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Special Difference Operators and the Constants in the Classical Jackson-Type Theorems

  • Alexander G. Babenko
  • Yuriy V. KryakinEmail author
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

In this survey we will show how one can use the operators
$$\displaystyle W_{2k}(f,h)(x) :=(-1)^k \frac 1{ h \binom {2k} k}\int _{-h}^h \ \widehat \Delta _t^{2k} f(x) \left ( 1 - \frac {|t|}h \right )\, dt, \qquad h>0,$$
where
$$\displaystyle \widehat \Delta _h^{m} f(x):=\sum _{j=0}^{m} (-1)^{j+m} \binom {m}{j} f(x+jh-mh/2),$$
to indicate the sharp order (with respect to k) of the Jackson–Stechkin constants in the main theorems of the classical approximation theory.

Notes

Acknowledgements

The work of A.G. Babenko was supported by the Russian Foundation for Basic Research (project no. 18-01-00336a) and by the Ural Federal University within the Russian Academic Excellence Project “5-100” (agreement no. 02.A03.21.0006).

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Copyright information

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Authors and Affiliations

  1. 1.N. N. Krasovskii Institute of Mathematics and MechanicsUral Branch of the Russian Academy of SciencesEkaterinburgRussia
  2. 2.Institute of Natural Sciences and MathematicsUral Federal University named after the First President of Russia B. N. YeltsinEkaterinburgRussia
  3. 3.Institute of MathematicsUniversity of WroclawWroclawPoland

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