Table of contents

The Pontryagin-van Kampen duality for locally compact Abelian groups can be generalized in two ways to wider classes of topological Abelian groups: in the first approach the dual group is endowed with the topology of uniform convergence on compact subsets of and in the second, with the topology of uniform convergence on totally bounded subsets of . The corresponding two classes of groups "reflexive in the sense of Pontryagin-van Kampen" are very wide and are so close to each other that it was unclear until recently whether they coincide or not. A series of counterexamples constructed in this paper shows that these classes do not coincide and also answer several other questions arising in this theory. The results of the paper can be interpreted as evidence that the second approach to the generalization of the Pontryagin duality is more natural.

For the description of extremal polynomials (that is, the typical solutions of least deviation problems) one uses real hyperelliptic curves. A partitioning of the moduli space of such curves into cells enumerated by trees is considered. As an application of these techniques the range of the period map of the universal cover of the moduli space is explicitly calculated. In addition, extremal polynomials are enumerated by weighted graphs.

The problem of the stabilization of a semilinear equation in the exterior of a bounded domain is considered. In view of the impossibility of an exponential stabilization of the form of the solution of a parabolic equation in an unbounded domain no matter what the boundary control is, one poses the problem of power-like stabilization by means of a boundary control. For a fixed initial condition and parameter of the rate of stabilization the existence of a boundary control such that the solution approaches zero at the rate is demonstrated.

Let be a system of bounded functions complete and orthonormal in and assume that , , for some . Then the elements of the system can be rearranged so that the resulting system has the -strong property: for each there exists a (measurable) subset of measure and a measurable function , , on such that for all and one can find a function coinciding with on such that its Fourier series in the system converges to in the -norm and the sequence of Fourier coefficients of this function belongs to all spaces , .

A certain new symmetric representation of Riemann's xi function is considered. A theorem on the zeros of trigonometric integrals analogous to Kakeya's theorem on the zeros of polynomials with monotonically non-decreasing coefficients is used. A modification of Polya's method is suggested, which allows one to obtain new assertions on the disposition of the zeros of the zeta function.

A system of ordinary differential equations with impulse action at fixed moments of time is considered. The system is assumed to have the zero solution. It is shown that the existence of a corresponding Lyapunov function is a necessary and sufficient condition for the uniform asymptotic stability of the zero solution. Restrictions on perturbations of the right-hand sides of differential equations and impulse actions are obtained under which the uniform asymptotic stability of the zero solution of the "unperturbed" system implies the uniform asymptotic stability of the zero solution of the "perturbed" system.

For Sobolev and Korobov spaces of functions of several variables a quadrature formula with explicitly defined coefficients and nodes is constructed. This formula is precise for trigonometric polynomials with harmonics from the corresponding step hyperbolic cross. The error of the quadrature formula in the classes , is , where is the number of nodes and is a parameter depending on the class. The problem of the approximate calculation of multiple integrals for functions in is considered in the case when this class does not lie in the space of continuous functions, that is, for .