scholarly journals On approximation of functions by algebraic polynomials on the average in real domain with Chebyshev-Hermite weight

2021 ◽  
Vol 19 ◽  
pp. 28
Author(s):  
S.B. Vakarchuk ◽  
M.B. Vakarchuk
Keyword(s):  
Best Approximation ◽  
Algebraic Polynomials ◽  
Approximation Of Functions ◽  
The Best Approximation ◽  
Real Domain

Exact inequalities of Jackson's type, connected with the best approximation of functions by algebraic polynomials, have been obtained in the space $L_{2,\rho}(\mathbb{R})$ at the Chebyshev-Hermite weight.

scholarly journals Sharp inequalities of Jackson type in weighted space $L_{2;\rho}({\mathbb{R}}^2)$

2014 ◽  
Vol 22 ◽  
pp. 17
Author(s):  
S.B. Vakarchuk ◽  
M.B. Vakarchuk
Keyword(s):  
Best Approximation ◽  
Weighted Space ◽  
Jackson Type ◽  
Algebraic Polynomials ◽  
The Best Approximation ◽  
Differentiable Functions ◽  
Functions Of Two Variables

Sharp inequalities of Jackson type, connected with the best approximation by "angles" of algebraic polynomials have been obtained on the classes of differentiable functions of two variables in the metric of space $L_{2;\rho}({\mathbb{R}}^2)$ of the Chebyshev-Hermite weight.

scholarly journals The best approximation of classes $W^r H^{\omega}$ by algebraic polynomials in $L_1$ space

1998 ◽  
Vol 6 ◽  
pp. 52
Author(s):  
A.M. Kogan
Keyword(s):  
Asymptotic Behavior ◽  
Best Approximation ◽  
Algebraic Polynomials ◽  
The Best Approximation

We consider asymptotic behavior of the best approximation of classes $W^r H^{\omega}$ by algebraic polynomials in $L_1$ space.

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 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 \(L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})\).

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.

On the Jackson-Type Inequality for the Best Sp-Approximations of Functions by Trigonometric Polynomials

2017 ◽  
Vol 8 ◽  
pp. 27-35
Author(s):  
Alexander N. Shchitov
Keyword(s):  
Best Approximation ◽  
Type Inequality ◽  
Trigonometric Polynomials ◽  
Sharp Constant ◽  
Jackson Type ◽  
Moduli Of Continuity ◽  
Approximation Of Functions ◽  
The Best Approximation

We find the sharp constant in the Jackson-type inequality between the value of the best approximation of functions by trigonometric polynomials and moduli of continuity of m-th order in the spaces Sp, 1 ≤ p < ∞. In the particular case we obtain one result which in a certain sense generalizes the result obtained by L.V. Taykov for m = 1 in the space L2 for the arbitrary moduli of continuity of m-th order (m Є N).

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