Vol 296, No Suppl 1 (2017)

We study the problem of optimal recovery of a function analytic in a doubly connected domain from its approximately given values on one of the two components of the boundary. An optimal recovery method is obtained in the case when the error is an integer power of the modulus of the doubly connected domain.

An estimation problem for a random set that is a reachable domain of the Ito differential equation with respect to its initial data is considered. The Markov property of the reachable set in the space of closed sets is proved. For the purposes of numerical solution, a random initial set of the differential equation is approximated by a finite set on an integer multidimensional grid, and the differential equation is replaced by a multistep Markov chain. Examples are considered.

We consider the properties of functions f from the space L²(T) on the period T = [−π, π) with lacunary Fourier series such that the size of each gap is not less than a given positive integer q − 1. We find two-sided estimates of the L² norms of such functions on T in terms of similar norms (more exactly, seminorms) on intervals I of length |I| = 2h < 2π. The estimates are obtained in terms of best one-sided integral approximations to the characteristic function of the interval (−h, h) by trigonometric polynomials of degree at most q−1. The issue considered in this paper appeared first in N. Wiener’s studies (1934). Important results in this area were obtained by A.E. Ingham (1936) and by A. Selberg in the 1970s.

Let Γ be an antipodal graph with intersection array {2r+1, 2r−2, 1; 1, 2, 2r+1}, where 2r(r + 1) ≤ 4096. If 2r + 1 is a prime power, then Mathon’s scheme provides the existence of an arc-transitive graph with this intersection array. Note that 2r + 1 is not a prime power only for r ∈ {7, 17, 19, 22, 25, 27, 31, 32, 37, 38, 42, 43}. We study automorphisms of hypothetical distance-regular graphs with the specified values of r. The cases r ∈ {7, 17, 19} were considered earlier. We prove that, if Γ is a vertex-symmetric graph with intersection array {2r + 1, 2r − 2, 1; 1, 2, 2r +1}, 2r + 1 is not a prime power, and r ≤ 43, then r = 25, 27, or 31.

A reachability problem with constraints of asymptotic nature is considered in a topological space. The properties of a rather general procedure that defines an extension of the problem are studied. In particular, we specify a rule that transforms an arbitrary extension scheme (a compactifier) into a similar scheme with the property that the continuous extension of the objective operator of the reachability problem is homeomorphic. We show how to use this rule in the case when the extension is realized in the ultrafilter space of a broadly understood measurable space. This version is then made more specific for the case of an objective operator defined on a nondegenerate interval of the real line.

We construct harmonic wavelets and give a bound for the convergence rate of their partial sums in the spaces of harmonic functions introduced in the paper. These wavelets can be used for the solution of the Neumann problem.

We establish sufficient conditions for the convergence of minimizers and minimum values of integral and more general functionals on sets of functions defined by bilateral obstacles in variable domains. The given obstacles are elements of the corresponding Sobolev space, and the degeneracy on a set of measure zero is admitted for the difference between the upper and lower obstacles. We show that a weakening of the condition of positivity of this difference on a set of full measure may lead to a certain violation of the established convergence result.

A method for the construction of biorthogonal bases of multiwavelets from known bases of multiscaling functions is given. It is similar to the method presented in the author’s 2014 paper joint with N.I. Chernykh and is based on the same principle: the construction of multiwavelets based on k multiscaling functions employs an analog of the vector product of vectors in a 2k-dimensional space.

The problem of reconstructing unknown external actions in a linear stochastic differential equation is investigated on the basis of the approach of the theory of dynamic inversion. We consider the statement when the simultaneous reconstruction of disturbances in the deterministic and stochastic terms of the equation is performed with the use of discrete information on a number of realizations of a part of coordinates of the stochastic process. The problem is reduced to an inverse problem for systems of ordinary differential equations describing the mathematical expectation and covariance matrix of the original process. A finite-step software-oriented solution algorithm based on the method of auxiliary controlled models is proposed. We derive an estimate for its convergence rate with respect to the number of measured realizations.

The problem of guaranteed closed-loop guidance by a given time under incomplete information on the initial state is studied for a dynamical control system with delay by means of the method of open-loop control packages. A solvability criterion is proved for this problem in the case of a finite set of admissible initial states. The proposed technique is illustrated by a specific linear control system of differential equations with delay.

We study a problem on solutions (V, p) of the Euler equation with solenoidal velocity field V in a torus D, which is similar to the problem considered in the authors’ previous paper in 2014. Now, the problem is considered in the class of vector fields V whose lines coincide with lines of latitude of tori embedded in D with the same circular axis. Conditions are obtained under which this problem is solvable, and solutions are found.