On the best approximation of classes of convolutions of periodic functions by trigonometric polynomials

1995 ◽  
Vol 47 (9) ◽  
pp. 1435-1440 ◽  
Author(s):  
A. S. Serdyuk
Keyword(s):  
Best Approximation ◽  
Trigonometric Polynomials ◽  
Periodic Functions ◽  
The Best Approximation

scholarly journals To the question of approximation of continuous periodic functions by trigonometric polynomials

2021 ◽  
pp. 39
Author(s):  
V.V. Shalaev
Keyword(s):  
Best Approximation ◽  
Modulus Of Smoothness ◽  
Trigonometric Polynomials ◽  
Periodic Functions ◽  
Bounded Operators ◽  
The Best Approximation

In the paper, it is proved that$$1 - \frac{1}{2n} \leqslant \sup\limits_{\substack{f \in C\\f \ne const}} \frac{E_n(f)_C}{\omega_2(f; \pi/n)_C} \leqslant \inf\limits_{L_n \in Z_n(C)} \sup\limits_{\substack{f \in C\\f \ne const}} \frac{\| f - L_n(f) \|_C}{\omega_2 (f; \pi/n)_C} \leqslant 1$$where $\omega_2(f; t)_C$ is the modulus of smoothness of the function $f \in C$, $E_n(f)_C$ is the best approximation by trigonometric polynomials of the degree not greater than $n-1$ in uniform metric, $Z_n(C)$ is the set of linear bounded operators that map $C$ to the subspace of trigonometric polynomials of degree not greater than $n-1$.

scholarly journals EESTIMATES OF BEST APPROXIMATIONS OF FUNCTIONS WITH LOGARITHMIC SMOOTHNESS IN THE LORENTZ SPACE WITH ANISOTROPIC NORM

2020 ◽  
Vol 6 (1) ◽  
pp. 16
Author(s):  
Gabdolla Akishev
Keyword(s):  
Lorentz Space ◽  
Best Approximation ◽  
Trigonometric Polynomials ◽  
Exact Order ◽  
Sufficient Condition ◽  
Periodic Functions ◽  
Best Approximations ◽  
Approximation Of Functions ◽  
The Best Approximation ◽  
Anisotropic Norm

In this paper, we consider the anisotropic Lorentz space \(L_{\bar{p}, \bar\theta}^{*}(\mathbb{I}^{m})\) of periodic functions of \(m\) variables. The Besov space \(B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}\) of functions with logarithmic smoothness is defined. The aim of the paper is to find an exact order of the best approximation of functions from the class \(B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}\) by trigonometric polynomials under different relations between the parameters \(\bar{p}, \bar\theta,\) and \(\tau\).The paper consists of an introduction and two sections. In the first section, we establish a sufficient condition for a function \(f\in L_{\bar{p}, \bar\theta^{(1)}}^{*}(\mathbb{I}^{m})\) to belong to the space \(L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})\) in the case \(1{<\theta^{2}<\theta_{j}^{(1)}},$ ${j=1,\ldots,m},\) in terms of the best approximation and prove its unimprovability on the class \(E_{\bar{p},\bar{\theta}}^{\lambda}=\{f\in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})\colon{E_{n}(f)_{\bar{p},\bar{\theta}}\leq\lambda_{n},}\) \({n=0,1,\ldots\},}\) where \(E_{n}(f)_{\bar{p},\bar{\theta}}\) is the best approximation of the function \(f \in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})\) by trigonometric polynomials of order \(n\) in each variable \(x_{j},\) \(j=1,\ldots,m,\) and \(\lambda=\{\lambda_{n}\}\) is a sequence of positive numbers \(\lambda_{n}\downarrow0\) as \(n\to+\infty\). In the second section, we establish order-exact estimates for the best approximation of functions from the class \(B_{\bar{p}, \bar\theta^{(1)}}^{(0, \alpha, \tau)}\) in the space \(L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})\).

