On embedding theorems of spaces of functions with mixed logarithmic smoothness

The article considers the Lorentz space $L_{p,\tau}(\mathbb{T}^{m})$, $2\pi$ of periodic functions of many variables and spaces with mixed logarithmic smoothness. Equivalent norms of a space with mixed logarithmic smoothness are found and embedding theorems are proved

On the exactness of the conditions of embedding theorems for spaces of functions with mixed logarithmic smoothness

The article considers the Lorentz space $L_{p,\tau}(\mathbb{T}^{m})$, $2\pi$ of periodic functions of many variables and $S_{p,\tau,\theta}^{0, \overline{b}}\mathbf{B}$, $S_{p, \tau, \theta}^{0, \overline{b}}B$ -- spaces of functions with mixed logarithmic smoothness. The article establishes necessary and sufficient conditions for embedding the spaces $S_{p, \tau, \theta}^{0, \overline{b}}\mathbf{B}$ and $S_{p, \tau, \theta}^{ 0, \overline{b}}B$ into each other.Comment: arXiv admin note: text overlap with arXiv:2308.0673

On the exactness of the inequality of different metrics for trigonometric polynomials in the generalized Lorentz space

We consider the generalized Lorentz space Lψ,τ(Tm) defined by some continuous concave function ψ such that ψ(0) = 0. For two spaces Lψ1,τ1(Tm) and Lψ2,τ2(Tm) such that αψ1 = limt→0ψ1(2t)/ψ1(t) = βψ2 = limt→0ψ2(2t)/ψ2(t), we prove an order-exact inequality of different metrics for multiple trigonometric polynomials. We also prove an auxiliary statement for functions of one variable with monotonically decreasing Fourier coefficients in a trigonometric system. In this statement we establish a two-sided estimate for the norm of the function f ∈ Lψ,τ(T) in terms of the series composed of the Fourier coefficients of this function. © 2019 Mofid University - Center for Human Rights Studies. All rights reserved.Ministry of Education and Science of the Russian Federation, MinobrnaukaUral Federal University, UrFUThis work was supported by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University)

Оценки билинейных приближений функций в пространстве Лоренца

В статье рассмотрены пространство Лоренца периодических функций многих переменных и класс Никольского-Бесова. В пространстве Лоренца дано определение билинейного приближения функции и приведена теорема Марцинкевича-Зигмунда для тригонометрического полинома. Установлены оценки наилучших билинейных приближений класса Никольского-Бесова в пространстве Лоренца

Estimations of the best M-term approximations of functions in the Lorentz space with constructive methods

This paper considers the Lorentz space of periodic functions of many variables with the anisotropic norm, of functional Nikol’skii-Besov’s class and of the best M -term approximation of function. We have established sufficient conditions for the function to belong to one of the Lorentz spaces in another. We obtain upper and lower bounds for the best M -member approximations of functions from the Nikol’skii-Besov class in the anisotropic Lorentz space To prove the upper bound, we used a new constructive method developed by V.N. Temlyakov

Some Fourier inequalities for orthogonal systems in Lorentz–Zygmund spaces

A number of classical inequalities and convergence results related to Fourier coefficients with respect to unbounded orthogonal systems are generalized and complemented. All results are given in the case of Lorentz–Zygmund spaces. © 2019, The Author(s).We thank the referees and Professors Dag Lukkasson and Annette Meidell for some good advice which improved the final version of the paper. Moreover, the first author is grateful for the support of this work given by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University)

Monadic Bounded Algebras

The object of study of the thesis is the notion of monadic bounded algebras (shortly, MBA's). These algebras are motivated by certain natural constructions in free (first-order) monadic logic and are related to free monadic logic in the same way as monadic algebras of P. Halmos to monadic logic (Chapter 1). Although MBA's come from logic, the present work is in algebra. Another important way of approaching MBA's is via bounded graphs, namely, the complex algebra of a bounded graph is an MBA and vice versa. The main results of Chapter 2 are two representation theorems: 1) every model is a basic MBA and every basic MBA is isomorphic to a model; 2) every MBA is isomorphic to a subdirect product of basic MBA's. As a consequence, every MBA is isomorphic to a subdirect product of models. This result is thought of as an algebraic version of semantical completeness theorem for free monadic logic. Chapter 3 entirely deals with MBA-varieties. It is proved by the method of filtration that every MBA-variety is generated by its finite special members. Using connections in terms of bounded morphisms among certain bounded graphs, it is shown that every MBA-variety is generated by at most three special (not necessarily finite) MBA's. After that each MBA-variety is equationally characterized. Chapter 4 considers finitely generated MBA's. We prove that every finitely generated MBA is finite (an upper bound on the number of elements is provided) and that the number of elements of a free MBA on a finite set achieves its upper bound. Lastly, a procedure for constructing a free MBA on any finite set is given

Global solvability of a nonlinear Boltzmann equation

In this paper, based on the splitting method scheme, the existence and uniqueness theorem on the whole time interval t ∈ [0 ,T ) ,T ≤∞ for the full nonlinear Boltzmann equation in the nonequilibrium case is proved where the intermolecular interactions are hard-sphere molecule and central forces. Considering the existence of a bounded solution in the space C, the strict positivity of the solution to the full nonlinear Boltzmann equation is proved when the initial function is positive. On the basis of this some mathematical justification of the H -theorem of Boltzmann is shown