Let $N$ be a natural number greater than $1$.
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 this article approximation of functions 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 $f_2(x)=\mathrm{sign}\, x$ for $x \in
[-\pi,\pi]$.
For the first function $f_1$ we show that instead of the estimation
$\left|f_{1}(x)-L_{n,N}(f_{1},x)\right| \leq c\ln n/n$ which follows from the
well-known Lebesgue inequality for the polynomials $L_{n,N}(f,x)$ we found an
exact order estimation $\left|f_{1}(x)-L_{n,N}(f_{1},x)\right| \leq c/n$ ($x
\in
\mathbb{R}$) which is uniform with respect to $1 \leq n \leq N/2$.
Moreover, we found a local estimation $\left|f_{1}(x)-L_{n,N}(f_{1},x)\right|
\leq c(\varepsilon)/n^2$ ($\left|x - \pi k\right| \geq \varepsilon$) which is
also uniform with respect to $1 \leq n \leq N/2$.
For the second function $f_2$ we found only a local estimation
$\left|f_{2}(x)-L_{n,N}(f_{2},x)\right| \leq c(\varepsilon)/n$ ($\left|x - \pi
k\right| \geq \varepsilon$) which is uniform with respect to $1 \leq n \leq
N/2$.
The proofs of these estimations are based on comparing of approximating
properties of discrete and continuous finite Fourier series.
