Estimates of the best m-term trigonometric approximations of classes of analytic functions

1Serdyuk, AS
2Stepanyuk, TA
1Institute of Mathematics of the NAS of Ukraine, Kyiv
2Lesya Ukrainka Eastern European National University, Lutsk
Dopov. Nac. akad. nauk Ukr. 2015, 2:32-37
https://doi.org/10.15407/dopovidi2015.02.032
Section: Mathematics
Language: Ukrainian
Abstract: 
In the metrics of spaces $L_{s}$, $1\leq s\leq\infty$, we obtain exact in order estimates of the best $m$-term
trigonometric approximations of classes of the convolutions of periodic functions that belong to a unit ball of the space $L_{p}$, $1\leq p\leq\infty$, with the generating kernel $\Psi_{\beta}(t)
=\textstyle\sum\limits_{k=1}^{\infty}\psi(k)\cos(kt-{\beta\pi}/{2})$, $\beta\in \mathbb{R}$, whose coefficients $\psi(k)$ tend to zero not slower than a geometric progression. The obtained estimates coincide in order with the approximation by Fourier sums of the given classes of functions in the Ls-metric. This fact allows us to write down exact order estimates of the best orthogonal trigonometric approximations and the trigonometric widths of the given classes.
Keywords: analytic function, trigonometric approximation
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