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:

Un critère sur les petites boules dans le théorème limite central

We show that a Banach space valued random variableX such that t} \right\} = 0$$ " align="middle" border="0"> satisfies the central limit theorem if and only if the following criterion on small balls is fulfilled:

Uniformly maldistributed sequences in a strict sense

It is shown that the following three limits

A Mean-Value Theorem for Multiplicative Functions on the Set of Shifted Primes

Let f be a complex-valued multiplicative function, letp denote a prime and let π(x) be the number of primes not exceeding x. Further put $$m_p (f): = \mathop {\lim }\limits_{x \to \infty } \frac{1}{{\pi (x)}}\sum\limits_{p \leqslant x} {f(p + 1)} {\text{, }}M(f): = \mathop {\lim }\limits_{x \to \infty } \frac{1}{x}\sum\limits_{n \leqslant x} {f(n)}$$ and suppose that $$\mathop {\lim \sup }\limits_{x \to \infty } \frac{1}{x}\sum\limits_{n \leqslant x} {\left| {f\left( n \right)} \right|^2 } < \infty ,\sum\limits_{p \leqslant x} {\left| {f\left( n \right)} \right|^2 } \ll x\left( {\ln x} \right)^{ - \varrho } ,$$ with some \varrho > 0. For such functions we prove: If there is a Dirichlet character χ_d such that the mean-value M(f χ_d) exists and is different from zero,then the mean-value m_p(f) exists. If the mean-value M(f) exists, then the same is true for the mean-valuem_p(f) .     

Analytic solution of vector model kinetic equations with constant kernel and their applications

Exact solutions are obtained for the first time for the half-space boundary-value problem for the vector model kinetic equations

Estimating the diameter of one class of functions in L2

Let

L p Boundedness for the Multilinear Singular Integral Operator

L^p (\mathbbRⁿ )L^{p} (\mathbb{R}^{n} ) boundedness is considered for the maximal multilinear singular integral operator which is defined by

A Generalized Measure Concentrated on a Closed Set of Measure Zero

When E is a closed set of measure zero in the dyadic group and the Walsh series satisfies

A modification of the uniqueness criterion for the solution of the Watson problem for a half-plane

It is proved that a known theorem yielding the solution of the Watson problem for a half-plane in terms of the Ostrovskii function remains valid if the Ostrovskii function is replaced by the function 0} r^x /m (x)$$ " align="middle" border="0"> , where for x [n, n+1) the function m(x)=m_n, or by the function .Translated from Matematicheskie Zametki, Vol. 14, No. 5, pp. 609–614, November, 1973.     

When all trajectories in the Liénard plane cross the vertical isocline?

In this paper we study the problem whether all trajectories of the system =y–F(x) and =–g(x) cross the vertical isocline which is very important for the existence of periodic solutions and oscillation theory. The problem has not been solved for the critical case:

On the Growth of the Denominators of Convergents

Let log . We prove that there exist non-denumerably many pairwise not equivalent irrational numbers such that and where qn() denotes the denominator of the nth convergent of .     

On the strong summation of Fourier series with variable exponents

Пустьf 2π-периодическ ая суммируемая функц ия, as _k (x) еë сумма Фурье порядк аk. В связи с известным ре зультатом Зигмунда о сильной суммируемости мы уст анавливаем, что если λn→∞, то сущес твует такая функцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _{2n} } } \right\}^{1/\lambda _{2n} } = \infty .$$ Отсюда, в частности, вы текает, что если λn?∞, т о существует такая фун кцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } } \right\}^{1/\lambda _n } = \infty .$$ Пусть, далее, ω-модуль н епрерывности и $$H^\omega = \{ f:\parallel f(x + h) - f(x)\parallel _c \leqq K_f \omega (h)\} .$$ . Мы доказываем, что есл и λ _n ?∞, то необходимым и достаточным условие м для того, чтобы для всехf∈H ^ω выполнялос ь соотношение $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _n } } \right\}^{1/\lambda _n } = 0(x \in [0;2\pi ])$$ является условие $$\omega \left( {\frac{1}{n}} \right) = o\left( {\frac{1}{{\log n}} + \frac{1}{{\lambda _n }}} \right).$$ Это же условие необхо димо и достаточно для того, чтобы выполнялось соотнош ение $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } = 0(f \in H^\omega ,x \in [0;2\pi ]).$$      

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 ,

Subexponential distributions and characterizations of related classes

LetL(),0, 0, denote the class of distributionsF satisfying

On the almost sure central limit theorem and domains of attraction

We give necessary and sufficient criteria for a sequence (X _n) of i.i.d. r.v.'s to satisfy the a.s. central limit theorem, i.e.,

Dichotomy laws for the behaviour near the unit element of group representations

A well known “zero-two law" shows that if is a strongly continuous one-parameter group of bounded operators on a Banach space X, and if then Here we discuss analogous problems for general unital representations θ of a topological group G on a unital Banach algebra A. Let 1 be the unit of G, and I the unit element of A. We show that either or if, moreover, θ admits “continuous division by any positive integer”, then, either or Our argument also gives automatic continuity results for representations of abelian Baire groups on a separable Banach algebra and representations of compact non abelian groups on a Banach algebra which are locally bounded and satisfy Received: 8 June 2005; revised: 13 October 2005     

Comparison results for some types of relaxation of variational integral functionals

A comparison between some relaxation methods of an integral functional is carried out. The following relaxed functionals of the variational integral I(, u)= :

An asymptotic formula for the minimal capacity among sets of equal area

We give an asymptotic formula for

Optimal ray sequences of rational functions connected with the Zolotarev problem

Given two compact disjoint subsetsE ₁,E ₂ of the complex plane, the third problem of Zolotarev concerns estimates for the ratio

On the almost sure (a.s.) central limit theorem for random variables with infinite variance

We give criteria for a sequence (X _n ) of i.i.d.r.v.'s to satisfy the a.s. central limit theorem, i.e.,