Search

We obtain the exact order estimates for approximation of the functions of many variables from the generalized Nikol'skii–Besov classes B p , θ Ω R d by de la Vallée Poussin sums in the metrics of the spaces L ∞ R d and L 1 R d . These classes of functions for some given Ω coincide with the well-known classical isotropic Nikol'skii–Besov classes. [ABSTRACT FROM AUTHOR]

The aim of this work is to derive a symbol calculus on L 2 (R n) for multidimensional Hausdorff operators. Two aspects of this activity result in two almost independent parts. While throughout the perturbation matrices are supposed to be self-adjoint and form a commuting family, in the second part they are additionally assumed to be positive definite. What relates these two parts is the powerful method of diagonalization of a normal Hausdorff operator elaborated earlier by the second named author. [ABSTRACT FROM AUTHOR]

Given a dilation matrix M, a so-called space of M-positive vectors in the Euclidean space is introduced and studied. An algebraic structure of this space is similar to the positive half-line equipped with the termwise addition modulo 2, which is used in the Walsh analysis. The role of harmonics is played by some analogues of the classical Walsh functions. The concept of Fourier transform is introduced, and the Poisson summation formula, Plancherel theorem, Vilenkin-Chrestenson formulas and so on are proved. A kind of analogue of the Schwartz class is studied. This class consists of functions such that both the function itself and its Fourier transform have compact support. [ABSTRACT FROM AUTHOR]

The Dirichlet problem for a multidimensional elliptic system in the half-space is considered. With the help of the Fourier transform, the problem of the solvability of the problem is reduced to the study of a second-order partial differential equation. [ABSTRACT FROM AUTHOR]

In this paper, estimates are obtained for the Fourier transform of smooth charges (measures) concentrated on some nonconvex hypersurfaces. The summability of the maximal Randall function is proved for a wide class of nonconvex hypersurfaces. In addition, in the three-dimensional case, estimates are obtained depending on the Varchenko height. The accuracy of the obtained estimates is proved. The proof of the estimate for oscillatory integrals is based on the Weierstrass preparatory theorem. [ABSTRACT FROM AUTHOR]

The work is devoted to integro-differential equations related to stochastic processes. We study the relationship between differential equations with random perturbations — stochastic differential equations (SDEs) — and deterministic equations for the probability characteristics of processes determined by random perturbations. The resulting deterministic pseudodifferential equations are investigated by semigroup methods and Fourier transform methods. [ABSTRACT FROM AUTHOR]