Estimates for the Entropy Numbers of the Classes B p , θ Ω $$ {B}_{p,\theta}^{\varOmega } $$ of Periodic Multivariable Functions in the Uniform Metric

Estimates for the entropy numbers of the Nikol'skii–Besov classes of functions with mixed smoothness in the space of quasi‐continuous functions

“…has a rich history and belong to the important concepts of approximation theory. The questions related to the estimates of entropy numbers of the classes of functions of one and many variables, in particular, the Sobolev classes $$W^{\bm{r}}_{p,\alpha }(\mathbb {T}^d)$$, Nikol'skii classes $$H^{\bm{r}}_{p}(\mathbb {T}^d)$$, Nikol'skii–Besov classes $$B^{\bm{r}}_{p,\theta }(\mathbb {T}^d)$$, and their analogs were studied in many works [2–4, 7, 8, 11–14, 19, 24, 25, 30–33, 35, 36, 38, 40–44, 47]. Entropy numbers are also used in probability theory, where their research is closely related to the “small ball” problems [16, 17, 34] (see also [6, section 6.3]).…”

Approximation Properties of the Generalized Abel-Poisson Integrals on the Weyl-Nagy Classes

“…At present, the extremal problems of the approximation theory, being related to the study of the approximation properties of linear methods for summing Fourier series, become increasingly relevant in applied mathematics, in particular, in the creation of mathematical models [15][16][17][18][19], in signal transmission [20,21], in the decision theory [22] and others. The problem considered in the paper, as well as those close to it [23][24][25] find practical application in the issues of coding, transmission and reproduction of images.…”