Table of contents

The dynamics of the behaviour of the absolute values of the Dirichlet kernels is described: the discrete dynamics for the Dirichlet kernel of the Walsh system and the continuous dynamics for the generalized Walsh-Dirichlet kernel. Estimates of the -norms of the Dirichlet kernels are obtained. The concept of generalized Lebesgue constant is introduced and the corresponding formulae are found, which generalize Fine's formulae for the Lebesgue constants. These results hold not only for the Walsh system in the Paley enumeration, but also for rearrangements of the Walsh systems, including linear and piecewise linear ones.

An effective method for finding the polynomial approximating the exponential function with order 3 at the origin and deviating from 0 by at most 1 on the longest interval of the real axis is put forward. This problem is reduced to the solution of four equations on a 4-dimensional moduli space of algebraic curves. A numerical realization of this method using summation of linear Poincaré series is described.

An explicit construction of simultaneous Padé approximations for generalized hypergeometric series and formulae for the quantities , , in terms of these series are used for estimates of irrationality measures of these multiples of . Other possible applications are also discussed.

In a cylindrical domain , where is an unbounded subdomain of , one considers the first mixed problem for a higher order equation

with homogeneous boundary conditions and compactly supported initial function. A new method of obtaining an upper estimate of the -norm  of the solution of this problem is put forward, which works in a broad class of domains and equations. In particular, in domains , , for the operator  with symbol satisfying a certain condition this estimate takes the following form:

The estimate is shown to be sharp in a broad class of unbounded domains for , that is, for second-order parabolic equations.

Elasticity problems on a plane plate reinforced with a thin periodic network or in a 3-dimensional body reinforced with a thin periodic box skeleton are considered. The composite medium depends on two parameters approaching zero and responsible for the periodicity cell and the thickness of the reinforcing structure. The parameters can be dependent or independent. For these problems Zhikov's method of 'two-scale convergence with variable measure' is used to derive the homogenization principle: the solution of the original problem reduces in a certain sense to the solution of the homogenized (or limiting) problem. The latter has a classical form. From the operator form of the homogenization principle, on the basis of the compactness principle in the L²-space, which is also established, one obtains for the composite structure the Hausdorff convergence of the spectrum of the original problem to the spectrum of the limiting problem.

Suppose that is a 3-dimensional terminal singularity of type or  defined in  by an equation that is non-degenerate with respect to its Newton diagram. We show that there exists at most one non-rational divisor  over with discrepancy . We also describe all the blow-ups of the singularity with non-rational exceptional divisors of discrepancy .