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 Eurasian Math. J., 2011, Volume 2, Number 3, Pages 98–124 (Mi emj65)

Methods of trigonometric approximation and generalized smoothness. I

K. Runovskia, H.-J. Schmeisserb

a V. I. Vernadskiy Taurida National University, Simferopol, Ukraine
b Mathematisches Institut, Friedrich-Schiller University, Jena, Germany

Abstract: We give a unified approach to trigonometric approximation and study its interrelation with smoothness properties of functions. In the first part our concern lies on convergence of the Fourier means, interpolation means and families of linear trigonometric polynomial operators in the scale of the $L_p$-spaces with $0<p\le+\infty$. We establish a general convergence theorem which allows to determine the ranges of convergence for approximation methods generated by classical kernels.
The second part will deal with the equivalence of the approximation errors for families of linear polynomial operators generated by classical kernels in terms of $K$-functionals generated by homogeneous functions and general moduli of smoothness. It will also be shown that the results of the classical approximation theory on the Fourier means and interpolation means in the case $1\le p\le+\infty$, classical differential operators and moduli of smoothness are direct consequences of our general approach.

Keywords and phrases: trigonometric approximation, Fourier means, families of linear polynomial operators, $L_p$-convergence ($0<p\leq+\infty$), classical kernels and summability methods, stochastic approximation.

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MSC: 42A10, 42A15, 42A24, 42B08, 42B35
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Citation: K. Runovski, H.-J. Schmeisser, “Methods of trigonometric approximation and generalized smoothness. I”, Eurasian Math. J., 2:3 (2011), 98–124

Citation in format AMSBIB
\Bibitem{RunSch11} \by K.~Runovski, H.-J.~Schmeisser \paper Methods of trigonometric approximation and generalized smoothness.~I \jour Eurasian Math. J. \yr 2011 \vol 2 \issue 3 \pages 98--124 \mathnet{http://mi.mathnet.ru/emj65} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2910844} \zmath{https://zbmath.org/?q=an:1266.42003} 

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This publication is cited in the following articles:
1. G. A. Yusupov, “Best polynomial approximations and widths of certain classes of functions in the space $L_2$”, Eurasian Math. J., 4:3 (2013), 120–126
2. K. V. Runovskii, “A Direct Theorem of Approximation Theory for a General Modulus of Smoothness”, Math. Notes, 95:6 (2014), 833–842
3. S. Yu. Artamonov, “Direct Jackson-Type Estimate for the General Modulus of Smoothness in the Nonperiodic Case”, Math. Notes, 97:5 (2015), 811–814
4. S. Yu. Artamonov, “Quality of Approximation by Fourier Means in Terms of General Moduli of Smoothness”, Math. Notes, 98:1 (2015), 3–10
5. S. Yu. Artamonov, “Nonperiodic Modulus of Smoothness Corresponding to the Riesz Derivative”, Math. Notes, 99:6 (2016), 928–931
6. S. Yu. Artamonov, “On some constructions of a non-periodic modulus of smoothness related to the Riesz derivative”, Eurasian Math. J., 9:2 (2018), 11–21
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