Order estimation of the best approximations and of the approximations by Fourier sums of classes of $(\psi,\beta)$--diferentiable functions

TL;DR

This paper derives precise order estimates for the best uniform and $L_p$-approximation of certain classes of periodic functions using Fourier sums, revealing when Fourier sums achieve optimal approximation orders.

Contribution

It provides exact-order estimates for best approximations of $(eta, ho)$-differentiable functions by Fourier sums in both uniform and $L_p$ metrics, extending previous results.

Findings

Fourier sums realize the best approximation orders in studied cases.

Exact order estimations are established for uniform and $L_p$-approximations.

Results apply to classes defined via convolutions with kernels having decreasing Fourier coefficients.

Abstract

There were established the exact-order estimations of the best uniform approximations by{\psi} the trigonometrical polynoms on the $C^{\psi}_{\beta,p}$ classes of $2\pi$-periodic continuous functions $f$, which are defined by the convolutions of the functions, which belong to the unit ball in $L_p$, $1\leq p <\infty$ spaces with generating fixed kernels $\Psi_{\beta}\subset|L_{p'}$, $\frac{1}{p}+\frac{1}{p'}=1$, whose Fourier coeficients decreasing to zero approximately as power functions. The exact order estimations were also established in $L_p$-metrics, $1 < p \leq\infty$ for $L^{\psi}_{\beta,1}$ classes of $2\pi$-periodic functions $f$, which are equivalent by means of Lebesque measure to the convolutions of $\Psi_{\beta}\subset|L_{p}$ kernels with the functions that belong to the unit ball in $L_1$ space. We showed that in investigating cases the orders of best approximations are realized by Fourier sums.