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Inequalities in Approximation Theory Involving Fractional Smoothness in Lp, 0 < p < 1

  • Yurii KolomoitsevEmail author
  • Tetiana Lomako
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

In the paper, we study inequalities for the best trigonometric approximations and fractional moduli of smoothness involving the Weyl and Liouville-Grünwald derivatives in Lp, 0 < p < 1. We extend known inequalities to the whole range of parameters of smoothness as well as obtain several new inequalities. As an application, the direct and inverse theorems of approximation theory involving the modulus of smoothness ωβ(f(α), δ)p, where f(α) is a fractional derivative of the function f, are derived. A description of the class of functions with the optimal rate of decrease of a fractional modulus of smoothness is given.

Keywords

Best approximation Trigonometric polynomials Fractional moduli of smoothness Fractional derivatives The spaces Lp, 0 < p < 1 

2000Mathematics Subject Classification

42A10 26A33 41A17 41A25 41A28 

Notes

Acknowledgements

This research was supported by the project AFFMA that has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 704030.

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Copyright information

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Authors and Affiliations

  1. 1.Institut für MathematikUniversität zu LübeckLübeckGermany
  2. 2.Institute of Applied Mathematics and Mechanics of NAS of UkraineSlov’yans’k, Donetsk RegionUkraine

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