On exact Bernstein-type inequalities for splines

  • V. A. Kofanov


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 $iπ/n, i ∈ Z, n ∈ N:$ $$\left\| {s^{(k)} } \right\|_q \leqslant n^{k + 1/p - 1/q} \frac{{\left\| {\varphi _{r - k} } \right\|_q }}{{\left\| {\varphi _r } \right\|_p }}\left\| s \right\|_p ,$$ where $k, r ∈ N, k < r, p = 1$ or $p = 2, q > p$, and $ϕr$ is the perfect Euler spline of order $r$.
How to Cite
Kofanov, V. A. “On Exact Bernstein-Type Inequalities for Splines”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, no. 10, Oct. 2006, pp. 1357–1367, https://umj.imath.kiev.ua/index.php/umj/article/view/3538.
Research articles