Approximation properties of some discrete Fourier sums for piecewise smooth discontinuous functions

Denote by $L_{n,\,N}(f, x)$ a trigonometric polynomial of order at most $n$ possessing the least quadratic deviation from $f$ with respect to the system $\left\{t_k = u + \frac{2\pi k}{N}\right\}_{k=0}^{N-1}$, where $u \in \mathbb{R}$ and $n \leq N/2$. Let $D^1$ be the space of $2\pi$-periodic piecewise continuously differentiable functions $f$ with a finite number of jump discontinuity points $-\pi = \xi_1 < \ldots < \xi_m = \pi$ and with absolutely continuous derivatives on each interval $(\xi_i, \xi_{i+1})$. In the present article, we consider the problem of approximation of functions $f \in D^1$ by the trigonometric polynomials $L_{n,\,N}(f, x)$. We have found the exact order estimate $\left|f(x) - L_{n,\,N}(f, x)\right| \leq c(f, \varepsilon)/n$, $\left|x - \xi_i\right| \geq \varepsilon$. The proofs of these estimations are based on comparing of approximating properties of discrete and continuous finite Fourier series.