Vol 30, No 149 (2025)

This paper is devoted to the study of the existence of extremal elements of the set of continuously differentiable concave extensions to the set $[0,1]^n$ of an arbitrary Boolean function $f_{B}(x_1,\ldots,x_n)$, as well as finding the cardinality of the set of continuously differentiable concave extensions to $[0,1]^n$ of the Boolean function $f_{B}(x_1,\ldots,x_n).$ As a result of the study, it is proved that, firstly, for any Boolean function $f_{B}(x_1,\ldots,x_n)$ among its continuously differentiable concave extensions to $[0,1]^n$ there is no maximal element, secondly, if the Boolean function $f_{B}(x_1,\ldots,x_n)$ has more than one essential variable, then among its continuously differentiable concave extensions to $[0,1]^n$ there is no minimal element, and if the Boolean function is constant or has only one essential variable, then among its continuously differentiable concave extensions to $[0,1]^n$ there is a unique minimal element, the explicit form of which is given in the paper. It was also established that the cardinality of the set of continuously differentiable concave extensions to $[0,1]^n$ of an arbitrary Boolean function $f_{B}(x_1,\ldots,x_n)$ is equal to the continuum.

The article considers the motions of dynamical system $g^t$ defined on a topological compact manifold $V.$

It is shown that the set $M_1$ of non-wandering points with respect to $V$ is the set of central motions $\fM$, and  the union of all compact minimal sets is everywhere dense in the set $\fM.$ It is established that for any motion $f(t,p),$ there exists a compact minimal set $\Om\subset V$ with the following property: for all values $t_0\in\R$ and every neighborhood $E_{\Om}$ of the set $\Om,$ the probability that the arc $\{f(t,p)\colon t\in[t_0,t_1]\}$ of the motion trajectory $f(t,p)$ belongs to the set $E_{\Om},$ tends to 1 as $t_1\to+\iy;$ a similar statement is true for the arc $\{f(t,p)\colon t\in[-t_1,t_0]\}.$

All statements of this article can be transferred without any changes to the system $g^t$ defined in a Hausdorff sequentially compact topological space.

We consider the set $L_{2}^{(r)}$ of $2\pi$-periodic functions $f\in L_{2}$ whose $(r-1)$-th order derivative is absolutely continuous, and the $r$-th order derivative $f^{(r)}\in L_{2}.$ We solve the extremal problem of finding an exact Jackson--Stechkin type constant that connects the best polynomial approximation of functions from $L_{2}^{(r)}$ with the average value of the generalized $m$-th order modulus of continuity of their derivative $f^{(r)}$ in the space $L_{2}.$ We also consider the classes $W_{m}^{(r)}(u)$ and $W_{m}^{(r)}(u,\Phi)$ of functions from $L_{2}^{(r)}$ such that the average value of the generalized $m$-th order modulus of continuity of their derivative $f^{(r)}$ is bounded from above by unity and, accordingly, by the value of some function $\Phi(u).$ We calculate the exact values of the known $n$-widths (according to Bernstein, to Gelfand, to Kolmogorov, linear, and projection) of the class $W_{m}^{(r)}(u).$ Then we solve the extremal problem of finding the exact value of the best approximation for the class $W_{m}^{(r)}(u,\Phi).$  The obtained results develop and complement some known results on the best approximation of various classes of functions in $L_{2}.$ In the paper, we use methods for solving extremal problems in normed spaces, as well as the method developed by V.\,M.~Tikhomirov
for estimating from below the $n$-widths of functional classes in Banach spaces.

Let $\Omega_0^r$ be a class of holomorphic functions $\omega$ in the unit disk $\Delta,$ with real coefficients, and such that $|\omega(z)|<1,$ $\omega(0)=0,$ $z\in\Delta.$ The coefficients problem in the class $\Omega_0^r$ is formulated as follows: find the necessary and sufficient conditions to be imposed on the real numbers $\{\omega\}_1, \{\omega\}_2,\ldots$ in order for the series $\{\omega\}_1 z+\{\omega\}_2 z^2+\ldots$ to be the Taylor series of a function in the class $\Omega_0^r.$

   The class $B^r$ consists of holomorphic functions $f$ in $\Delta$ with real coefficients and such that    $0<|f(z)|\leq 1,$ $z\in\Delta.$ The classes $B_t^r,$ $t\geq 0,$ are defined as the sets of functions $f\in B^r$ such that $f(0)=e^{-t}.$ The problem of obtaining a sharp estimation of $|\{f\}_n|,$ $n\in\mathbb N,$ on the class $B^r$ or $B_t^r$ is commonly referred to as the Krzyz problem (for the class $B^r$ or $B_t^r$). It~is clear that the union of all classes $B_t^r$ exhausts the class $B^r$ up to rotations in the plane of variable $w$ ($w=f(z)$).

 Based on the solution of the coefficients problem for the class $\Omega_0^r,$ the problem of obtaining a sharp estimation of the functional $|\{f\}_3|$ on the classes $B_t^r$ for every $t\geq 0$ is solved by transitioning to the functional over the class $\Omega_0^r,$ after which the problem is reduced to finding the global constrained extremum of a function of two real variables with inequality-type constraints.

The extreme functions are found in two forms: as a convex combination of Schwartz kernels related to the Caratheodory class, and as Blaschke products related to the class
$\Omega_0^r.$