On 7 September 2007 the well-known mathematician Professor Alexandr L’vovich Garkavi, doctor of mathematics and physics, passed away in his 83rd year. He was born on 10 November 1924 in Rostov-on-Don. His father was an outstanding power engineering specialist and his mother an accountant. After finishing secondary school, Garkavi was mobilized into the Red Army in 1942. At the end of the war he remained in the ranks of the defenders of the homeland and was demobilized late in 1950, with many combat medals. The outstanding abilities of Garkavi enabled him, after a nine-year break from school, to enter the Faculty of Mathematics and Mechanics at L’vov University and to graduate with distinction in 1956. The same year he was accepted for graduate work at the Steklov Mathematical Institute of the USSR Academy of Sciences, where his research advisor was Professor S. B. Stechkin. In 1959 Garkavi defended his Ph.D. dissertation and in 1966 his D.Sc. dissertation. Most of Garkavi’s works are related to the theory of approximations and the geometry of Banach spaces. He obtained interesting results on simultaneous approximation of periodic functions and their derivatives by trigonometric polynomials. In 1961 Garkavi published duality relations in the problem of best approximation by elements of an arbitrary convex set in a Banach space. These results essentially generalized the Nikol’skii–Krein duality relations, well-known around that time, relating to approximation by linear subspaces. The duality relations provide a convenient and uniform approach to questions in the Chebyshev circle of ideas in approximation theory: criteria for an element of best approximation, the uniqueness or plurality of such elements, the calculation or estimation of the size of the best approximation, and algorithms for constructing a best approximation. Theorems on “refinement” play a significant role in approximation theory. In the case of approximation in the space of continuous functions on some set S, such theorems show that approximation on S can be replaced by approximation on a “refined” set, in most cases a finite set. The first such theorem was obtained by de la Vallée Poussin and related to polynomial approximation on an interval. In the general case of approximation in a Banach space, S is the unit ball of the

It is shown that a straightforward generalization of Cauchy's integral formula is possible only in domains with boundary of finite length (in some sense or other). An example of a simply connected domain with boundary of infinite length is constructed such that for fairly general functionals on no extremal function (including the Ahlfors function) can be represented as a Cauchy potential.

The problem of possible weakening of conditions imposed on the boundary functions in the existence problem for automorphic analytic functions with boundary values of given modulus is considered. An interpretation of earlier results in terms of the Wiener and Martin compactifications is presented.

The flow of the current along the magnetic field lines in the thin plasma directed opposite to the electric field is considered. The particles moving to the equatorial plane are supposed to have mirror points above the region of absorption (the ionosphere) and the particles moving to the ionosphere are supposed to have mirror points below the region of absorption. The current, therefore, flows. The functions of the distribution of the electrons and ions are considered to be mono-energetic. The energies of the electrons and the ions and their densities on the boundary of absorption are estimated for the potential difference and for the current density which are typical for the auroral field lines.

Using the well-known equation for the normal component of the current which exist near the tangential discontinuity in the plasma in the case of the frozen-in magnetic field, and supposing that the current closes in the ionosphere in the auroral oval in the region 1, one calculates and compares with the data of observations the dependence of the density of the field-aligned current at the level of the ionosphere on the local time.

Domains and Riemann surfaces of Parreau-Widom type are considered, along with the extension to these domains of classical results on the Hardy spaces H(p) and on representing measures and orthogonal measures (the theorems of the Riesz brothers), and so on.

For low velocities of convection, the normal component of the current near the magnetopause is calculated in a case when the magnetopause is a tangential discontinuity. It is shown that for the great pressure of the magnetospheric plasma this component of the current, closing through the ionosphere, create the global system of field-aligned currents which is consistent with the Triad data on the value, the direction and the distribution with the local time.

It is shown that in an ideal frozen-in plasma which can move in an arbitary direction across the magnetic field there is a definite current which does not depend on the flow velocity. The pattern of the distributed currents in the inner magnetosphere and its value, the global system of field-aligned currents and the influence of the components B(z) and B(y) of the interplanetary magnetic field on the field-aligned current and on the convection potential are explained with help of this current.

Riemann surfaces of Parreau-Widom type have recently been studied intensively in connection with the generalization to them of results of Bjorling on the description of subspaces invariant under a shift. The history of the problem and the main results are expounded in [1 ]. In the definition of a Parreau-Widom type surface an important role is played by the concept of a multiplicative function—a multivalued analytic function on a Riemann whose module single-valued function on R. Every function / defines, in a natural manner, a character ry on the fundamental group π^Λ) of Λ. The set of all multiplicative (bounded multiplicative) functions on R, inducing a given character r, will be denoted by H(R, Γ) (#""(/?, Γ)).

In order to understand the reason of the existence of the electric field in the magnetosphere, and for the theoretical evaluation of its value, it is necessary to find the solution of the problem of determination of the magnetosphere boundary form in the frameworks of the continuum medium model which takes into account part of the magnetospheric plasma movement in supporting the magnetospheric boundary equilibrium. A number of problems for finding the distribution of the pressure, the density, the magnetic field and the electric field on the particular tangential discontinuity is considered in the case when the form of discontinuity is set (the direct problem) and a number of problems for finding the form of the discontinuity and the distribution of the above-mentioned physical quantities on the discontinuity is considered when the law of the change of the external pressure along the boundary is set (for example, with the help of the approximate Newton equation). The problem which is considered here, which deals with the calculation of the boundary form and with the calculation of the distribution of the corresponding physical quantities on the discontinuity of the 1st kind for the compressible fluid with the magnetic field with field lines which are perpendicular to the plane of the flow in question, concerns the last sort of problems. The comparison of the results of the calculation with the data in the equatorial cross-section of the magnetosphere demonstrates that the calculated form of the boundary, the value of the velocity of the return flow and the value of the electric field on the magnetopause, agree satisfactorily with the observational data.

The plasma flow in the equatorial plane of the magnetosphere is examined within the framework of a one-dimensional model in which all quantities are supposed to depend only on the distance along the Sun-Earth axis. The following models are considered: (1) the gasdynamical model in which the Ampère force is ignored, (2) the magnetohydrodynamical model in which the normal component of the Ampère force on the magnetopause is taken into account. The flow regime is calculated in the region including two regions: (1) the layer of the return flow where flow velocity is directed from the Sun, (2) the region of convection where the velocity is directed toward the Sun - on the assumption that the form of the magnetopause and the distribution of the solar wind pressure on the magnetopause are known.