Approximation properties of dicrete Fourier sums for some piecewise linear functions

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 other words, the greatest lower bound of the sums $\sum_{k=0}^{N-1}|f(t_k)-T_n(t_k)|^2$ on the set of trigonometric polynomials $T_n$ of order $n$ is attained by $L_{n,N}(f)$. In the present article the problem of function approximation by the polynomials $L_{n,N}(f,x)$ is considered. Using some example functions we show that the polynomials $L_{n,N}(f,x)$ uniformly approximate a piecewise-linear continuous function with a convergence rate $O(1/n)$ with respect to the variables $x \in \mathbb{R}$ and $1 \leq n \leq N/2$. These polynomials also uniformly approximate the same function with a rate $O(1/n^2)$ outside of some neighborhood of function's “crease” points. Also we show that the polynomials $L_{n,N}(f,x)$ uniformly approximate a piecewise-linear discontinuous function with a rate $O(1/n)$ with respect to the variables $x$ and $1 \leq n \leq N/2$ outside some neighborhood of discontinuity points. 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 estimate $\left|f_{1}(x)-L_{n,N}(f_{1},x)\right| \leq c\ln n/n$ which follows from the well-known Lebesgue inequality for the polynomials $L_{n,N}(f,x)$ we found an exact order estimate $\left|f_{1}(x)-L_{n,N}(f_{1},x)\right| \leq c/n$ ($x \in \mathbb{R}$) which is uniform relative 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 relative 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 relative 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.