A New Modulus of Smoothness for Uniform Approximation

Eman Samir Bhaya,Bushra Khudhair Hussein

doi:10.29196/jubpas.v26i9.1827

Abstract

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

Highlights

The moduli of smoothness that have the most attention in the recent years are the Ditizian–Totik modulus of smoothness [3], and Ivanov τ modulus of smoothness
Sometimes we need to improve the degree of best approximation near the end points, Hear come the needness of uniform approximation if we want to get uniform estimate we choose modulus of smoothness other than the ordinary modulus of smoothness
Definition 3.10 Let E(∆kLp, Πn ∩ ∆k)p = supf∈∆kLpinfqn∈Πn∩∆k‖f − qn‖p is the degree of best approximation of function from ∆kLp using k-monotone polynomial in Πn

Summary

Introduction

The estimates of best approximation using classical modulus of smoothness is not uniform. Sometimes we need to improve the degree of best approximation near the end points, Hear come the needness of uniform approximation if we want to get uniform estimate we choose modulus of smoothness other than the ordinary modulus of smoothness. We define several types of moduli of smoothness of kmonotone function in Lp, spaces for p < 1. We introduce estimates for these moduli of smoothness and applications for these estimates. Let us define the kth order modulus of smoothness for functions f in lp(j), as ωk(f, δ, j)p = sup 0

The Auxiliary Results

The Main Results

Conclusions