On various moduli of smoothness and $K$-functionals

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 $\S 2$ 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-τ_{r,n}(f)\big\|_p\asympω_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 ${\mathbb{R}}^d$ ($d\ge 2$), 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.

33 pages, the paper is in Russian, with the title, abstract and key words in English