(i)-Convergence and its application to a sequence of functions

Let , where A is a directed set containing cofinal chains — a generalized sequence in a complete chain. It is established that every such sequence contains a monotonic cofinal sub-sequence. For a monotonically increasing (decreasing) bounded sequence , by definition, we put . For an arbitrary sequence the (i)-limit is defined as the common (i)-limit of its monotonic cofinal sub-sequences. The properties of (i)-convergence and some of its applications to generalized sequences of mappings are discussed.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 809–820, December, 1973.     

Completion of an additive measure to a σ-additive measure by an extension of the space

For an algebra of subsets of a set X there is constructed a set and an algebra of its subsets so that the mapping is a one-to-one correspondence between and and for each additive measure on the measure on defined by the equation is countably additive.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 71–76, January, 1968.The author wishes to express his deep appreciation to S. V. Fomin, under whose guidance this paper was written.     

An asymptotic method for calculating limits of maximal means

For a continuous almost periodic function , we show that the function

Kernels of sequences of complex numbers and their regular transformations

It is proved that , where U(a, r) is the ball of radius r with center at the pointa, is the smallest closed convex set containing the kernel of any sequence {y_n} obtained from the sequence {x_n} by means of a regular transformation (c_nk) satisfying the condition , where x, x_n, c_nk (n, k=1, 2,...) are complex numbers.Traslated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 815–823, December, 1977.     

On the tail behaviour of the distribution function of multiple stochastic integrals

Summary Let F _n (u) denote the empirical distribution function of a sample of i.i.d. random variables with uniform distribution on [0, 1]. Define , and consider the integrals where f is a bounded measurable function. We give a good upper bound on the probability . An analogous estimate is given for multiple integrals with respect to a Poisson process.     

A cyclic inequality

The existence of , where is proved, and a simple method of calculating it is derived.Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 113–119, February, 1971.I must thank V. I. Levin for posing this problem and for his interest in the work.     

The kernel of a sequence obtained from a given sequence with the aid of a regular transformation

The smallest set is found that contains the kernel of a sequence obtained from a sequence of elements {x_n} of a Banach space with the aid of a regular transformation of the class T(C, C). Here T(C, C) is the set of complex matrices (c_nk) (a_nk+ib_nk) satisfying the conditions .Translated from Matematicheskie Zametki, Vol. 19, No. 5, pp. 707–716, May, 1976.In conclusion the author would like to express his thanks to A. A. Melentsov for his assistance and attention to the work.     

Estimate of the upper bound on the Gaussian curvature of certain surfaces with boundary

In this article it is shown that if S is a complete, regular (of class C⁴) surface with geodesic boundary along which the normal curvature does not change sign, then the Gaussian curvature of the surface satisfies the condition: .Translated from Matematicheskie Zametki, Vol. 20, No. 1, pp. 113–120, July, 1976.The author wishes to thank N. V. Efimov for introducing him to this problem and for several valuable pieces of advice given when the work was being written up.     

Uniform Approximation of Nonperiodic Functions Defined on the Entire Axis

Using the following notation: C is the space of continuous bounded functions f equipped with the norm , V is the set of functions f such that , the set E consists of fCV and possesses the following property:

On periodic groups of automorphisms of extremal groups

It is proved that if a periodic group has an extremal normal divisor , determining a complete abelian factor group , then the center of the group contains a complete abelian subgroup , satisfying the relation and intersecting on a finite subgroup. It is also established with the aid of this proposition that every periodic group of automorphisms of an extremal group is a finite extension of a contained in it subgroup of inner automorphisms of the group .Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 91–96, July, 1968.     

Sharp Constant in a Jackson-Type Inequality for Approximation by Positive Linear Operators

In what follows, C is the space of -periodic continuous real-valued functions with uniform norm, is the first continuity modulus of a function with step h, H _n is the set of trigonometric polynomials of order at most n, is the set of linear positive operators (i.e., of operators such that for every ), is the space of square-integrable functions on ,

Subdifferentials with respect to dualities

LetX andW be two sets and: ¯R^X ¯R^W a duality (i.e., a mapping such that for all and all index setsI). We introduce and study the subdifferential of a function at a pointx _o X, with respect to. We also consider the particular cases when is a (Fenchel-Moreau) conjugation, or a -duality, or a -duality, in the sense of [8].     

