Том 231, № 1 (2018)

The problems of extremal decomposition with free poles on a circle are well known in the geometric theory of functions of a complex variable. One of such problems is the problem of maximum of the functional

\( {I}_n\left(\upgamma \right)={r}^{\upgamma}\left({B}_0,0\right)\prod \limits_{k=1}^nr\left({B}_k{,}_{ak}\right), \)

where γ ∈ (0, n], B₀, B₁, B₂,...,B_n, n ≥ 2, are pairwise disjoint domains in \( \overline{\mathrm{C}},{a}_0=0,\left|{a}_k\right|=1,k=\overline{1,n} \) are different points of the circle, r(B, a) is the inner radius of the domain B ⊂ \( \overline{\mathrm{C}} \) relative to the point a ∈ B. We consider a more general problem, in which the restriction \( \left|{a}_k\right|=1,k=\overline{1,n}, \) is replaced by a more general condition.

We obtain exact-order estimates of the approximation of the classes \( {B}_{p,\theta}^{\Omega} \) of periodic functions of several variables in the space L_∞, by using operators of orthogonal projection, as well as linear operators subjected to some conditions.

Solutions of the Itô stochastic differential equation in a random environment are considered. The random environment is formed by the generalized telegraph process. It is proved that the initial problem is equivalent to a system of two stochastic differential equations with nonrandom coefficients. The first equation is the Itô equation, and the initial process is its solution. The second equation is an equation with Poisson process, and its solution is a generalized telegraph process. The theorems of existence and uniqueness of strong and weak solutions are proved.

Let P, Q, and W be real functions of bounded variation on [0, l], and let W be nondecreasing. The integral system

\( J\overrightarrow{f}(x)-J\overrightarrow{a}=\underset{0}{\overset{x}{\int }}\left(\begin{array}{cc}\uplambda dW- dQ& 0\\ {}0& dP\end{array}\right)\overrightarrow{f}(t),\kern1em J=\left(\begin{array}{cc}0& -1\\ {}1& 0\end{array}\right) \)

on a finite compact interval [0, l] was considered in [6]. The maximal and minimal linear relations A_max and A_min associated with the integral system (0.1) are studied in the Hilbert space L²(W). It is shown that the linear relation A_min is symmetric with deficiency indices n_±(A_min) = 2 and A_max = \( {A}_{min}^{\ast }. \) Boundary triples for A_max are constructed, and the the corresponding Weyl functions are calculated.