2017
Том 69
№ 9

All Issues

On exact Bernstein-type inequalities for splines

Kofanov V. A.

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Abstract

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 $r$.

English version (Springer): Ukrainian Mathematical Journal 58 (2006), no. 10, pp 1538-1551.

Citation Example: Kofanov V. A. On exact Bernstein-type inequalities for splines // Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1357–1367.

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