Parabolic spline on the non-homogeneous mesh (Q2739966)

Let us consider partitions \(\Delta_{t}:\;a=t_0<t_1<\ldots<t_{N+2}=b\), \(\Delta_{\tau}:\;a=\tau_0<\tau_1<\ldots<\tau_{N+1}=b\), \(t_{i}< \tau_{i}<t_{i+1}, i=1,\ldots,N\). Let us denote for some function \(f(t):\;f_{i}=f(t_{i}), i=0,\ldots,N+2\). If \(f(t)\in C[a,b]\), then there exists a spline \(s(t)\) such that \(s(t_{i}) =f(t_{i}), i=0,\ldots,N+2\), \(s(\tau_{i}-0)=s(\tau_{i}+0)\), \(s'(\tau_{i}-0)=s'(\tau_{i}+0)\), \(i=1,2,\ldots,N\). One of the proved results is the following. If \(f(t)\in C^2[a,b]\) and interpolation parabolic spline \(s(t)\) satisfies the above-mentioned conditions, then \(\|f^{(v)}(t)-s^{(v)}(t)\|_{C[a,b]}\leq \gamma_{v}\omega(f'',\Delta_{\tau}), v=0,1,2\), where \(\gamma_{v}, v=0,1,2\) are some constants; \(\omega(\Delta_{\tau})\) is the continuity modulus.