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Том 222, № 5 (2017)
- Год: 2017
- Статей: 6
- URL: https://ogarev-online.ru/1072-3374/issue/view/14821
Article
Growth of Norms in L2 of Derivatives of the Steklov Functions and Properties Defined by the Best Approximations and Fourier Coefficients
Аннотация
In the paper, for periodic functions, a connection between integrals of norms in L2 of derivatives of the Steklov functions and series constructed from Fourier coefficients and the best approximations in L2 is established, and the question on their simultaneous convergence or divergence is examined. Similar investigations are carried out for even and odd periodic functions. Bibliography: 13 titles.
525-543
Differentiation of Induced Toric Tiling and Multidimensional Approximations of Algebraic Numbers
Аннотация
The paper considers the induced tilings \( \mathcal{T} \) = \( \mathcal{T} \) |Kr of the D-dimensional torus \( \mathbb{T} \)D generated by embedded karyons Kr. On \( \mathcal{T} \) , differentiation operations σ : \( \mathcal{T} \) −→\( \mathcal{T} \)σ are defined, which provide the induced tilings \( \mathcal{T} \)σ = \( \mathcal{T} \) |Krσ of the same torus \( \mathbb{T} \)D with the derivative karyon Krσ. They are used for approximation of 0 ∈ \( \mathbb{T} \)D by an infinite sequence of points xj ≡ jα mod ℤD, j = 0, 1, 2, . . . , where α = (α1, . . . , αD) is a vector whose coordinates α1, . . . , αD belong to an algebraic field ℚ(θ) of degree D+1 over the rational field ℚ. To this end, an infinite sequence of convex parallelohedra T (i) ⊂ \( \mathbb{T} \)D, i = 0, 1, 2, . . ., is constructed, and natural orders m(0) < m(1) < · · · < m(i) < · · · for T (i) are defined. Then the above parallelohedra contain a subsequence of points \( {\left\{{x}_{j^{\prime }}\right\}}_{j^{\prime }=1}^{\infty } \) that are the best approximations of 0 ∈ \( \mathbb{T} \)D. Bibliography: 25 titles.
544-584
Bounded Remainder Sets
Аннотация
The paper considers the category (\( \mathcal{T} \), S, X) consisting of mappings S :\( \mathcal{T} \) −→\( \mathcal{T} \) of spaces \( \mathcal{T} \) with distinguished subsets X ⊂ \( \mathcal{T} \). Let rX (i, x0) be the distribution function of points of an S-orbit x0, x1 = S(x0), . . . , xi−1 = Si−1(x0) getting into X, and let δX (i, x0) be the deviation defined by the equation rX (i, x0) = aX i + δX (i, x0), where aX i is the average value. If δX (i, x0) = O(1), then such sets X are called bounded remainder sets. In the paper, bounded remainder sets X are constructed in the following cases: (1) the space \( \mathcal{T} \) is the circle, torus, or the Klein bottle; (2) the map S is a rotation of the circle, a shift or an exchange mapping of the torus; (3) X is a fixed subset X ⊂ \( \mathcal{T} \) or a sequence of subsets depending on the iteration number i = 0, 1, 2, . . .. Bibliography: 27 titles.
585-640
Inner Radius, Polarization, and Circular Truncation of a Set
Аннотация
The paper considers the difference between the reduced module m(B, 0) of an open set B, 0 ∈ B, and the reduced module m(Br, 0) of its circular truncation Br, where Br = B ∩ {|z| < r}. It is proved that this difference does not decrease under polarization and circular symmetrization. Bibliography: 6 titles.
641-644
Geometric Function Theory. Jenkins’ Results. The method of Modules of Curve Families
Аннотация
Results and applications of the method of modules in geometric function theory are presented. The method was originally created by J. A. Jenkins, and further developed in works of the Leningrad–St. Petersburg mathematical school. A retrospective description of the origin of the method is given, and the determining role of Jenkins in the development of the method of the extremal metric is pointed out.
645-689
Extreme Values of the Epstein Zeta Functions
690-702