Uniform approximations by Fourier sums on the sets of convolutions of periodic functions of high smoothness

TL;DR

This paper derives optimal Lebesgue-type inequalities for uniform approximations of high-smoothness periodic functions using Fourier sums, linking deviations to best approximations of related functions.

Contribution

It establishes the best possible uniform approximation bounds for Fourier sums on classes of high-smoothness functions defined via $(eta, eta)$-integrals, with asymptotic equalities.

Findings

Derived Lebesgue-type inequalities for Fourier sum deviations.

Proved the estimates are optimal for rapidly decreasing $\\psi(k)$.

Established asymptotic equalities for approximation boundaries.

Abstract

On the sets of $2\pi$-periodic functions $f$, which are defined with a help of $(\psi, \beta)$-integrals of the functions $\varphi$ from $L_{1}$, we establish Lebesgue-type inequalities, in which the uniform norms of deviations of Fourier sums are expressed via the best approximations by trigonometric polynomials of the functions $\varphi$. We prove that obtained estimates are best possible, in the case when the sequences $\psi(k)$ decrease to zero faster than any power function. In some important cases we establish the asymptotic equalities for the exact upper boundaries of uniform approximations by Fourier sums on the classes of $(\psi, \beta)$-integrals of the functions $\varphi$, which belong to the unit ball of the space $L_{1}$.