scholarly journals A New Modulus of Smoothness for Uniform Approximation

2018 ◽  
Vol 26 (9) ◽  
pp. 8-15
Author(s):  
Eman Samir Bhaya ◽  
Bushra Khudhair Hussein
Keyword(s):  
Uniform Approximation ◽  
Best Approximation ◽  
Modulus Of Smoothness ◽  
Monotone Functions ◽  
End Points ◽  
Uniform Estimates ◽  
Classical Modulus

The estimates of best approximation using classical modulus of smoothness is not uniform. Also we sometimes need to improve the degree of best approximation near the end points. Thus we need to improve this classical modulus of smoothness. Here we define a new modulus of smoothness to achieve uniform estimates of  best approximation and an improvement of  a degree of such version of best approximation.  Our modulus of smoothness is for k-monotone functions. Estimates for using our  modulus of smoothness are introduced. Applications for these estimates are also introduced

Note on Best Approximation of |x|

1979 ◽  
Vol 22 (3) ◽  
pp. 363-366
Author(s):  
Colin Bennett ◽  
Karl Rudnick ◽  
Jeffrey D. Vaaler
Keyword(s):  
General Problem ◽  
Uniform Approximation ◽  
Best Approximation ◽  
Best Uniform Approximation ◽  
Linear Fractional Transformations ◽  
Symmetric Complex ◽  
Image Position ◽  
Complex Valued ◽  
Special Case

In this note the best uniform approximation on [—1,1] to the function |x| by symmetric complex valued linear fractional transformations is determined. This is a special case of the more general problem studied in [1]. Namely, for any even, real valued function f(x) on [-1,1] satsifying 0 = f ( 0 ) ≤ f (x) ≤ f (1) = 1, determine the degree of symmetric approximationand the extremal transformations U whenever they exist.

scholarly journals Bivariate-Schurer-Stancu operators based on (p,q)-integers

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1251-1258 ◽  
Author(s):  
Nadeem Rao ◽  
Abdul Wafi
Keyword(s):  
Rate Of Convergence ◽  
Uniform Approximation ◽  
Type Theorem ◽  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Second Order ◽  
Korovkin Type Theorem ◽  
Stancu Operators

The aim of this article is to introduce a bivariate extension of Schurer-Stancu operators based on (p,q)-integers. We prove uniform approximation by means of Bohman-Korovkin type theorem, rate of convergence using total modulus of smoothness and degree of approximation via second order modulus of smoothness, Peetre?s K-functional, Lipschitz type class.

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 $n-1$.

Classical Approximation

1995 ◽  
Author(s):  
Marek A. Kowalski ◽  
Krzystof A. Sikorski ◽  
Frank Stenger
Keyword(s):  
20Th Century ◽  
Uniform Approximation ◽  
Best Approximation ◽  
Inner Product ◽  
Normed Spaces ◽  
Remez Algorithm ◽  
Best Uniform Approximation ◽  
Converse Theorems ◽  
Inner Product Spaces ◽  
Finite Dimensional

In this chapter we acquaint the reader with the theory of approximation of elements of normed spaces by elements of their finite dimensional subspaces. The theory of best approximation was originated between 1850 and 1860 by Chebyshev. His results and ideas have been extended and complemented in the 20th century by other eminent mathematicians, such as Bernstein, Jackson, and Kolmogorov. Initially, we present the classical theory of best approximation in the setting of normed spaces. Next, we discuss best approximation in unitary (inner product) spaces, and we present several practically important examples. Finally, we give a reasonably complete presentation of best uniform approximation, along with examples, the Remez algorithm, and including converse theorems about best approximation. The goal of this section is to present some general results on approximation in normed spaces.

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 Spherical Approximation on Unit Sphere

2018 ◽  
Vol 26 (3) ◽  
pp. 61-68
Author(s):  
Eman Samir Bhaya ◽  
Ekhlas Annon Musa
Keyword(s):  
Unit Sphere ◽  
Best Approximation ◽  
Type Theorem ◽  
Modulus Of Smoothness ◽  
Central Problem ◽  
Spherical Approximation ◽  
Jackson Type ◽  
Approximation Of Functions ◽  
Lp Spaces ◽  
Jackson Type Theorem

In this paper we introduce a Jackson type theorem for functions in LP spaces on sphere And study on best approximation of  functions in  spaces defined on unit sphere. our central problem is to describe the approximation behavior of functions in    spaces for  by modulus of smoothness of functions.

scholarly journals On the uniform convergence of a q-series

2016 ◽  
Vol 32 (2) ◽  
pp. 141-146
Author(s):  
OCTAVIAN AGRATINI ◽  
◽  
VIJAY GUPTA ◽  
Keyword(s):  
Uniform Convergence ◽  
Upper Bound ◽  
Uniform Approximation ◽  
Approximation Error ◽  
Modulus Of Smoothness ◽  
Positive Operators ◽  
Linear Positive Operators

The paper deals with a class of linear positive operators expressed by q-series. By using modulus of smoothness an upper bound of approximation error is determined. We identify functions for which these operators provide uniform approximation over noncompact intervals. A particular case is delivered.

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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