DISCRETE LEAST SQUARES APPROXIMATION OF PIECEWISE-LINEAR FUNCTIONS BY TRIGONOMETRIC POLYNOMIALS

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DISCRETE LEAST SQUARES APPROXIMATION OF PIECEWISE-LINEAR FUNCTIONS BY TRIGONOMETRIC POLYNOMIALS

Journal Title: Проблемы анализа-Issues of Analysis - Year 2017, Vol 6, Issue 2

Abstract

Let N be a natural number greater than 1. Select N uniformly distributed points t_k = 2πk/N (0 ≤ k ≤ N − 1) on [0, 2π]. Denote by L_{n,N} (f) = L_{n,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 {t_k}^(N−1)_k=0 . In this article approximation of functions by the polynomials L_{n,N} (f, x) is considered. Special attention is paid to approximation of 2π-periodic functions f_1 and f_2 by the polynomials L_{n,N} (f, x), where f_1(x) = |x| and f_2(x) = sign x for x ∈ [−π, π]. For the first function f_1 we show that instead of the estimation |f_1(x) − L_{n,N} (f_1, x)| ≤ c ln n/n which follows from the well-known Lebesgue inequality for the polynomials L_{n,N} (f, x) we found an exact order estimation |f_1(x) − L_{n,N} (f_1, x)| ≤ c/n (x ∈ R) which is uniform with respect to 1 ≤ n ≤ N/2. Moreover, we found a local estimation |f_1(x) − L_{n,N} (f_1, x)| ≤ c(ε)/n2 (|x − πk| ≥ ε) which is also uniform with respect to 1 ≤ n ≤ N/2. For the second function f_2 we found only a local estimation |f_2(x) − L_{n,N} (f_2, x)| ≤ c(ε)/n (|x − πk| ≥ ε) which is uniform with respect to 1 ≤ n ≤ N/2. The proofs of these estimations are based on comparing of approximating properties of discrete and continuous finite Fourier series.

Authors and Affiliations

G. G. Akniyev

Keywords

function approximation. trigonometric polynomials. Fourier series

EP ID EP256503
DOI 10.15393/j3.art.2017.4070
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