslan ◽  
Türkan Yeliz Gökçer
Keyword(s):  
Error Estimation ◽  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Discrete Operator ◽  
Generalized Sampling ◽  
Sampling Series ◽  
Discrete Operators ◽  
Generator Function ◽  
Uniformly Continuous

<p style='text-indent:20px;'>In this note, we construct a pseudo-linear kind discrete operator based on the continuous and nondecreasing generator function. Then, we obtain an approximation to uniformly continuous functions through this new operator. Furthermore, we calculate the error estimation of this approach with a modulus of continuity based on a generator function. The obtained results are supported by visualizing with an explicit example. Finally, we investigate the relation between discrete operators and generalized sampling series.</p>

Approximation by max-product sampling Kantorovich operators with generalized kernels

2019 ◽  
pp. 1-26 ◽  
Author(s):  
Lucian Coroianu ◽  
Danilo Costarelli ◽  
Sorin G. Gal ◽  
Gianluca Vinti
Keyword(s):  
Uniform Convergence ◽  
Real Axis ◽  
Modulus Of Continuity ◽  
Higher Dimensions ◽  
Jackson Type ◽  
Convergence Properties ◽  
Type Estimate ◽  
Quantitative Estimates ◽  
Uniform Modulus Of Continuity ◽  
Kantorovich Operators

In a recent paper, for max-product sampling operators based on general kernels with bounded generalized absolute moments, we have obtained several pointwise and uniform convergence properties on bounded intervals or on the whole real axis, including a Jackson-type estimate in terms of the first uniform modulus of continuity. In this paper, first, we prove that for the Kantorovich variants of these max-product sampling operators, under the same assumptions on the kernels, these convergence properties remain valid. Here, we also establish the [Formula: see text] convergence, and quantitative estimates with respect to the [Formula: see text] norm, [Formula: see text]-functionals and [Formula: see text]-modulus of continuity as well. The results are tested on several examples of kernels and possible extensions to higher dimensions are suggested.

Regular completions of uniform convergence spaces

1974 ◽  
Vol 11 (3) ◽  
pp. 413-424 ◽  
Author(s):  
R.J. Gazik ◽  
D.C. Kent ◽  
G.D. Richardson
Keyword(s):  
Uniform Convergence ◽  
Real Line ◽  
Dual Space ◽  
Function Algebra ◽  
Continuous Functions ◽  
Convergence Space ◽  
Uniform Spaces ◽  
Convergence Spaces ◽  
Uniform Convergence Space ◽  
Uniformly Continuous

A regular completion with universal property is obtained for each member of the class of u–embedded uniform convergence spaces, a class which includes the Hausdorff uniform spaces. This completion is obtained by embedding each u-embedded uniform convergence space (X, I) into the dual space of a complete function algebra composed of the uniformly continuous functions from (X, I) into the real line.

scholarly journals Approximation properties of multivariate exponential sampling series

2021 ◽  
Vol 13 (3) ◽  
pp. 666-675
Author(s):  
S. Kurşun ◽  
M. Turgay ◽  
O. Alagöz ◽  
T. Acar
Keyword(s):  
Asymptotic Behaviour ◽  
Type Theorem ◽  
Continuous Functions ◽  
Approximation Properties ◽  
Multivariate Functions ◽  
Voronovskaja Type Theorem ◽  
Sampling Series ◽  
The Family ◽  
Uniformly Continuous ◽  
Multivariate Exponential

In this paper, we generalize the family of exponential sampling series for functions of $n$ variables and study their pointwise and uniform convergence as well as the rate of convergence for the functions belonging to space of $\log$-uniformly continuous functions. Furthermore, we state and prove the generalized Mellin-Taylor's expansion of multivariate functions. Using this expansion we establish pointwise asymptotic behaviour of the series by means of Voronovskaja type theorem.