Approximative characteristics and properties of operators of the best approximation of classes of functions from the Sobolev and Nikol'skii-Besov spaces

2020 ◽  
Vol 17 (3) ◽  
pp. 372-395
Author(s):  
Anatolii Romanyuk ◽  
Viktor Romanyuk
Keyword(s):  
Besov Spaces ◽  
Best Approximation ◽  
Linear Operators ◽  
Trigonometric Polynomials ◽  
Exact Order ◽  
Periodic Functions ◽  
Sobolev Classes ◽  
The Best Approximation ◽  
Several Variables

We have obtained the exact-order estimates for some approximative characteristics of the Sobolev classes $\mathbb{W}^{\boldsymbol{r}}_{p,\boldsymbol{\alpha}}$ and Nikоl'skii--Besov classes $\mathbb{B}^{\boldsymbol{r}}_{p,\theta}\ $ of periodic functions of one and several variables in the norm of the space $B_{\infty, 1}$. Properties of the linear operators realizing the orders of the best approximation for the classes $\mathbb{B}^{\boldsymbol{r}}_{\infty, \theta}$ in this space by trigonometric polynomials generated by a set of harmonics with $``$numbers$"$ from step hyperbolic crosses are investigated.

scholarly journals On the approximation by trigonometric polynomials in weighted Lorentz spaces

2010 ◽  
Vol 8 (1) ◽  
pp. 67-86 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Yunus E. Yildirir
Keyword(s):  
Best Approximation ◽  
Structural Characteristics ◽  
Modulus Of Smoothness ◽  
Lorentz Spaces ◽  
Trigonometric Polynomials ◽  
Periodic Functions ◽  
The Best Approximation ◽  
Weighted Lorentz Spaces

We obtain estimates of structural characteristics of 2π-periodic functions by the best trigonometric approximations in weighted Lorentz spaces, and show that the order of generalized modulus of smoothness depends not only on the rate of the best approximation, but also on the metric of the spaces. In weighted Lorentz spacesLps, this influence is expressed not only in terms of the parameterp, but also in terms of the second parameters.

NOTE ON JACKSON AND BERNSTE N TYPE APPROXIMATION THEOREMS IN THE CASE OF APPROXIMATION BY ALGEBRAIC POLYNOMIALS N THE SPACES L AND C

2000 ◽  
Vol 36 (3-4) ◽  
pp. 353-358 ◽  
Author(s):  
S. Pawelke
Keyword(s):  
Periodic Function ◽  
Best Approximation ◽  
Trigonometric Polynomials ◽  
Algebraic Polynomials ◽  
Cient Condition ◽  
Approximation Theorems ◽  
Type Approximation ◽  
The Best Approximation ◽  
Approximation By Algebraic Polynomials

We con ider the best approximation E (n,f)by algebraic polynomials of degree at most n for function f in L 1 (-1, 1)or C [-1, 1]and give imple necessary and u .cient condition for E (n,f)=O (n-.),n ›.,u ing the well-known results in the ca e of ap- proximation of periodic function by trigonometric polynomials.

scholarly journals Upper Estimates of the angle best approximations of generalized Liouville-Weyl derivatives

2021 ◽  
Vol 103 (3) ◽  
pp. 54-67
Author(s):  
A.E. Jetpisbayeva ◽  
◽  
A.A. Jumabayeva ◽  
Keyword(s):  
Best Approximation ◽  
Three Dimensional ◽  
Continuous Functions ◽  
Trigonometric Polynomials ◽  
Dimensional Case ◽  
Best Approximations ◽  
Approximation Of Functions ◽  
The Best Approximation

In this article we consider continuous functions f with period 2π and their approximation by trigonometric polynomials. This article is devoted to the study of estimates of the best angular approximations of generalized Liouville-Weyl derivatives by angular approximation of functions in the three-dimensional case. We consider generalized Liouville-Weyl derivatives instead of the classical mixed Weyl derivative. In choosing the issues to be considered, we followed the general approach that emerged after the work of the second author of this article. Our main goal is to prove analogs of the results of in the three-dimensional case. The concept of general monotonic sequences plays a key role in our study. Several well-known inequalities are indicated for the norms, best approximations of the r-th derivative with respect to the best approximations of the function f. The issues considered in this paper are related to the range of issues studied in the works of Bernstein. Later Stechkin and Konyushkov obtained an inequality for the best approximation f^(r). Also, in the works of Potapov, using the angle approximation, some classes of functions are considered. In subsection 1 we give the necessary notation and useful lemmas. Estimates for the norms and best approximations of the generalized Liouville-Weyl derivative in the three-dimensional case are obtained.

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