On summability of pairs of convolution sequences of probability measures on locally compact semigroups

In this paper we study the problem of convergence in the weak and the vague topology of the sequence

Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.     

An asymptotic method for calculating limits of maximal means

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

Solutions to Some Open Problems in Fluid Dynamics

Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.     

On the space l P(β)

Let be a sequence of positive numbers and 1 ≤p < ∞. We consider the spacel ^P(β) of all power series such that . We give a necessary and sufficient condition for a polynomial to be cyclic inl ^P(β) and a point to be bounded point evaluation onl ^P(β).     

Precise asymptotics in the law of the iterated logarithm*

Let X₁, X₂, ... be i.i.d. random variables with EX₁ = 0 and positive, finite variance σ², and set S_n = X₁ + ... + X_n. For any α > −1, β > −1/2 and for κ_n(ε) a function of ε and n such that κ_n(ε) log log n → λ as n ↑ ∞ and , we prove that

On the estimation of the regression coefficients of a continuous parameter process with stationary residual

The asymptotic expressions of the covariance matrices for both the least square estimates _L α _T and Markov (best linear) estimates are obtained, based on a sample in a finite interval (0, T) of the regression co-efficients α = (α ₁, …, α _{m 0})′ of a parameter-continuous process with a stationary residual. We assume that the regression variables φ _ν(t), t ⩾ 0, ν = 1, …, m ₀, are continuous in t, and satisfy conditions (3.1)–(3.3). For the residual, we assume that it is a stationary process that possesses a bounded continuous spectral density f(λ). Under these assumptions, it is proven that

Trigonometric series of classesL^p (\mathbb{T}), p \in ]1;\infty [ and their conservative meansand their conservative means

Suppose that a lower triangular matrix μ:[μ _m ⁽ⁿ⁾ ] defines a conservative summation method for series, i.e.,

Precise rate in the law of iterated logarithm for <Emphasis Type="Italic">ρ</Emphasis>-mixing sequence

Let {X, X _n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set . Suppose lim _n→∞ and , where d=2, if −1<b<0 and d>2(b+1), if b≥0. It is proved that, for any b>−1,

Integration and Lipschitz functions

The aim of the paper is to prove that every f ∈ L ¹([0,1]) is of the form f = , where j _n,k is the characteristic function of the interval [k- 1 / 2 ⁿ , k / 2 ⁿ ) and Σ _n=0^∞Σ _k=1²ⁿ |a _n,k | is arbitrarily close to ||f|| (Theorem 2). It is also shown that if μ is any probabilistic Borel measure on [0,1], then for any ɛ > 0 there exists a sequence (b _n,k ) _n≧0 ^k=1,...,2n of real numbers such that and for each Lipschitz function g: [0,1] → ℝ (Theorem 3).      

多线性分数次积分算子交换子的有界性

Ibαf ( x) =∫R ∏mj=1( bj( x) - bj( y) ) 1| x - y| n-αf ( y) dyare considered.The following priori estimates are proved.For 1 01Φ1t| {y∈Rn:| Ibαf( y) | >t}| 1q ≤csupt>01Φ1t| {y∈Rn:ML( log L) 1r ,α(‖b‖f ) ( y) >t}| 1q,where‖b‖=∏mj=1‖bj‖Oscexp Lrj,Φ( t) =t( 1 + log+t) 1r,1r =1r1+ ...+ 1rm,ML(…     

About entire functions with specialL 2-properties on one-dimensional subspaces of ℂ n

In this paper we consider special elements of the Fock space #x2131; _n . That is the space of entire functionsf:ℂ: ⁿ →ℂ, such that the followingL ²- condition is satisfied: . Here we show that there exists an entire functiong:ℂ ⁿ →ℂ such that for every one-dimensional subspace Π⊂ℂ ⁿ and for all 0<∈<2 we have , but in the limit case ∈=0 we have . This result is analogue to a result from [1]. There holomorphic functions on the unit-ball are investigated. Furthermore the proof — as the one in [1] — uses a theorem from [2]. Therefore we give another application of the results from [2] — namely for spaces of entire functions.     

Approximating periodic functions in Hölder type metrics by singular integrals with positive Kernels

Let M be either the space of 2π-periodic functions L_p, where 1 ≤ p < ∞, or C; let ω_r(f, h) be the continuity modulus of order r of the function f, and let

Equivalent definition of some weighted Hardy spaces

We present an equivalent definition of functions analytic in the half-plane ℂ₊ = {z: Re z > 0} for which

On the decay rate of (p, A)-lacunary series

A power series with radius of convergence equal 1 is called a (p,A)-lacunary one if n_k ≥ Ak^p, A > 0, 1 < p < ∞. It is proved that if 1 < p < 2 and f(x) is a (p,A)-lacunary series that satisfies the condition

Spherical means and the restriction phenomenon

Let Γ be a smooth compact convex planar curve with arc length dm and let dσ=ψ dm where ψ is a cutoff function. For Θ∈SO (2) set σ_Θ(E) = σ(ΘE) for any measurable planar set E. Then, for suitable functions f in ℝ², the inequality.

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.     

Precise asymptotics in Chung’s law of the iterated logarithm

Let X, X1, X2,... be i.i.d, random variables with mean zero and positive, finite variance σ^2, and set Sn = X1 ＋... ＋ Xn, n≥1. The author proves that, if EX^2I{｜X｜≥t} = 0（（log log t）^-1） as t→∞, then for any a〉-1 and b〉 -1,lim ε↑1/√1＋a（1/√1＋a-ε）b＋1 ∑n=1^∞（logn）^a（loglogn）^b/nP{max κ≤n｜Sκ｜≤√σ^2π^2n/8loglogn（ε＋an）}=4/π（1/2（1＋a）^3/2）^b＋1 Г（b＋1）,whenever an = o（1/log log n）. The author obtains the sufficient and necessary conditions for this kind of results to hold.     

Homological Connectivity Of Random 2-Complexes

Let Δ_n−1 denote the (n − 1)-dimensional simplex. Let Y be a random 2-dimensional subcomplex of Δ_n−1 obtained by starting with the full 1-dimensional skeleton of Δ_n−1 and then adding each 2−simplex independently with probability p. Let denote the first homology group of Y with mod 2 coefficients. It is shown that for any function ω(n) that tends to infinity

An error bound of the Ritz method for a singular second-order differential equation

The paper presents an error bound of the Ritz method for the problem of minimizing the functional