Hausdorff Distance and a Compactness Criterion for Continuous Functions

1986 ◽  
Vol 29 (4) ◽  
pp. 463-468 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Uniform Convergence ◽  
Hausdorff Distance ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Compactness Criterion ◽  
Uniformly Continuous

AbstractLet 〈X, dx〉 and 〈Y, dY〉 be metric spaces and let hp denote Hausdorff distance in X x Y induced by the metric p on X x Y given by p[(x1, y1), (x2, y2)] = max ﹛dx(x1, x2),dY(y1, y2)﹜- Using the fact that hp when restricted to the uniformly continuous functions from X to Y induces the topology of uniform convergence, we exhibit a natural compactness criterion for C(X, Y) when X is compact and Y is complete.

scholarly journals Shape-Preserving and Convergence Properties for theq-Szász-Mirakjan Operators for Fixedq∈(0,1)

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Heping Wang ◽  
Fagui Pu ◽  
Kai Wang
Keyword(s):  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Bernstein Operator ◽  
Convergence Properties ◽  
Shape Preserving ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous ◽  
Good Shape ◽  
Variation Diminishing ◽  
Shape Preserving Properties

We introduce aq-generalization of Szász-Mirakjan operatorsSn,qand discuss their properties for fixedq∈(0,1). We show that theq-Szász-Mirakjan operatorsSn,qhave good shape-preserving properties. For example,Sn,qare variation-diminishing, and preserve monotonicity, convexity, and concave modulus of continuity. For fixedq∈(0,1), we prove that the sequence{Sn,qf}converges toB∞,q(f)uniformly on[0,1]for eachf∈C[0, 1/(1-q)], whereB∞,qis the limitq-Bernstein operator. We obtain the estimates for the rate of convergence for{Sn,qf}by the modulus of continuity off, and the estimates are sharp in the sense of order for Lipschitz continuous functions.

scholarly journals The Rate of Convergence of Lupasq-Analogue of the Bernstein Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Heping Wang ◽  
Yanbo Zhang
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Bernstein Operators ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous

We discuss the rate of convergence of the Lupasq-analogues of the Bernstein operatorsRn,q(f;x)which were given by Lupas in 1987. We obtain the estimates for the rate of convergence ofRn,q(f)by the modulus of continuity off, and show that the estimates are sharp in the sense of order for Lipschitz continuous functions.

scholarly journals Lattices of uniformly continuous functions

2013 ◽  
Vol 160 (1) ◽  
pp. 50-55 ◽  
Author(s):  
Félix Cabello Sánchez ◽  
Javier Cabello Sánchez
Keyword(s):  
Continuous Functions ◽  
Uniformly Continuous

APPROXIMATIVE PROPERTIES OF ABEL–POISSON-TYPE OPERATORS ON THE GENERALIZED HÖLDER CLASSES

2021 ◽  
Vol 1 ◽  
pp. 76-83
Author(s):  
Yuri I. Kharkevich ◽  
◽  
Alexander G. Khanin ◽  
Keyword(s):  
Modulus Of Continuity ◽  
Applied Mathematics ◽  
Continuous Functions ◽  
Elliptic Type ◽  
Periodic Functions ◽  
French Mathematician ◽  
First Order ◽  
Function Classes ◽  
In The Beginning ◽  
Poisson Type

The paper deals with topical issues of the modern applied mathematics, in particular, an investigation of approximative properties of Abel–Poisson-type operators on the so-called generalized Hölder’s function classes. It is known, that by the generalized Hölder’s function classes we mean the classes of continuous -periodic functions determined by a first-order modulus of continuity. The notion of the modulus of continuity, in turn, was formulated in the papers of famous French mathematician Lebesgue in the beginning of the last century, and since then it belongs to the most important characteristics of smoothness for continuous functions, which can describe all natural processes in mathematical modeling. At the same time, the Abel-Poisson-type operators themselves are the solutions of elliptic-type partial differential equations. That is why the results obtained in this paper are significant for subsequent research in the field of applied mathematics. The theorem proved in this paper characterizes the upper bound of deviation of continuous -periodic functions determined by a first-order modulus of continuity from their Abel–Poisson-type operators. Hence, the classical Kolmogorov–Nikol’skii problem in A.I. Stepanets sense is solved on the approximation of functions from the classes by their Abel–Poisson-type operators. We know, that the Abel–Poisson-type operators, in partial cases, turn to the well-known in applied mathematics Poisson and Jacobi–Weierstrass operators. Therefore, from the obtained theorem follow the asymptotic equalities for the upper bounds of deviation of functions from the Hölder’s classes of order from their Poisson and Jacobi–Weierstrass operators, respectively. The obtained equalities generalize the known in this direction results from the field of applied mathematics.

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