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Vol 210, No 10 (2019)
- Year: 2019
- Articles: 7
- URL: https://journal-vniispk.ru/0368-8666/issue/view/7458
An analogue of the two-constants theorem and optimal recovery of analytic functions
Abstract
Several related extremal problems for analytic functions in a simply connected domain $G$ with rectifiable Jordan boundary $\Gamma$ are treated. The sharp inequality $$|f(z)|\le\mathscr C^{r,q}(z;\gamma_0,\varphi_0;\gamma_1,\varphi_1)\|f\|^\alpha_{L^q_{\varphi_1}(\gamma_1)}\|f\|^{1-\alpha}_{L^r_{\varphi_0}(\gamma_0)}$$is established between a value of an analytic function in the domain and the weighted integral norms of the restrictions of its boundary values to two measurable subsets $\gamma_1$ and $\gamma_0=\Gamma\setminus\gamma_1$ of the boundary of the domain. It is an analogue of the F. and R. Nevanlinna two-constants theorem. The corresponding problems of optimal recovery of a function from its approximate boundary values on $\gamma_1$ and of the best approximation to the functional of analytic extension of a function from the part of the boundary $\gamma_1$ into the domain are solved. Bibliography: 35 titles.
Matematicheskii Sbornik. 2019;210(10):3-16
3-16
Some properties of embeddings of rearrangement invariant spaces
Abstract
Let $E$ and $F$ be rearrangement invariant spaces on $[0,1]$, and let $E\subset F$. This embedding is said to be strict if the functions in the unit ball of the space $E$ have absolutely equicontinuous norms in $F$. For the main classes of rearrangement invariant spaces necessary and sufficient conditions are obtained for an embedding to be strict, and also the relationships this concept has with other properties of embeddings are studied, especially the property of disjoint strict singularity. In the final part of the paper, a characterization of the property of strict embedding in terms of interpolation spaces is obtained. Bibliography: 23 titles.
Matematicheskii Sbornik. 2019;210(10):17-36
17-36
Harmonic analysis of functions in homogeneous spaces and harmonic distributions that are periodic or almost periodic at infinity
Abstract
Vector-valued functions in homogeneous spaces and harmonic distributions that are periodic or almost periodic at infinity are investigated. The concept of the Fourier series of a function (distribution), periodic or almost periodic at infinity, with coefficients that are functions (distributions) slowly varying at infinity, is introduced. The properties of the Fourier series are investigated and an analogue of Wiener's theorem on absolutely convergent Fourier series is obtained for functions periodic at infinity. Special attention is given to criteria ensuring that solutions of differential or difference equations are periodic or almost periodic at infinity. The central results involve theorems on the asymptotic behaviour of a bounded operator semigroup whose generator has no limit points on the imaginary axis. In addition, the concept of an asymptotically finite-dimensional operator semigroup is introduced and a theorem on the structure of such a semigroup is proved. Bibliography: 39 titles.
Matematicheskii Sbornik. 2019;210(10):37-90
37-90
On equivariant fibrations of $G$-CW-complexes
Abstract
It is proved that if $G$ is a compact Lie group, then an equivariant Serre fibration of $G$-CW-complexes is an equivariant Hurewicz fibration in the class of compactly generated $G$-spaces. In the nonequivariant setting, this result is due to Steinberger, West and Cauty. The main theorem is proved using the following key result: a $G$-CW-complex can be embedded as an equivariant retract in a simplicial $G$-complex. It is also proved that an equivariant map $p\colon E\to B$ of $G$-CW-complexes is a Hurewicz $G$-fibration if and only if the $H$-fixed point map $p^H\colon E^H \to B^H$ is a Hurewicz fibration for any closed subgroup $H$ of $G$. This gives a solution to the problem of James and Segal in the case of $G$-CW-complexes. Bibliography: 9 titles.
Matematicheskii Sbornik. 2019;210(10):91-98
91-98
Lifting of parallelohedra
Abstract
A parallelohedron is a polyhedron that can tessellate the space via translations without gaps and overlaps. Voronoi conjectured that any parallelohedron is affinely equivalent to a Dirichlet-Voronoi cell of some lattice. Delaunay used the term displacement parallelohedron in his paper “Sur la tiling regulière de l'espace à 4 dimensions. Première partie”, where the four-dimensional parallelohedra are listed. In our work, such a parallelohedron is called a lifted parallelohedron, since it is obtained as an extension of a parallelohedron to a parallelohedron of dimension larger by one. It is shown that the operation of lifting yields precisely parallelohedra whose Minkowski sum with some nontrivial segment is again a parallelohedron. It is proved that Voronoi's conjecture holds for parallelohedra admitting lifts and lifted in general position. Bibliography: 20 titles.
Matematicheskii Sbornik. 2019;210(10):99-121
99-121
Free products of groups are strongly verbally closed
Abstract
In a number of recent papers it was established that many almost free groups, fundamental groups of almost all connected surfaces, and all groups that are nontrivial free products of groups with identities are algebraically closed in any group in which they are verbally closed. In the present paper we establish that any group that is a nontrivial free product of groups is algebraically closed in any group in which it is verbally closed. Bibliography: 13 titles.
Matematicheskii Sbornik. 2019;210(10):122-160
122-160
A sliceness criterion for odd free knots
Abstract
The problem of concordance and cobordism of knots is a well-known classical problem in low-dimensional topology. The purpose of this paper is to show that for odd free knots, that is, free knots with all intersections odd, the question of whether the knot is slice (concordant to a trivial knot) can be answered effectively by analysing pairing of the chords in a knot diagram. Bibliography: 8 titles.
Matematicheskii Sbornik. 2019;210(10):161-178
161-178