On approximation of continuous functions by piecewise-continuous ones

Let $f(x) \in W^r H_{\omega} [-\pi, \pi]$ ($r = 0;1$) and $x_k = \frac{k\pi}{n} = h \cdot k$ ($k = 0, \pm 1, \ldots, \pm n$). We call $2\pi$-periodic function $S_2(f, x)$ an interpolation spline of order 2 if, in any segment $[x_k - \frac{h}{2}, x_k + \frac{h}{2}]$, it is the Lagrange polynomial of second degree that interpolates the function $f(x)$ in the points $x_{k-1}$, $x_k$, $x_{k+1}$.We establish that for any concave modulus of continuity $\omega (t)$ the equalities hold:$$\sup\limits_{f \in H_{\omega}[-\pi, \pi]} \| f - S_2(f) \|_{\infty} = \omega(\frac{h}{2}) + \frac{1}{8} \omega (h),$$\sup\limits_{f \in W^1 H_{\omega}[-\pi, \pi]} \| f - S_2(f) \|_{\infty} = \frac{65}{192} \int\limits_0^{\frac{4}{5}h} \omega (t) dt + \frac{5}{48} \int\limits_{\frac{4}{5}h}^{\frac{6}{5}h} \omega (t) dt$$