Том 222, № 5 (2017)

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.

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 r_X (i, x₀) be the distribution function of points of an S-orbit x₀, x₁ = S(x₀), . . . , x_i−1 = Sⁱ⁻¹(x₀) getting into X, and let δ_X (i, x₀) be the deviation defined by the equation r_X (i, x₀) = aX i + δ_X (i, x₀), where aX ⁱ is the average value. If δ_X (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.

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.