Volume 7, Issue 1

For the Fourier coefficients of the generalized Wiener class BV( p ( n ) ↑ ∞), the precise order , where k ( n ) is an integer for which 1 + log 2 n < k ( n ) ≤ 2 + log 2 n , is established. A problem concerning the number of Fourier coefficients having exactly the order is studied.

The basic boundary-contact problems of oscillation are considered for a two-dimensional piecewise-homogeneous isotropic elastic medium bounded by several closed curves. Asymptotic formulas for the distribution of eigenfunctions and eigenvalues of the considered problems are derived using the correlation method.

We give a new and simpler proof of Sawyer's theorem characterizing the weights governing the two-weight, strong-type norm inequality for the Hardy-Littlewood maximal operator and the fractional maximal operator. As a further application of our techniques, we give new proofs of two sufficient conditions for such weights due to Wheeden and Sawyer.

A sufficient condition for the existence of an almost everywhere convergent rearrangement of series in Orlicz spaces of vector-valued functions is obtained. The results of this paper cover most of the related statements known previously for the series of scalar functions.

The convergence and summability of N -dimensional Walsh–Fourier series in the metrics L P ([0, 1] N ), p ∈ [1, + ∞], are studied by the negative order Césaro method. Getsadze's theorem and a generalization of Moricz' theorem are obtained as special cases.

The problem of a Stokes flow in the presence of a solid particle, a rigid wall and a viscous cell is formulated as a system of Fredholm integral equations of the second kind, with the surface force on the boundary of the solid particle and the velocity on the interface as unknowns. The particularity of the problem consists in the fact that the total force and the total torque of the flow on the solid particle are zero. The existence and the uniqueness result of solution is obtained when the boundaries are curves of the class C 2 .

The a.s. unconditionally convergent random series are investigated. The connection of the a.s. unconditionally convergence with the geometry of spaces is established.

We consider a heat equation with the non-linear right-hand side which depends on certain Volterra-type functionals acting on the unknown function and on its gradient. We give some natural sufficient conditions for the existence and uniqueness of solutions to this equation. The solution is obtained as a limit of a fast convergent sequence of successive approximations obtained by the quasi-linearisation method.

A state space representation of a (singular) AR-model is defined. It is shown, in particular, that this representation determines the model uniquely up to an isomorphism and respects its behavior. An explicit construction is presented for a ”standard” AR-model.

Criteria for the existence and uniqueness of a solution of the boundary value problem are established, where ƒ :] a, b [× R 2 → R satisfies the local Carathéodory conditions, and μ : [ a, b ] → R is the function of bounded variation. These criteria apply to the case where the function ƒ has nonintegrable singularities in the first argument at the points a and b .

If ( W t ) t ∈[ 0, 1] is a Wiener process in an arbitrary separable Banach space X, ψ : [0, 1] × X → Y is a continuous function with values in another separable Banach space, and ψ has continuous Frechet derivatives , and , then the Ito formula is obtained for ψ ( t , W t ). The method is based on the concept of covariance operator and a special construction of the Ito stochastic integral.

A condition is established for the sequence { N k } when there exists a rearrangement of functions of the block-orthonormal system { φ n } such that the sequence {log 2 n } ({(log log n ) 2 }) is a Weyl multiplier for the convergence (for the summability by the methods ( C, α ) ( α > 0)) almost everywhere of series with respect to the rearranged system { φ ν n }.

For certain polynomial functions from to ℂ the behavior at infinity is studied. Compactified fibers of the function, which have, in general, singularities at infinity are considered. A classification of polynomials of degree 4 is given with respect to the behavior at infinity.

The piecewise linear canonical Chua's system of ordinary differential equations is investigated. We consider the case when this problem can be reduced to a two-dimensional system with three parameters. For the reduced system, the qualitative behavior of solutions is analyzed for all parameter values. In particular, we indicate the parameters for which at least one, two or three nontrivial cycles exist.