Approximation theory

Daghestan Electronic Mathematical Reports: Issue 8 (2017)

Approximation of piecewise linear functions by discrete Fourier sums

UDK: 517.521.2

Pages: 21 - 26

Let N be a natural number greater than 1. We select N uniformly distributed points t_k = 2\pi k / N (0 \leq k \leq N - 1) on [0,2\pi]. Denote by L_{n,N}(f)=L_{n,N}(f,x) (1\leq n\leq N/2) the trigonometric polynomial of order n possessing the least quadratic deviation from f with respect to the system \{t_k\}_{k=0}^{N-1}. In the present article the problem of function approximation by the polynomials L_{n,N}(f,x) is considered. Special attention is paid to approximation of 2\pi-periodic functions f_1 and f_2 by the polynomials L_{n,N}(f,x), where f_1(x)=|x| and f_2(x)=\mbox{sign\,} x for x \in [-\pi,\pi]. For the first function f_1 we show that instead of the estimate \left|f_{1}(x)-L_{n,N}(f_{1},x)\right| \leq c\ln n/n which follows from well-known Lebesgue inequality for the polynomials L_{n,N}(f,x) we found an exact order estimate \left|f_{1}(x)-L_{n,N}(f_{1},x)\right| \leq c/n (x \in \mathbb{R}) which is uniform with respect to 1 \leq n \leq N/2. Moreover, we found a local estimate \left|f_{1}(x)-L_{n,N}(f_{1},x)\right| \leq c(\varepsilon)/n^2 (\left|x - \pi k\right| \geq \varepsilon) which is also uniform with respect to 1 \leq n \leq N/2. For the second function f_2 we found only a local estimate \left|f_{2}(x)-L_{n,N}(f_{2},x)\right| \leq c(\varepsilon)/n (\left|x - \pi k\right| \geq \varepsilon) which is uniform with respect to 1 \leq n \leq N/2. The proofs of these estimations based on comparing of approximating properties of discrete and continuous finite Fourier series.

Keywords: function approximation, trigonometric polynomials, Fourier series.

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