A note on a theorem of Daboussi

The following assertion is proved. Let be the set of integers the number of the prime power of which is . Let be the size of . Then for each irrational , uniformly in , \begin{equation*} \frac{1}{\pi_k(x)} \bigg|\sum _{\alul{n\le x}{n\in\cN_k}} f(n) e^{2\pi in\alpha}\bigg|\to 0, \end{equation*} where is an arbitrary multiplicative function with , is a positive constant. This revised version was published online in June 2006 with corrections to the Cover Date.     

Precise asymptotics in the Baum-Katz and davis law of large numbers for positively associated sequences

§ 1　 Introduction and resultsL et { X,Xi;i≥ 1} be a sequence of i.i.d.random variables,and set Sn= ni=1 Xi,n≥1.Hsu and Robbins[1 ] introduced the conceptof complete convergence.They together withErdos[2 ] proved n≥ 1 P(|Sn|≥εn) <∞ ,ε>0 (1)if and only if EX=0 and EX2 <∞ .L ater,Spitzer[3] proved n≥ 11n P(|Sn|≥εn) <∞ ,ε>0if and only if EX =0 and E|X|<∞ .More generally,it was shown by Baum and Katz[4 ]that,for 0 0 (…     

Marcinkiewicz-\theta-Summability of Fourier Transforms

A general summability method of two-dimensional Fourier transforms is given with the help of an integrable function . Under some conditions on we show that the maximal operator of the Marcinkiewicz- -means of a tempered distribution is bounded from to for all and, consequently, is of weak type , where depends only on . As a consequence we obtain a generalization for Fourier transforms of a summability result due to Marcinkievicz and Zhizhiashvili, more exactly, the Marcinkiewicz- -means of a function converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz- -means are uniformly bounded on the spaces and so they converge in norm . Some special cases of the Marcinkievicz- -summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, de la Vallée-Poussin, Rogosinski and Riesz summations. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Jackson Type Estimate for Shepard Operators in L p Spaces for p≧ 1

This paper considers the approximation of the Kantorovich–Shepard operators in spaces for . For the Kantorovich–Shepard operators are defined by (1.1). Then

On Rounding Error Sums Related to the Circle Problem. ii

For a positive real parameter t, real numbers , , and , we consider sums , where is the rounding error function, i.e.\ . Generalizing and improving the main result of Part I of the paper we show that there exists an absolute constant such that for all , and all . Further, we give applications concerning the circle problem with linear, polynomial, and general weight.     

Finite rank operators in Jacobson radical \mathcal{R}_{\mathcal{N} \otimes \mathcal{M}}

In this paper we investigate finite rank operators in the Jacobson radical of Alg( ), where are nests. Based on the concrete characterizations of rank one operators in Alg( ) and , we obtain that each finite rank operator in can be written as a finite sum of rank one operators in and the weak closure of equals Alg( ) if and only if at least one of is continuous.     

A Littlewood-Paley type inequality

In this note we prove the following theorem: Let u be a harmonic function in the unit ball and . Then there is a constant C = C(p, n) such that

Isometries and direct decompositions of pseudo MV-algebras

In the paper isometries in pseudo MV-algebras are investigated. It is shown that for every isometry f in a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) there exists an internal direct decomposition of with commutative such that and for each x ∈ A. On the other hand, if is an internal direct decomposition of a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) with commutative, then the mapping g: A → A defined by is an isometry in and .      

On small zeros of quadratic forms over finite fields (II)

Let be a quadratic form with integer coefficients and letp denote a prime. Cochrane^[1] proved that ifn≥4 then has a solution satisfying , where . The aim of the present paper is to generalize the above result to finite fields.     

On Sign Changes in Weighted Polynomial L^1 -Approximation

For denote by a polynomial of best weighted -approximation to on of degree . In the present paper, we are dealing with the asymptotic distribution of sign changes of the error functions .     

