Table of contents

The behaviour of rectangular partial sums of the Fourier series of functions of several variables having bounded -variation is considered. It is proved that if a continuous function is also continuous in harmonic variation, then its Fourier series uniformly converges in the sense of Pringsheim. On the other hand, it is demonstrated that in dimensions greater than 2 there always exists a continuous function of bounded harmonic variation with Fourier series divergent over cubes at the origin.

A hierarchy of extremal polynomials described in terms of real hyperelliptic curves of genus is constructed. These polynomials depend on integer-valued and continuous parameters. The classical Chebyshëv polynomials are obtained for and the Zolotarëv polynomials for .

Even positive-definite splines with support in [–1,1] that are equal to real algebraic polynomials on [0,1] are investigated. Examples of such splines are presented. Under consideration are the -splines, which have several extremal properties, and the positive-definite -splines, which have the maximum possible smoothness on . An estimate of the approximation by a linear combination of shifts of an -spline is indicated. New relations for the hypergeometric function are found.

The stationary Navier-Stokes system of equations is considered in a domain coinciding for large  with the layer . A theorem is proved about the asymptotic behaviour of the solutions as . In particular, it is proved that for arbitrary data of the problem the solutions having non-zero flux through a cylindrical cross-section of the layer behave at infinity like the solutions of the linear Stokes system.

A non-linear system of differential equations ("generalized Dubrovin system") is obtained to describe the behaviour of the zeros of polynomials orthogonal on several intervals that lie in lacunae between the intervals. The same system is shown to describe the dynamical behaviour of zeros of this kind for more general orthogonal polynomials: the denominators of the diagonal Padé approximants of meromorphic functions on a real hyperelliptic Riemann surface. On the basis of this approach several refinements of Rakhmanov's results on the convergence of diagonal Padé approximants for rational perturbations of Markov functions are obtained.

A Coxeter decomposition of a polyhedron in a hyperbolic space is a decomposition of it into finitely many Coxeter polyhedra such that any two tiles having a common facet are symmetric with respect to it. The classification of Coxeter decompositions is closely related to the problem of the classification of finite-index subgroups generated by reflections in discrete hyperbolic groups generated by reflections. All Coxeter decompositions of simplexes in the hyperbolic spaces  with are described in this paper.