Inequalities of Kolmogorov type and estimates of spline interpolation on periodic classes \(W^ m_ 2\) (Q1813363)

Making use of precise inequalities of Kolmogorov type for periodic functions, the author estimates the exact constant \(C_{m,k}(\Delta_ n)\) in the inequality \[ \| f^{(k)}-S^{(k)}_{n,2m-1}\|_ \infty\leq C_{m,k}(\Delta_ n)\|\Delta_ n\|^{m-k-1/2}\| f^{(m)}\|_ 2 \] (here \(S_{n,2m-1}=S_{n,2m-1}(f,\Delta_ n)\) is a periodic polynomial spline of degree \(2m-1\) and \(\Delta_ n=\{0=t_ 0<t_ 1<\cdots<t_ n=2\pi\}\) is a partitioning of the interval \([0,2\pi])\). The mentioned result is used to derive estimates of the quantity \[ \ell^{(k)}_{n,2m-1}(W^ \mu_ 2)_ \infty=\sup_{f\in W^ \mu_ 2}\| f^{(k)}-S^{(k)}_{n,2m-1}\|_ \infty \] in the case of a uniform partitioning \(\Delta_ n=\delta_ n=\{2\pi i/n\}^ n_{i=0}\).