On approximation of continuous functions by piecewise-continuous ones

Abstract

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:
$$\sup\limits_{f \in H_{\omega}[-\pi, \pi]} \| f - S_2(f) \|_{\infty} = \omega(\frac{h}{2}) + \frac{1}{8} \omega (h),$$
$$\sup\limits_{f \in W^1 H_{\omega}[-\pi, \pi]} \| f - S_2(f) \|_{\infty} = \frac{65}{192} \int\limits_0^{\frac{4}{5}h} \omega (t) dt + \frac{5}{48} \int\limits_{\frac{4}{5}h}^{\frac{6}{5}h} \omega (t) dt$$