2019
Том 71
№ 11

All Issues

Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness

Babenko V. F., Kofanov V. A., Pichugov S. A.

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Abstract

We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for r = 2, k = 1 or r = 3, k = 1, 2 and arbitrary q, p ∈ [1, ∞], the following unimprovable inequality holds for functions \(x \in L_\infty ^r \) : $$\left\| {x^{\left( k \right)} } \right\|_q \leqslant \frac{{\left\| {{\phi }_{r - k} } \right\|_q }}{{\left\| {{\phi }_r } \right\|_p^\alpha }}\left\| x \right\|_p^\alpha \left\| {x^{\left( k \right)} } \right\|_\infty ^{1 - \alpha } $$ where \(\alpha = \min \left\{ {1 - \frac{k}{r},\frac{{r - k + {1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-0em} q}}}{{r + {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-0em} p}}}} \right\}\) and ϕ r is the perfect Euler spline of order r.

English version (Springer): Ukrainian Mathematical Journal 53 (2001), no. 10, pp 1569-1582.

Citation Example: Babenko V. F., Kofanov V. A., Pichugov S. A. Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness // Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1299-1308.

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