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
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- EP ID EP256503
- DOI 10.15393/j3.art.2017.4070
- Views 91
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