卷 300, 编号 Suppl 1 (2018)

We consider the anisotropic Lorentz space of periodic functions. Sufficient conditions are proved for a function to belong to the anisotropic Lorentz space. Estimates for the order of approximation by trigonometric polynomials of the Nikol’skii–Besov class in the anisotropic Lorentz space are established.

We consider the generalized Poisson kernel Π_q,α = cos(απ/2)P + sin(απ/2)Q with q ∈ (−1, 1) and α ∈ ℝ, which is a linear combination of the Poisson kernel \(P(t) = 1/2 + \sum\nolimits_{k = 1}^\infty {{q^k}} \cos kt\) and the conjugate Poisson kernel \(Q(t) = \sum\nolimits_{k = 1}^\infty {{q^k}} \sin kt\) . The values of the best integral approximation to the kernel Π_q,α from below and from above by trigonometric polynomials of degree not exceeding a given number are found. The corresponding polynomials of the best one-sided approximation are obtained.

We consider the problem of visiting closed sets in a compact metric space complicated by constraints in the form of precedence constraints and a possible dependence of the cost function on a set of tasks. We study a variant of the approximate realization of the extremum by applying models that involve problems of sequential visits to megalopolises (nonempty finite sets). This variant is naturally embedded into a more general construction that implements sequential visits to nonempty closed sets (NCSs) from a finite system in a metrizable compact space. The space of NCSs is equipped with the Hausdorff metric, which is used to estimate (under the corresponding condition that the sections of the cost functions are continuous) the proximity of the extrema in the problem of sequential visits for any two systems of NCSs (it is assumed that the numbers or NCSs in the systems are the same). The constraints in the form of precedence constraints are preserved in this variant.

We give a solution to the Bohman extremal problem for nonnegative even entire functions of exponential type that are Jacobi transforms of compactly supported functions. We prove that the extremal function is unique. The Gauss quadrature formula on the half-line over zeros of the Jacobi function is used.

It is well known that every control that steers the trajectory of a control system to the boundary of the reachable set satisfies the Pontryagin maximum principle. This fact is valid for systems with pointwise constraints on the control. We consider a system with quadratic integral constraints on the control. The system is nonlinear in the state variables and linear in the control. It is shown that any admissible control that steers the system to the boundary of its reachable set is a local solution of some optimal control problem with quadratic integral functional if the corresponding linearized system is completely controllable. The proof of this fact is based on the Graves theorem on covering mappings. This implies the maximum principle for the controls that steer the trajectories to the boundary of the reachable set. We also discuss an algorithm for constructing the reachable set based on the maximum principle.

We find exact values for the uniform Lebesgue constants of interpolating L-splines that are bounded on the real axis, have equidistant knots, and correspond to the linear thirdorder differential operator L₃(D) = D(D₂ + α₂) with constant real coefficients, where α > 0. We compare the obtained result with the Lebesgue constants of other L-splines.

We consider the problem of retaining the motions of an abstract dynamic system in a given constraint set. Constructions from the programmed iteration method are extended to problems whose dynamics, in general, does not possess any topological properties. The weaker requirements are compensated by introducing transfinite iterations of the programmed absorption operator. The technique of fixed points of mappings in chain-complete partially ordered sets is used in the proofs. The proposed procedure produces a set where the retention problem is solved in the class of quasistrategies. The control interval is not assumed to be bounded.

We propose and validate a simple numerical method that finds an approximate solution with any given accuracy to the Dirichlet boundary value problem in a disk for a homogeneous equation with the Laplace operator. There are many known numerical methods that solve this problem, starting with the approximate calculation of the Poisson integral, which gives an exact representation of the solution inside the disk in terms of the given boundary values of the required functions. We employ the idea of approximating a given 2π-periodic boundary function by trigonometric polynomials, since it is easy to extend them to harmonic polynomials inside the disk so that the deviation from the required harmonic function does not exceed the error of approximation of the boundary function. The approximating trigonometric polynomials are constructed by means of an interpolation projection to subspaces of a multiresolution analysis (approximation) with basis 2π-periodic scaling functions (more exactly, their binary rational compressions and shifts). Such functions were constructed by the authors earlier on the basis of Meyer-type wavelets; they are either orthogonal and at the same time interpolating on uniform grids of the corresponding scale or only interpolating. The bounds on the rate of approximation of the solution to the boundary value problem are based on the property ofMeyer wavelets to preserve trigonometric polynomials of certain (large) orders; this property was used for other purposes in the first two papers listed in the references. Since a numerical bound of the approximation error is very important for the practical application of the method, a considerable portion of the paper is devoted to this issue, more exactly, to the explicit calculation of the constants in the order bounds of the error known earlier.

We study the general problem of interpolation by polynomial splines and consider the construction of such splines using the coefficients of expansion of a certain derivative in B-splines. We analyze the properties of the obtained systems of equations and estimate the interpolation error.