Asymptotic estimates for the widths of classes of functions of high smoothness

We find two-sided estimates for Kolmogorov, Bernstein, linear and projection widths of the classes of convolutions of $2\pi$-periodic functions $\varphi$, such that $\|\varphi\|_2\le1$, with fixed generated kernels $\Psi_{\bar{\beta}}$, which have Fourier series of the form $\sum\limits_{k=1}^\infty \psi(k)\cos(kt-\beta_k\pi/2),$ where $\psi(k)\ge0,$ $\sum\psi^2(k)<\infty, \beta_k\in\mathbb{R},$ in the space $C$. It is shown that for rapidly decrising sequences $\psi(k)$ (in particular, if $\lim\limits_{k\rightarrow\infty}{\psi(k+1)}/{\psi(k)}=0$) obtained estimates are asymptotic equalities. We establish that asymptotic equalities for widths of this classes are realized by trigonometric Fourier sums.Comment: 14 page

Social security as a subject of human geography research study

This article deals social security as a subject of human geography research study. The publication focuses on the social, economic, geoecological, information, mental and cultural components of social development of Ukraine. The main social problems and scientific aspects of social security in Ukraine are considere