scholarly journals An Inverse Result of Approximation by Sampling Kantorovich Series

2018 ◽  
Vol 62 (1) ◽  
pp. 265-280 ◽  
Author(s):  
Danilo Costarelli ◽  
Gianluca Vinti
Keyword(s):  
Real Line ◽  
Representation Formula ◽  
Order Of Approximation ◽  
Generalized Sampling ◽  
Representation Formulas ◽  
Sampling Series ◽  
Bounded Functions ◽  
Saturation Theorem ◽  
Uniformly Continuous ◽  
Kantorovich Operators

AbstractIn the present paper, an inverse result of approximation, i.e. a saturation theorem for the sampling Kantorovich operators, is derived in the case of uniform approximation for uniformly continuous and bounded functions on the whole real line. In particular, we prove that the best possible order of approximation that can be achieved by the above sampling series is the order one, otherwise the function being approximated turns out to be a constant. The above result is proved by exploiting a suitable representation formula which relates the sampling Kantorovich series with the well-known generalized sampling operators introduced by Butzer. At the end, some other applications of such representation formulas are presented, together with a discussion concerning the kernels of the above operators for which such an inverse result occurs.

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>

scholarly journals Quantitative Estimates for Nonlinear Sampling Kantorovich Operators

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Nursel Çetin ◽  
Danilo Costarelli ◽  
Gianluca Vinti
Keyword(s):  
Orlicz Spaces ◽  
General Setting ◽  
Modulus Of Smoothness ◽  
Direct Approach ◽  
Order Of Approximation ◽  
Lipschitz Classes ◽  
General Frame ◽  
Quantitative Estimates ◽  
Nonlinear Sampling ◽  
Kantorovich Operators

AbstractIn this paper, we establish quantitative estimates for nonlinear sampling Kantorovich operators in terms of the modulus of smoothness in the setting of Orlicz spaces. This general frame allows us to directly deduce some quantitative estimates of approximation in $$L^{p}$$ L p -spaces, $$1\le p<\infty $$ 1 ≤ p < ∞ , and in other well-known instances of Orlicz spaces, such as the Zygmung and the exponential spaces. Further, the qualitative order of approximation has been obtained assuming f in suitable Lipschitz classes. The above estimates achieved in the general setting of Orlicz spaces, have been also improved in the $$L^p$$ L p -case, using a direct approach suitable to this context. At the end, we consider the particular cases of the nonlinear sampling Kantorovich operators constructed by using some special kernels.

Approximation Par des Fonctions Lipschitziennes et Critere de Convergence Etroite D'une Suite de Probabilites

1984 ◽  
Vol 27 (4) ◽  
pp. 514-516 ◽  
Author(s):  
I. Rihaoui
Keyword(s):  
Weak Convergence ◽  
Metric Space ◽  
Bounded Function ◽  
Uniform Limit ◽  
Bounded Functions ◽  
New Criterion ◽  
Uniformly Continuous

AbstractIn this paper, we prove that a real valued bounded function, defined on a metric space and uniformly continuous is the uniform limit of a sequence of Lipschitzian bounded functions.As a consequence, a new criterion for the weak convergence of probabilities is given.

scholarly journals Inequalities and representation formulas for functions of exponential type

1995 ◽  
Vol 58 (1) ◽  
pp. 15-26
Author(s):  
C. Frappier ◽  
P. Olivier
Keyword(s):  
Exponential Type ◽  
Entire Functions ◽  
Representation Formula ◽  
Bernstein’S Inequality ◽  
Representation Formulas ◽  
Functions Of Exponential Type ◽  
Bernstein's Inequality

AbstractWe generalise the classical Bernstein's inequality: . Moreover we obtain a new representation formula for entire functions of exponential type.

scholarly journals Remarks on uniformly symmetrically continuous functions

2016 ◽  
Vol 09 (03) ◽  
pp. 1650069
Author(s):  
Tammatada Khemaratchatakumthorn ◽  
Prapanpong Pongsriiam
Keyword(s):  
Real Line ◽  
Nonempty Subset ◽  
Continuous Functions ◽  
The Real ◽  
Definition Of ◽  
Symmetrically Continuous Functions ◽  
Uniformly Continuous

We give the definition of uniform symmetric continuity for functions defined on a nonempty subset of the real line. Then we investigate the properties of uniformly symmetrically continuous functions and compare them with those of symmetrically continuous functions and uniformly continuous functions. We obtain some characterizations of uniformly symmetrically continuous functions. Several examples are also given.

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.

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