scholarly journals Trigonometric Approximation by Modulus of Smoothness in Lp,α (X)

2021 ◽  
Vol 32 (3) ◽  
pp. 20
Author(s):  
Mohammed Hamad Fayyadh ◽  
Alaa Adnan Auad
Keyword(s):  
Best Approximation ◽  
Modulus Of Smoothness ◽  
Trigonometric Approximation ◽  
Weighted Spaces ◽  
Trigonometric Polynomials ◽  
The Best Approximation ◽  
The Relationship ◽  
Derivatives Of

In this paper, we study the approximate properties of functions by means of trigonometric polynomials in weighted spaces. Relationships between modulus of smoothness of function derivatives and those of the jobs themselves are introduced. In the weighted spaces we also proved of theorems about the relationship between the derivatives of the polynomials for the best approximation and the best approximation of the functions

scholarly journals Derivatives of a polynomial of best approximation and modulus of smoothness in generalized Lebesgue spaces with variable exponent

2017 ◽  
Vol 50 (1) ◽  
pp. 245-251 ◽  
Author(s):  
Sadulla Z. Jafarov
Keyword(s):  
Best Approximation ◽  
Variable Exponent ◽  
Modulus Of Smoothness ◽  
Lebesgue Spaces ◽  
The Best Approximation ◽  
Polynomial Of Best Approximation ◽  
The Relationship ◽  
Derivatives Of ◽  
De La Vallée Poussin

Abstract The relation between derivatives of a polynomial of best approximation and the best approximation of the function is investigated in generalized Lebesgue spaces with variable exponent. In addition, the relationship between the fractional modulus of smoothness of the function and its de la Vallée-Poussin sums is studied.

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 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.

scholarly journals Approximation by trigonometric polynomials in the variable exponent weighted Morrey spaces

2021 ◽  
Vol 13 (3) ◽  
pp. 750-763
Author(s):  
Z. Cakir ◽  
C. Aykol ◽  
V.S. Guliyev ◽  
A. Serbetci
Keyword(s):  
Best Approximation ◽  
Variable Exponent ◽  
Modulus Of Smoothness ◽  
Morrey Spaces ◽  
Trigonometric Polynomials ◽  
The Best Approximation ◽  
Weighted Morrey Spaces ◽  
Direct And Inverse Theorems ◽  
Inverse Theorems

In this paper we investigate the best approximation by trigonometric polynomials in the variable exponent weighted Morrey spaces ${\mathcal{M}}_{p(\cdot),\lambda(\cdot)}(I_{0},w)$, where $w$ is a weight function in the Muckenhoupt $A_{p(\cdot)}(I_{0})$ class. We get a characterization of $K$-functionals in terms of the modulus of smoothness in the spaces ${\mathcal{M}}_{p(\cdot),\lambda(\cdot)}(I_{0},w)$. Finally, we prove the direct and inverse theorems of approximation by trigonometric polynomials in the spaces ${\mathcal{\widetilde{M}}}_{p(\cdot),\lambda(\cdot)}(I_{0},w),$ the closure of the set of all trigonometric polynomials in ${\mathcal{M}}_{p(\cdot),\lambda(\cdot)}(I_{0},w)$.

Best Trigonometric Approximation, Fractional Order Derivatives and Lipschitz Classes

1977 ◽  
Vol 29 (4) ◽  
pp. 781-793 ◽  
Author(s):  
P. L. Butzer ◽  
H. Dyckhoff ◽  
E. Görlich ◽  
R. L. Stens
Keyword(s):  
Fractional Order ◽  
Best Approximation ◽  
Continuous Functions ◽  
Trigonometric Approximation ◽  
Trigonometric Polynomials ◽  
Lipschitz Classes ◽  
The Best Approximation ◽  
Image Position ◽  
Best Trigonometric Approximation

Let C2π denote the space of 2π-periodic continuous functions and πn the set of trigonometric polynomials of degree ≦ n, where n ϵ P = {0, 1, … } . Given θ > 0, the well-known theorem of Stečkin and its converse state that the best approximation of an ƒ ϵ C2π with respect to the max-norm satisfies

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

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