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Ukrainian Mathematical Journal

, Volume 54, Issue 12, pp 2055–2059 | Cite as

On Kolmogorov-Type Inequalities with Integrable Highest Derivative

  • V. F. Babenko
  • V. A. Kofanov
  • S. A. Pichugov
Article
  • 18 Downloads

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.

Keywords

High Derivative Integrable High 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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REFERENCES

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    V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “On exact inequalities of Kolmogorov-type in the case of small smoothness,” Dopov. Akad. Nauk Ukrainy, No. 6, 11–15 (1998).Google Scholar
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    A. N. Kolmogorov, Selected Works. Mathematics and Mechanics [in Russian], Nauka, Moscow (1985).Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • V. F. Babenko
    • 1
  • V. A. Kofanov
    • 1
  • S. A. Pichugov
    • 1
  1. 1.Dnepropetrovsk UniversityDnepropetrovsk

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