Том 100, № 3-4 (2016)

Two sharp results for best approximations of periodic functions are established in this paper. We prove the sharpness of the step of the modulus of continuity in Jackson’s inequality with least possible constant for approximations by trigonometric polynomials. We also prove the sharpness of the constants in a Jackson-type inequality for approximations by Haar polynomials in several variables.

The notion of metric regularity can be extended to multivalued mappings acting in the products of metric spaces. A vector analog of Arutyunov’s coincidence-point theorem for two multivalued mappings is proved. Statements on the existence and estimates of solutions of systems of inclusions of special form occurring in the multiple fixed-point problem are obtained. In particular, these results imply some well-known double-point theorems.

The paper deals with conditions under which the Green function of a multipoint boundary-value problem for fourth-order equations describing small strains of a rod fastened to a solid elastic basement and additionally fixed by “concentrated” elastic supports at separate points has the oscillation property. It is shown that the condition that the Green function is positive is necessary and sufficient for the Green function to have the oscillation property.

A parallel between physical derivation and mathematical proof in classical thermodynamics is drawn. A relationship between thermodynamics and analytic number theory is demonstrated.

In the paper, the invariance property of characteristics (the order and type) of an operator and of a sequence of operators with respect to a topological isomorphism is proved. These characteristics give precise upper and lower bounds for the expressions ‖A_n(x)‖_p and enable one to state and solve problems of operator theory in locally convex spaces in a general setting. Examples of such problems are given by the completeness problem for the set of values of a vector function in a locally convex space, the structure problem for a subspace invariant with respect to an operator A, the problem of applicability of an operator series to a locally convex space, the theory of holomorphic operator-valued functions, the theory of operator and differential-operator equations in nonnormed spaces, and so on. However, the immediate evaluation of characteristics of operators (and of sequences of operators) directly by definition is practically unrealizable in spaces with more complicated structure than that of countably normed spaces, due to the absence of an explicit form of seminorms or to their complicated structure. The approach that we use enables us to find characteristics of operators and sequences of operators using the passage to the dual space, by-passing the definition, and makes it possible to obtain bounds for the expressions ‖A_n(x)‖_p even if an explicit form of seminorms is unknown.

The notion of general mixed modulus of smoothness of periodic functions of several variables in the spaces L_p is introduced. The proposed construction is, on the one hand, a natural generalization of the general modulus of smoothness in the one-dimensional case, which was introduced in a paper of the first author and in which the coefficients of the values of a given function at the nodes of a uniform lattice are the Fourier coefficients of a 2π-periodic function called the generator of the modulus; while, on the other hand, this construction is a generalization of classical mixed moduli of smoothness and of mixed moduli of arbitrary positive order. For the modulus introduced in the paper, in the case 1 ≤ p ≤ +∞, the direct and inverse theorems on the approximation by the “angle” of trigonometric polynomials are proved. The previous estimates of such type are obtained as direct consequences of general results, new mixed moduli are constructed, and a universal structural description of classes of functions whose best approximation by “angle” have a certain order of convergence to zero is given.

Vanishing theorems for harmonic and infinitesimal harmonic transformations of complete Riemannian manifolds are proved. The proof uses well-known Liouville theorems on subharmonic functions on noncompact complete Riemannian manifolds.

We consider the quantile function of a fuzzy random variable and obtain expressions for some expectations related to fuzzy random variables via integrals of quantile functions.

Let τ be a faithful normal semifinite trace on a von Neumann algebra M, let p, 0 < p < ∞, be a number, and let L_p(M, τ) be the space of operators whose pth power is integrable (with respect to τ). Let P and Q be τ-measurable idempotents, and let A ≡ P − Q. In this case, 1) if A ≥ 0, then A is a projection and QA = AQ = 0; 2) if P is quasinormal, then P is a projection; 3) if Q ∈ M and A ∈ L_p(M, τ), then A² ∈ L_p(M, τ). Let n be a positive integer, n > 2, and A = Aⁿ ∈ M. In this case, 1) if A ≠ 0, then the values of the nonincreasing rearrangement μ_t(A) belong to the set {0} ∪ [‖Aⁿ⁻²‖⁻¹, ‖A‖] for all t > 0; 2) either μ_t(A) ≥ 1 for all t > 0 or there is a t₀ > 0 such that μ_t(A) = 0 for all t > t₀. For every τ-measurable idempotent Q, there is aunique rank projection P ∈ M with QP = P, PQ = Q, and PM = QM. There is a unique decomposition Q = P + Z, where Z² = 0, ZP = 0, and PZ = Z. Here, if Q ∈ L_p(M, τ), then P is integrable, and τ(Q) = τ(P) for p = 1. If A ∈ L₁(M, τ) and if A = A³ and A − A² ∈ M, then τ(A) ∈ R.

The theory of Riordan arrays studies the properties of formal power series and their sequences. The notion of generalized Lagrange series proposed in the present paper is intended to fill the gap in the methodology of this theory. Generalized Lagrange series appear in it implicitly, as various equalities. No special notation is provided for these series, although particular cases of these series are generalized binomial and generalized exponential series. We give the definition of generalized Lagrange series and study their relationship with ordinary Riordan arrays and, separately, with Riordan exponential arrays.

The stability of closed invariant sets of semidynamical systems defined on an arbitrary metric space is analyzed. The main theorems of Lyapunov’s second method for the uniform stability and uniform asymptotic stability (local and global) are stated. Illustrative examples are given.

We introduce the notion of disinformation as an object annihilated by the corresponding information and analyze the relationships of this object with abstract analytic number theory and thermodynamics. For the entropy in this statistics, we present Nazaikinskii’s model, which contains only the statistics of Bose and Fermi gases (and excludes parastatistics).

An algebra obtained by the external adjoining a unit to a nilalgebra is said to be unitary. It is proved that every simple finite-dimensional right alternative superalgebra with unitary even part over a field of characteristic 0 is associative.

Various realizations of the Hilbert state space for quantum systems with linear constraints are considered. One of the realizations is the completion of a Schwartz space with inner product differing from L². Another realization is a special class of distributions with inner product. An isomorphism between these realizations is constructed, and the properties of the operators are studied.