Mixed generalized modulus of smoothness and approximation by the “angle” of trigonometric polynomials

2016 ◽  
Vol 100 (3-4) ◽  
pp. 448-457 ◽  
Author(s):  
K. V. Runovskii ◽  
N. V. Omel’chenko
Keyword(s):  
Modulus Of Smoothness ◽  
Trigonometric Polynomials

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 ${\mathcal{M}}_{p(\cdot),\lambda(\cdot)}(I_{0},w)$.

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

Approximation of Unbounded Functions by Trigonometric Polynomials

2017 ◽  
Vol 13 (4) ◽  
pp. 106-116
Author(s):  
Alaa A. Auad ◽  
◽  
Mousa M. Khrajan
Keyword(s):  
Trigonometric Polynomials

Approximate Solution Of A Singular Integral Equation With A Multiplicative Cauchy Kernel In The Half-Plane

2008 ◽  
Vol 8 (2) ◽  
pp. 143-154 ◽  
Author(s):  
P. KARCZMAREK
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Half Plane ◽  
Singular Integral ◽  
Trigonometric Polynomials ◽  
Cauchy Kernel

AbstractIn this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

An extension of q-starlike and q-convex error functions endowed with the trigonometric polynomials

2020 ◽  
Vol 70 (3) ◽  
pp. 599-604
Author(s):  
Şahsene Altinkaya
Keyword(s):  
Error Function ◽  
Trigonometric Polynomials ◽  
Error Functions ◽  
The Family ◽  
Coefficient Inequalities

AbstractIn this present investigation, we will concern with the family of normalized analytic error function which is defined by$$\begin{array}{} \displaystyle E_{r}f(z)=\frac{\sqrt{\pi z}}{2}\text{er} f(\sqrt{z})=z+\overset{\infty }{\underset {n=2}{\sum }}\frac{(-1)^{n-1}}{(2n-1)(n-1)!}z^{n}. \end{array}$$By making the use of the trigonometric polynomials Un(p, q, eiθ) as well as the rule of subordination, we introduce several new classes that consist of 𝔮-starlike and 𝔮-convex error functions. Afterwards, we derive some coefficient inequalities for functions in these classes.

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