A study of approximation of functions of bounded variation by Faber-Schauder partial sums

Nikolaj Mormul`,Alexander Shchitov

doi:10.15587/1729-4061.2019.176595

Abstract

The Faber-Schauder system of functions was introduced in 1910 and became the first example of a basis in the space of continuous on [0, 1] functions. A number of results are known about the properties of the Faber-Schauder system, including estimations of errors of approximation of functions by polynomials and partial sums of series in the Faber-Schauder system.It is known that obtaining new estimates of errors of approximation of an arbitrary function by some given class of functions is one of the important tasks in the theory of approximation. Therefore, investigation of the approximation properties of polynomials and partial sums in the Faber-Schauder system is of considerable interest for the modern approximation theory.The problems of approximation of functions of bounded variation by partial sums of series in the Faber-Schauder system of functions are studied. The estimate of the error of approximation of functions from classes of functions of bounded variation Cp (1≤p<∞) in the space metric Lp using the values of the modulus of continuity of fractional order ϖ2-1/p(f, t) is obtained. From the obtained inequality, the estimate of the error of approximation of continuous functions in terms of the second-order modulus of continuity follows.Also, in the class of functions Cp (1<p<∞), the estimate of the error of approximation of functions in the space metric Lp using the modulus of continuity of fractional order ϖ1-1/p(f, t) is obtained.For classes of functions of bounded variation KCV(2,p) (1≤p<∞), the estimate of the error of approximation of functions in the space metric Lp by Faber-Schauder partial sums is obtained.Thus, several estimates of the errors of approximation of functions of bounded variation by their partial sums of series in the Faber-Schauder system are obtained. The obtained results are new in the theory of approximation. They generalize in a certain way the previously known results and can be used for further practical applications.

Highlights

The Faber-Schauder system of functions was introduced in the paper [1] and became the first example of a basis of the space of functions continuous on [0, 1]
Considering the approximation of functions f from the classes Cp (1≤ p < ∞) by polynomials in the Faber-Schauder system, several estimates of upper bounds are obtained using the modulus of continuity of fractional orders ( ) ωk−1/p f, δ
– to obtain estimates of errors of approximation of functions from classes of functions of bounded variation Cp (1≤ p < ∞) in the space metric Lp using the values of the moduli of continuity of fractional orders π1−1/p ( f, t ) and π2−1/p ( f, t );

Summary

Introduction

The Faber-Schauder system of functions was introduced in the paper [1] and became the first example of a basis of the space of functions continuous on [0, 1]. In [2] an estimate of the error of approximation of a continuous function by its Faber-Schauder partial sum is obtained. This result is specified in [4] using the second-order modulus of continuity. In [6,7,8], the exact estimates of errors of approximation of functions from some function classes by Faber-Schauder partial sums in uniform and integral metrics are obtained. Considering the approximation of functions f from the classes Cp (1≤ p < ∞) by polynomials in the Faber-Schauder system, several estimates of upper bounds are obtained using the modulus of continuity of fractional orders ( ) ωk−1/p f , δ. It is appropriate to use moduli of continuity of fractional orders ( ) ωk−1/p f , δ for obtaining estimates of errors of approximation of functions by series in the Faber-Schauder system

Literature review and problem statement

The aim and objectives of the study

Definitions and notations necessary for further presentation of the results

Conclusions