2017
Том 69
№ 9

All Issues

On Kolmogorov-Type Inequalities with Integrable Highest Derivative

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

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Abstract

We obtain the new exact Kolmogorov-type inequality $$\left\| {x^{\left( k \right)} } \right\|_2 \leqslant K\left\| x \right\|_2^{\frac{{r - k - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}{{r - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}} \left\| {x^{\left( r \right)} } \right\|_1^{\frac{k}{{r{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}}$$ for 2π-periodic functions \(x \in L_1^r\) and any k, rN, k < r. We present applications of this inequality to problems of approximation of one class of functions by another class and estimates of K-functional type.

English version (Springer): Ukrainian Mathematical Journal 54 (2002), no. 12, pp 2055-2059.

Citation Example: Babenko V. F., Kofanov V. A., Pichugov S. A. On Kolmogorov-Type Inequalities with Integrable Highest Derivative // Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1694-1697.

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