An index theorem for gauge-invariant families: The case of solvable groups

We define the gauge-equivariant index of a family of elliptic operators invariant with respect to the free action of a family of Lie groups (these families are called ``gauge-invariant families' in what follows). If the fibers of are simply-connected and solvable, we compute the Chern character of the gauge-equivariant index, the result being given by an Atiyah–Singer type formula that incorporates also topological information on the bundle . The algebras of invariant pseudodifferential operators that we study, and , are generalizations of ``parameter dependent' algebras of pseudodifferential operators (with parameter in R q), so our results provide also an index theorem for elliptic, parameter dependent pseudodifferential operators. We apply these results to study Fredholm boundary conditions on a simplex. This revised version was published online in June 2006 with corrections to the Cover Date.     

On a Binary Diophantine Inequality Involving Prime Numbers

Let 1 < c < 15/14 and N a sufficiently large real number. In this paper we prove that, for all (N, 2N\ A with , the inequality has solutions in primes .     

Markov- and Bernstein-Type Inequalities for Polynomials with Restricted Coefficients

The Markov-type inequality is proved for all polynomials of degree at most n with coefficients from {-1,0,1} with an absolute constant c. Here ·[0,1] denotes the supremum norm on [0,1]. The Bernstein-type inequality is shown for every polynomial p of the form The inequality is also proved for every analytic function p on the open unit disk D that satisfies the growth condition      

On One Extremal Problem for a Seminorm on the Space l<Subscript>1</Subscript> with Weight

Let be a nondecreasing sequence of positive numbers and let l _1,α be the space of real sequences for which . We associate every sequence ξ from l _1,α with a sequence , where ϕ(·) is a permutation of the natural series such that , j ∈ ℕ. If p is a bounded seminorm on l _1,α and , then

On the Uniqueness of the Recovery of Parameters of the Maxwell System from Dynamical Boundary Data

The paper deals with the problem of recovering the parameters (functions) and of the Maxwell dynamical system

On the Order of Magnitude of the Divisor Function

Let D be an increasing sequence of positive integers, and consider the divisor functions： d（n, D） =∑d｜n,d∈D,d≤√n1, d2（n,D）=∑[d,δ]｜n,d,δ∈D,[d,δ]≤√n1, where [d,δ]=1.c.m.（d,δ）. A probabilistic argument is introduced to evaluate the series ∑n=1^∞and（n,D） and ∑n=1^∞and2（n,D）.     

Effects Connected with the Coincidence of Velocities in the Two-Velocity Dynamical System

The paper deals with the system

Furstenberg family and chaos

A Furstenberg family F is a family,consisting of some subsets of the set of positive integers,which is hereditary upwards,i.e.A■B and A∈F imply B∈F.For a given system (i.e.,a pair of a complete metric space and a continuous self-map of the space) and for a Furstenberg family F, the definition of F-scrambled pairs of points in the space has been given,which brings the well-known scrambled pairs in Li-Yorke sense and the scrambled pairs in distribution sense to be F-scrambled pairs corresponding respectively to suitable Furstenberg family F.In the present paper we explore the basic properties of the set of F-scrambled pairs of a system.The generically F-chaotic system and the generically strongly F-chaotic system are defined.A criterion for a generically strongly F-chaotic system is showed.     

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 .     

AN ANALOGUE OF A THEOREM OF FERENC LUKÁCS FOR DOUBLE WALSH—FOURIER SERIES

A theorem of Ferenc Lukács states that if a periodic function is integrable in Lebesgue"s sense and has a discontinuity of first kind at some point , then the th partial sum of the conjugate series to its trigonometric Fourier series at divided by converges to as . An analogue of this theorem for Walsh–Fourier series was proved by Rafat Riad. The main aim of the present paper is to extend the latter result from single to double Wals–Fourier series. We consider also functions of two variables which are of bounded variation over a rectangle in the sense of Hardy and Krause. Among others, we present a proof of the existence of the so-called sector limits of such functions at each point.