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Theorem 2. Let v(z) be a subharmonic function of finite V -type σv < ∞. The following two statements are equivalent:

https://doi.org/10.33184/dokbsu-2020.1.1

2. Rubel Lee A. Entire and Meromorphic Functions /Lee A. Rubel. — New York Berlin Heidelberg : Springer-Verlag, 1995. — 196 p.

https://doi.org/10.1007/978-1-4612-0735-1

ABOUT ONE EMBEDDING THEOREM V.R. Misiuk (Grodno, Yanka Kupala State University)

misiuk@grsu.by

Let D be the circle |z| < 1 in the complex plane. For 0 < p ∞ we denote by Lp(D) the Lebesgue space of complex functions on D with respect to the flat Lebesgue measure with the usual quasi-norm f Lp(D). Spaces Sobolev Wps(D) are well known and quite deeply studied. The following Sobolev embedding theorem is well-known [1]:

Wq1(D) Lp(D),

where 2 p < ∞ and

1q = p1 + 12.

It turns out that the following analogue of the inversion of this theorem holds for rational functions of a given degree.

Theorem 1. Let p > 2 and

1q = p1 + 12.

Then for any rational function r of degree at most n with poles outside the circle D √

r Wq1(D) c n r Lp(D) ,

© Misiuk V.R., 2024

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where c > 0 and depends only on p.

Note that this relation is exact in the sense of the parameters p and n included in it. Note that the accuracy with respect to the growth of the factor √n is easily confirmed by the example of the functions r(z) = zn, n = 1, 2, . . . . The quasi-norm r Wq1 cannot be replaced respectively by the quasi-norm r Wu1 and for no u > q. This can be verified by the

example of the simplest rational function r(z) = (z0 − z)−1, for |z0| > 1. It should be noted that various aspects of these relations and their

applications were previously studied by the author in [2], [3].

References

1.Stein E.M. Singular integrals and differentiability propeties of functions / E.M. Stein. — Princeton Univ. Press, 1970. — 371 p.

2.Misiuk V.R. Refinement of inequalities and theorems of Bernstein type theory of rational approximations with respect to the plane Lebesgue measure / V.R. Misiuk // Vesnik of Yanka Kupala State University of Grodno. — 2008. № 2 (68). — P. 22—31.

3.Misiuk V.R. On the inverse theorem of the theory of rational approximations for Bergman spaces / V.R. Misiuk // Problems of physics, mathematics and technics. — 2010. — №.1(2). — С. 34–37.

APPLICATION OF A MAP APPROXIMATING THE PHASE FLOW TO STUDY THE ATTITUDE MOTION OF A SATELLITE IN A GRAVITATIONAL

FIELD

V.V. Sidorenko (Moscow, KIAM) vvsidorenko@list.ru

The attitude motion of an axisymmetric satellite under the influence of a gravitational torque is studied. The satellite’s center of mass moves in a circular orbit in a central gravitational field. The symmetry of the satellite allows us to reduce the investigation of its dynamics to the consideration of a Hamiltonian system with two degrees of freedom. If the projection of the satellite’s angular momentum vector onto its axis of symmetry is zero, then so-called «planar» motions are possible. In planar motions the axis of symmetry moves in the orbital plane and, therefore, the angular velocity vector of the satellite is perpendicular to this plane. Planar motions are divided into rotations and oscillations relative to the local vertical. The limiting case of these motions are aperiodic motions,

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in which the axis of symmetry of the satellite asymptotically approaches an unstable position of relative equilibrium.

To analyze the properties of the motions of the satellite, which are close to planar asymptotic ones, perturbation theory is applied. A mapping is constructed that approximates the mapping generated by the phase flow of the system. The idea to construct such a mapping for studying Hamiltonian systems with a separatrix contour on an invariant manifold was proposed by L.M. Lerman and C. Grotto-Rogazzo. But the system under consideration is somewhat different from the systems discussed by these specialists. Therefore, a significant modification of their approach was required.

Using this mapping, we were able to establish some previously unknown properties of the satelliteпїЅs attitude motion in a gravitational field. In particular, conditions were found for the phase trajectories of the problem to remain for a long time in the vicinity of the separatrix contour.

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Ушхо Д.С., 256 В.А. Костин, 21 Васильев В.Б., 74 Ведюшкина В.В., 76 Вирченко Ю.П., 79 Вотякова М.М., 82 Яремко Н.Н., 145 Захарова Т.А., 265 Зарипова Е.Ф., 167 Завьялов В.Н., 99 Зизов В.С., 104

Золотухина А.А., 109 Зубова С.П., 110 Звягин А.В., 101, 102 Жалукевич Д.С., 95 Жуйков К.Н., 97 Журба А.В., 128

Нестеров А.В., 180

C.Е. Пастухова, 191

Kabanko M.V., 308

Kolokoltsov V.N., 305

Korzyuk V.I., 306

Malyutin K.G., 308

Misiuk V.R., 311

Rudzko J.V., 306

Sidorenko V.V., 312

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