Asymptotic behavior of best approximations of classes of infinitely differentiable functions defined by moduli of continuity

TL;DR

This paper derives asymptotic estimates for the best approximation of certain classes of infinitely differentiable periodic functions by trigonometric polynomials, focusing on the behavior in $C$ and $L_1$ spaces.

Contribution

It provides new asymptotic estimates for best approximations of classes of smooth functions defined by moduli of continuity, with exactness in specific spaces.

Findings

Asymptotic estimates are obtained for best approximations.

Estimates are asymptotically exact for convex moduli of continuity.

Results apply to functions represented as convolutions with rapidly decaying Fourier coefficient kernels.

Abstract

We obtain asymptotic estimates for the best approximations by trigonometric polynomials in the metric space $C$ $(L_p)$ of classes of periodic functions that can be represented as a convolution of kernels $\Psi_\beta$, which Fourier coefficients tend to zero faster than any power sequence, with functions $\varphi\in C (\varphi\in L_p),$ which moduli of continuity do not exceed a fixed majorant $\omega(t)$. It is proved that in the spaces $C$ and $L_1$ the obtained estimates are asymptotically exact for convex moduli of continuity $\omega(t)$.