Order estimations of the best approximations and approximations of the Fourier sums on the classes of infinitely differentiable functions

TL;DR

This paper derives order estimations for best uniform and Fourier sum approximations of classes of periodic functions with specific derivative properties, extending results to various metrics and function classes.

Contribution

It provides new order estimations for approximation errors of classes of infinitely differentiable functions with $(eta, ext{psi})$-derivatives, including in $L_s$ metrics.

Findings

Order estimations for uniform approximations are established.

Analogous estimations in $L_s$-metrics are derived.

Results apply to classes with derivatives decreasing faster than any power function.

Abstract

We obtained order estimations for the best uniform approximations by trigonometric polynomials and approximations by Fourier sums of classes of $2\pi$-periodic continuous functions, which $(\psi,\beta)$-derivatives $f_{\beta}^{\psi}$ belong to unit balls of spaces $L_{p}, 1\leq p<\infty$ in case at consequences $\psi(k)$ decrease to nought faster than any power function. We also established the analogical estimations in $L_{s}$-metric, $1<s\leq\infty$, for classes of the summable $(\psi,\beta)$-differentiable functions, such that $\parallel f_{\beta}^{\psi}\parallel_{1}\leq1$.