On various moduli of smoothness and $K$-functionals

TL;DR

This paper explores the exact rates of approximation of functions using Fourier series and integrals, expressing these rates via special moduli of smoothness and $K$-functionals, with results applicable to functions on the real line, higher dimensions, and Banach spaces.

Contribution

It provides new explicit relationships between approximation rates and moduli of smoothness, extending classical results to multiple settings and including partial survey content.

Findings

Exact approximation rates are expressed via special moduli of smoothness.

Results are established for functions on the real line, higher-dimensional spaces, and Banach spaces.

The paper includes proofs for key theorems and discusses open problems.

Abstract

In this paper, exact rate of approximation of functions by linear means of Fourier series and Fourier integrals and corresponding $K$-functionals are expressed via special moduli of smoothness. . Introduction is given in $\S 1$. In $\S2$ functions on the line $\mathbb{R}$ are studied. A typical (well-known) result is as follows: for each $2\pi$-periodic function in $L_p$ on the period, for any $p\in[1,+\infty]$ ($L_\infty=C$) and $r\in\mathbb{N}$, there is a trigonometric polynomial $\tau_{r,n}(f)$ of degree not greater than $n$ such that \big\|f-\tau_{r,n}(f)\big\|_p\asymp\omega_r\Big(f;\frac{1}{n}\Big)_p\asymp \inf\limits_{g}\Big\{\|f-g\|_p+\frac{1}{n^r}\big\|g^{(r)}\big\|_p\Big\}, where the positive constants in these bilateral inequalities depend only on $r$. In $\S 3$ we deal with functions on $\mathbb{R}^d$ ($d\geq2$), while in $\S 4$ with functions on Banach spaces. The paper is partially of survey nature. The proofs are given only for Theorems 2.2, 3.9 and those in $\S 4$. Related open problems are formulated in $\S 5$. The list of references contains 52 items.