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Ukrainian Mathematical Journal

, Volume 62, Issue 3, pp 343–357 | Cite as

Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

  • V. F. Babenko
  • N.V. Parfinovych
  • S. A. Pichugov
Article
Let \( C\left( {{\mathbb{R}^m}} \right) \) be the space of bounded and continuous functions \( x:{\mathbb{R}^m} \to \mathbb{R} \) equipped with the norm
$$ \left\| x \right\|C = {\left\| x \right\|_{C\left( {{\mathbb{R}^m}} \right)}}: = \sup \left\{ {\left| {x(t)} \right|:t \in {\mathbb{R}^m}} \right\} $$
and let e j , j = 1,…,m, be a standard basis in \( {\mathbb{R}^m} \): Given moduli of continuity ω j , j = 1,…, m, denote
$$ {H^{j,{\omega_j}}}: = \left\{ {x \in C\left( {{\mathbb{R}^m}} \right):\left\| x \right\|{\omega_j} = \left\| x \right\|{H^{j,{\omega_j}}} = \mathop {{\sup }}\limits_{{t_j} \ne 0} \frac{{\left\| {\Delta {t_j}{e_j}x\left( \cdot \right)} \right\|C}}{{{\omega_j}\left( {\left| {{t_j}} \right|} \right)}} < \infty } \right\}. $$
We obtain new sharp Kolmogorov-type inequalities for the norms \( \left\| {D_\varepsilon^\alpha x} \right\|C \) of mixed fractional derivatives of functions \( x \in \cap_{j = 1}^m{H^{j,{\omega_j}}} \). Some applications of these inequalities are presented.

Keywords

Fractional Order Fractional Derivative Multivariate Function Successive Derivative Extremal Operator 
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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Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  • V. F. Babenko
    • 1
    • 2
  • N.V. Parfinovych
    • 1
  • S. A. Pichugov
    • 3
  1. 1.Dnepropetrovsk National UniversityDnepropetrovskUkraine
  2. 2.Institute of Applied Mathematics and MechanicsUkrainian National Academy of SciencesDonetskUkraine
  3. 3.Dnepropetrovsk State Technical University of Railway TransportDnepropetrovskUkraine

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