The main aim of this paper is to study the geometric and metric properties of $B$- and $C$-capacities related to problems of uniform approximation of functions by solutions of homogeneous second-order elliptic equations with constant complex coefficients on compact subsets of Euclidean spaces. In the harmonic case this problem is well known, and it was studied in detail in the framework of classical potential theory in the first half of the 20th century. For a wide class of equations mentioned above, we obtain two-sided estimates between the corresponding $B_+$- and $C_+$-capacities (defined in terms of potentials of positive measures) and the harmonic capacity in the same dimension. Our research method is based on new simple explicit formulae obtained for the fundamental solutions of the equations under consideration. Bibliography: 12 titles.

We obtain several new sharp C-m-continuity conditions, both necessary and sufficient, for operators of harmonic reflection of functions over boundaries of simple Carath ' eodory domains in R-N. These results are based on a new criterion (also obtained in this paper) for C-m-continuity of the Poisson operator in the aforesaid domains. As corollaries, we give new sufficient conditions for C-m-approximability of functions by harmonic polynomials on boundaries of simple Carath ' eodory domains in R-N.

Criteria for approximability of functions by solutions of homogeneous second order elliptic equations (with constant complex coefficients) in the norms of the Whitney C^1 -spaces on compact sets in ℝ^2 are obtained in terms of the respective C^1 -capacities. It is proved that the mentioned C^1 -capacities are comparable to the classic C -analytic capacity, and so have a proper geometric measure characterization.

We obtain several new sharp necessary and sufficient $${{\,\mathrm{\textit{Lip}}\,}}^m$$ -continuity conditions for operators of harmonic reflection of functions over boundaries of simple Carathéodory domains in $${\mathbb {R}}^N$$ . These results are based on our $${{\,\mathrm{\textit{Lip}}\,}}^m$$ -continuity criterion for the Poisson operator in the aforementioned domains.

In this review we present the main results jointly obtained by the authors and André Boivin (1955–2014) during the last 20 years. We also recall some important theorems obtained with colleagues and give new applications of the above mentioned results. Several open problems are also formulated.

If a closed subset of a Riemann surface is a set of uniform meromorphic approximation, then its boundary is shown to be a set of tangential meromorphic approximation.

Criteria for the individual approximability of functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients in the norms of Whitney-type C-m-spaces on compact subsets of R-N, N is an element of {2, 3,...}, are obtained for m. (0, 1) boolean OR (0, 2). These results, which are analogues of Vitushkin's celebrated criteria for uniform rational approximation, were previously established by Mazalov for harmonic approximations (for m is an element of (0, 1) and N >= 3) and by Mazalov and Paramonov for bi-analytic approximation.

New uniform approximability criteria formulated in terms of logarithmic capacity are obtained for approximations by harmonic functions on compact sets in ℝ 2 . A relationship between these approximations and analogous approximations on compact sets in ℝ 3 is established.

A proof of the classical theorem on a simple closed curve (Jordan's theorem) is discussed; this proof is given by a Norwegian mathematician H. Tverberg and is little known to specialists. The proof has a metric nature and makes it possible to obtain an important metric refinement of Jordan's theorem, which is interesting on its own.

In this paper we study several settings of the C-m-subharmonic extension problem on open Riemann surfaces. The problem is completely solved (for all m is an element of[ 0,+infinity)) for so-called Runge-type extensions. Several (in some sense sharp) sufficient conditions and counterexamples are found also for Walsh-type extensions. As applications, these results allow us to prove the existence of C-m-subharmonic extensions, automorphic with respect to some appropriate groups of automorphisms of an open Riemann surface.

The paper puts forward criteria for approximability by bianalytic functions in the norms of the Whitney-type spaces on planar compact sets with . These results, which are analogues of Vitushkin's well-known criteria for uniform rational approximation, together with results of O'Farrell and Verdera (the case ) and Mazalov (the case ), provide a complete set of criteria for approximability by bianalytic functions for all . These conditions for approximability are obtained for both individual functions and (as corollaries) for classes of functions, using the terminology of geometric measure theory. Bibliography: 21 titles.

Numerical modeling of optical wave propagation in atmospheric turbulence is traditionally performed with using the so-called split-operator method, when the influence of the propagation medium's refractive index inhomogeneities is accounted for only within a system of infinitely narrow layers (phase screens) where phase is distorted. Commonly, under certain assumptions, such phase screens are considered as mutually statistically uncorrelated. However, in several important applications including laser target tracking, remote sensing, and atmospheric imaging, accurate optical field propagation modeling assumes upper limitations on interscreen spacing. The latter situation can be observed, for instance, in the presence of large-scale turbulent inhomogeneities or in deep turbulence conditions, where interscreen distances become comparable with turbulence outer scale and, hence, corresponding phase screens cannot be statistically uncorrelated. In this paper, we discuss correlated phase screens. The statistical characteristics of screens are calculated based on a representation of turbulent fluctuations of three-dimensional (3D) refractive index random field as a set of sequentially correlated 3D layers displaced in the wave propagation direction. The statistical characteristics of refractive index fluctuations are described in terms of the von Karman power spectrum density. In the representation of these 3D layers by corresponding phase screens, the geometrical optics approximation is used.