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Topics in Classical and Modern Analysis pp 35–46Cite as

Birkhäuser

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

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Part of the book series: Applied and Numerical Harmonic Analysis ((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.

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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).

Author information

Authors and Affiliations

  1. N. N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia

    Alexander G. Babenko

  2. Institute of Natural Sciences and Mathematics, Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg, Russia

    Alexander G. Babenko

  3. Institute of Mathematics, University of Wroclaw, Wroclaw, Poland

    Yuriy V. Kryakin

Authors
  1. Alexander G. Babenko
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Corresponding author

Correspondence to Yuriy V. Kryakin .

Editor information

Editors and Affiliations

  1. Georgia Southern University, Statesboro, GA, USA

    Martha Abell

  2. Georgia Southern University, Statesboro, GA, USA

    Emil Iacob

  3. Georgia Southern University, Statesboro, GA, USA

    Alex Stokolos

  4. Georgia Southern University, Statesboro, GA, USA

    Sharon Taylor

  5. ICREA, Passeig de Lluís Companys, Barcelona, Spain ; Centre de Recerca Matemàtica, Barcelona, Spain

    Sergey Tikhonov

  6. Georgia Southern University, Statesboro, GA, USA

    Jiehua Zhu

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Babenko, A.G., Kryakin, Y.V. (2019). Special Difference Operators and the Constants in the Classical Jackson-Type Theorems. In: Abell, M., Iacob, E., Stokolos, A., Taylor, S., Tikhonov, S., Zhu, J. (eds) Topics in Classical and Modern Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-12277-5_2

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