On exact Bernstein-type inequalities for splines - Kofanov, V. A. (Ukrains’kyi Matematychnyi Zhurnal, 2006)
We establish new exact Bernstein-type and Kolmogorov-type inequalities. The main result of this work is the following exact inequality for periodic splines $s$ of order $r$ and defect 1 with nodes at the points $iπ/n, i ∈ Z, n ∈ N:$$$\left\| {s^{(k)} } \right\|_q \leqslant n^{k + 1/p - 1/q} \frac{{\left\| {φ _{r - k} } \right\|_q }}{{\left\| {φ _r } \right\|_p }}\left\| s \right\|_p ,$$where $k, r ∈ N, k < r, p = 1$ or $p = 2, q > p$, and $ϕr$ is the perfect Euler spline of order $r$.