On the uniform modulus of continuity of the operator of best approximation in the space of periodic functions

1979 ◽  
Vol 34 (1-2) ◽  
pp. 185-203 ◽  
Author(s):  
A. Kroó
Keyword(s):  
Modulus Of Continuity ◽  
Best Approximation ◽  
Periodic Functions ◽  
Uniform Modulus Of Continuity

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.

scholarly journals On multidimensional Jackson's theorem

2021 ◽  
pp. 30
Author(s):  
S.A. Pichugov
Keyword(s):  
Modulus Of Continuity ◽  
Periodic Functions ◽  
Linear Polynomial ◽  
Multiple Variables ◽  
Polynomial Methods ◽  
Methods Of Approximation

We have found the best linear polynomial methods of approximation of continuous periodic functions of multiple variables in uniform metric with concave modulus of continuity.

Best approximation by splines on classes of periodic functions in the metric of L

1976 ◽  
Vol 20 (5) ◽  
pp. 927-933
Author(s):  
N. P. Korneichuk
Keyword(s):  
Best Approximation ◽  
Periodic Functions

scholarly journals On approximation by trigonometrical polynomials in $L_2$ space

2016 ◽  
Vol 24 ◽  
pp. 89
Author(s):  
O.V. Polyakov
Keyword(s):  
Modulus Of Continuity ◽  
Best Approximation ◽  
Generalized Modulus Of Continuity ◽  
Jackson Type ◽  
The Best Approximation ◽  
Differentiable Functions

We obtain certain inequalities of Jackson type, connecting the value of the best approximation of periodic differentiable functions and the generalized modulus of continuity of the highest derivative.

scholarly journals Approximation by algebraic polynomials in metric spaces $L_{\psi}$

2013 ◽  
Vol 21 ◽  
pp. 3
Author(s):  
T.A. Agoshkova
Keyword(s):  
Modulus Of Continuity ◽  
Metric Spaces ◽  
Jackson Inequality ◽  
Algebraic Polynomials ◽  
Periodic Functions ◽  
Jackson Theorem ◽  
Approximation By Algebraic Polynomials

In the space $L_{\psi}[-1;1]$ of non-periodic functions with metric $\rho(f,0)_{\psi} = \int\limits_{-1}^1 \psi(|f(x)|)dx$, where $\psi$ is a function of the type of modulus of continuity, we study Jackson inequality for modulus of continuity of $k$-th order in the case of approximation by algebraic polynomials. It is proved that the direct Jackson theorem is true if and only if the lower dilation index of the function $\psi$ is not equal to zero.

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.

Sign in / Sign up

Export Citation Format

Share Document