Estimates of best $m$-term trigonometric approximation of classes of analytic functions

TL;DR

This paper derives precise order estimates for the best m-term trigonometric approximations of classes of convolutions of periodic functions in various L_s spaces, showing these estimates match Fourier sum approximations.

Contribution

It provides exact order estimates for m-term trigonometric approximations of convolution classes with geometrically decaying coefficients, extending understanding of approximation quality.

Findings

Estimates match Fourier sum approximations in order

Exact order estimates for orthogonal trigonometric approximation

Determination of trigonometric widths for the classes

Abstract

In metric of spaces $L_{s}, \ 1\leq s\leq\infty$, we obtain exact in order estimates of best $m$-term trigonometric approximations of classes of convolutions of periodic functions, that belong to unit all of space $L_{p}, \ 1\leq p\leq\infty$, with generated kernel $\Psi_{\beta}(t)=\sum\limits_{k=1}^{\infty}\psi(k)\cos(kt-\frac{\beta\pi}{2})$, $\beta\in \mathbb{R}$, whose coefficients $\psi(k)$ tend to zero not slower than geometric progression. Obtained estimates coincide in order with approximation by Fourier sums of the given classes of functions in $L_{s}$-metric. This fact allows to write down exact order estimates of best orthogonal trigonometric approximation and trigonometric widths of given classes.