Approximation properties of Fourier sums for $2\pi$-periodic piecewise linear continuous functions

In various areas of applications the problem of approximation of continuous function $f = f(x)$, whose values are known in nodes of some grid $\Omega_m = \{\xi_i\}_{i=0}^{m}$ arises. Usually, to solve this problem the polynomial spline $l_m^r(x)$ of given degree $r$ is used [ahlberb_splines,zavyalov_splain_functions,stechkin_splines_in_math], which in simplest case $r = 1$ is polyline $l_m = l_m(x) = l_m^1(x)$, that coincides in grid nodes with function $f$. If we want to store this polyline, we should store all the pairs $(\xi_0, y_0), \ldots, (\xi_m, y_m)$, where $y_i = f(\xi_i)$ $(i = 0, \ldots, m)$, which may take o lot of storage space if number of nodes is big. In this connection the interim problem of compression of information $(\xi_0, y_0), \ldots, (\xi_m, y_m)$ so that we can restore original polyline with given precision arises. To solve such problems the so called spectral method is usually applied, which is based on expansion of function $l_m$ in a series of a given orthonormal system and saving a minimal amount of the obtained expansion coefficients provided an ability to restore this function with given precision. In the present paper we attempted to solve this problem for $2\pi$-periodic continuous polylines by expansion them into trigonometric Fourier series.