On the exactness of a theorem of Mazur and Orlicz

For a preassigned unbounded sequence {S_n} of complex numbers, and preassigned complex numbers z₁ and z₂z₁ we construct: 1) regular matrices A=a_nk and B=b_nk such that the same bounded sequences are summable by these matrices and that , and ; 2) regular matrices A⁽¹⁾)=a _nk ⁽¹⁾ and B⁽¹⁾=b _nk ⁽¹⁾ such that B⁽¹⁾ A⁽¹⁾, and, . Our results show that the well known theorem of MazurOrlicz on the bounded consistency of two regular matrices, one of which is boundedly stronger than the other, is exact.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 431–436, April, 1972.     

Exact constants in inequalities of the jackson type for quadrature formulas

We prove that if is the error of a simple quadrature formula and ω(ε, δ)₁ is the integral modulus of continuity, then, for any δ ≥/π andn,r = 1, 2, …, the following equality is true: whereD _r is the Bernoulli kernel.     

An interesting result for positive series

For arbitrary given positive term series and , the present paper provides an interesting necessary and sufficient condition for . In Memory of Professor M. T. Cheng The Research Project of the Mathematical Institute, Ningbo University. Supported in part by Zhejiang Provincial Natural Science Foundation, and by State Key Laboratory of Southwest Institute of Petroleum.     

New Proof of the Semmes Inequality for the Derivative of the Rational Function

In the open disk of the complex plane, we consider the following spaces of functions: the Bloch space ; the Hardy--Sobolev space ; and the Hardy--Besov space . It is shown that if all the poles of the rational function R of degree n, , lie in the domain , then , where and depends only on . The second of these inequalities for the case of the half-plane was obtained by Semmes in 1984. The proof given by Semmes was based on the use of Hankel operators, while our proof uses the special integral representation of rational functions.     

Absolute Continuity of the Spectrum of a Periodic Schrödinger Operator

We prove the absolute continuity of the spectrum of the Schrödinger operator in , , with periodic (with a common period lattice ) scalar and vector potentials for which either , , or the Fourier series of the vector potential converges absolutely, , where is an elementary cell of the lattice , for , and for , and the value of is sufficiently small, where and otherwise, , and .     

Partially stabilizing LQ-optimal control for stabilizable semigroup systems

The Linear-Quadratic optimal control problem with a partial stabilization constraint (LQPS) is considered for exponentially stabilizable infinite dimensional semigroup state-space systems with bounded sensing and control (having their transfer function with entries in the algebra . It is reported that the LQPS-optimal state-feedback operator is related to a nonnegative self-adjoint solution of an operator Riccati equation and it can be identified (1) by solving a spectral factorization problem delivering a bistable spectral factor with entries in the distributed proper-stable transfer function algebra _, and (2) by obtaining any constant solution of a diophantine equation over _. These theoretical results are applied to a simple model of heat diffusion, leading to an approximation procedure converging exponentially fast to the LQPS-optimal state feedback operator.     

On the representation of differentiations by Hamiltonians,operating from an algebra of local observable spin systems

Let and be algebras of local and quasilocal observable spin systems corresponding to the group Zr, be a differentiation invariant with respect to displacements. The question of representation of D in the form of formal Hamiltonian formed by the displacements of an elementx ε is considered. It is shown that such a representation exists if the condition holds, where means an element obtained from the elements [T_kX,a] by some r-multiple process of summation. Translated from Matematicheskii Zametki, Vol. 21, No. 1, pp. 93–98, January, 1977.     

A criterion for the existence of bounded nonoscillatory solutions for nonlinear retarded differential equatory

Summary For the equation (⋆) below we give a condition which guarantees the existence of a bounded nonoscillatory solution x with Entrata in Redazione il 19 luglio 1977.