Приближение непрерывных 2π-периодических кусочно-гладких функций дискретными суммами Фурье

G G Akniyev

doi:10.18500/1816-9791-2019-19-1-4-15

Abstract

Приближение непрерывных 2π-периодических кусочно-гладких функций дискретными суммами Фурье

Highlights

In the present article we consider the problem of approximation of functions f ∈ CΩ2 by the polynomials Ln,N (f, x)
We show that instead of the estimate |f (x) − Ln,N (f, x)| c ln n/n, which follows from the well-known Lebesgue inequality, we found an exact order estimate |f (x) − Ln,N (f, x)| c/n (x ∈ R) which is uniform with respect to n (1 n N/2)
The proofs of these estimations are based on comparing of approximating properties of discrete and continuous finite Fourier series

Summary

Piecewise Smooth Functions by Discrete Fourier Sums

Let N be a natural number greater than 1. Select N uniformly distributed points tk = 2πk/N + u (0 k N − 1), and denote by Ln,N (f ) = Ln,N (f, x) (1 n N/2) the trigonometric polynomial of order n possessing the least quadratic deviation from f with respect to the system {tk}kN=−01. In the present article we consider the problem of approximation of functions f ∈ CΩ2 by the polynomials Ln,N (f, x). We found a local estimate |f (x) − Ln,N (f, x)| c(ε)/n2 (|x − ai| ε) which is uniform with respect to n (1 n N/2) The proofs of these estimations are based on comparing of approximating properties of discrete and continuous finite Fourier series.

INTRODUCTION

In the similar way we can get

Hence we have