On the connection between the best approximation by algebraic polynomials and the modulus of smoothness of order r

2008 ◽  
Vol 155 (1) ◽  
pp. 153-169
Author(s):  
M. K. Potapov ◽  
F. M. Berisha
Keyword(s):  
Best Approximation ◽  
Modulus Of Smoothness ◽  
Algebraic Polynomials ◽  
The Best Approximation ◽  
Approximation By Algebraic Polynomials

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 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.

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 Weighted Approximation for Jackson-Matsuoka Polynomials on the Sphere

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Guo Feng
Keyword(s):  
Spherical Harmonics ◽  
Unit Sphere ◽  
Best Approximation ◽  
Weighted Approximation ◽  
Modulus Of Smoothness ◽  
The Best Approximation

We consider the best approximation by Jackson-Matsuoka polynomials in the weightedLpspace on the unit sphere ofRd. Using the relation betweenK-functionals and modulus of smoothness on the sphere, we obtain the direct and inverse estimate of approximation by these polynomials for theh-spherical harmonics.

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 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.

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