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 $\S2$ 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.