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On exact Bernstein-type inequalities for splines

Authors:

V A Kofanov

Institution:

(1) Dnepropetrovsk National University, Dnepropetrovsk

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 ϕ_r is the perfect Euler spline of order r. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1357–1367, October, 2006.

Keywords:

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