Approximation of continuous $2\pi$-periodic piecewise smooth functions by discrete Fourier sums

Let $N$ be a natural number greater than $1$. Select $N$ uniformly distributed points $t_k = 2\pi k / N + u$ $(0 \leq k \leq N - 1)$, and 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 $\{t_k\}_{k=0}^{N-1}$. Select $m+1$ points $-\pi=a_{0}<a_{1}<\ldots<a_{m-1}<a_{m}=\pi$, where $m\geq 2$, and denote $\Omega = \left\{a_i\right\}_{i=0}^{m}$. Denote by $C_{\Omega}^{r}$ a class of $2\pi$-periodic continuous functions $f$, where $f$ is $r$-times differentiable on each segment $\Delta_{i}=[a_{i},a_{i+1}]$ and $f^{(r)}$ is absolutely continuous on $\Delta_{i}$. In the present article we consider the problem of approximation of functions $f\in C_{\Omega}^{2}$ by the polynomials $L_{n,N}(f,x)$. We show that instead of the estimate $\left|f(x)-L_{n,N}(f,x)\right| \leq c\ln n/n$, which follows from the well-known Lebesgue inequality, we found an exact order estimate $\left|f(x)-L_{n,N}(f,x)\right| \leq c/n$ ($x \in \mathbb{R}$) which is uniform with respect to $n$ ($1 \leq n \leq N/2$). Moreover, we found a local estimate $\left|f(x)-L_{n,N}(f,x)\right| \leq c(\varepsilon)/n^2$ ($\left|x - a_i\right| \geq \varepsilon$) which is also uniform with respect to $n$ ($1 \leq n \leq N/2$). The proofs of these estimations are based on comparing of approximating properties of discrete and continuous finite Fourier series.