On approximation of smooth functions from null spaces of optimal linear differential operators with constant coefficients

For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln(ψ) = n ∑ k=0 akψ(k), where the constant coefficients ak ∈ R may be adapted to f . We prove that for each f ∈ C(n)(I), there is a selection of coefficients {a1, ,an} and a corresponding linear combination Sn( f ,t) = n ∑ k=1 bkeλkt of functions ψk(t) = eλkt in the nullity of L which satisfies the following Jackson’s type inequality: f (m) Sn(m )( f ,t) ∞≤ |an|2n|Im|1/1q/ep|λ|λn|n|I||nm1 Ln( f ) p, where |λn| = mka x|λk|, 0 ≤ m ≤ n 1, p,q ≥ 1, and 1p + q1 = 1. For the particular operator Mn(f) = f + 1/(2n) f(2n) the rate of approximation by the eigenvalues of Mn for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.     

ON SIMULTANEOUS APPROXIMATION TO A DIFFERENTIABLE FUNCTION AND ITS DERIVATIVE BY INVERSE PAL-TYPE INTERPOLATION POLYNOMIALS

Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P'n-1(x). By the theory of the inverse Pal-Type interpolation, for a function f(x) ∈ C[-1 1], there exists a unique polynomial Rn(x) of degree 2n - 2 (if n is even) satisfying conditions Rn(f,ξk) = f(∈ek)(1≤ k≤ n - 1) ;R'n(f,xk) = f'(xk)(1≤ k≤ n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f,x)} (n is even) and the main result of this paper is that if f ∈ C'[1,1], r≥2, n≥ + 2> and n is even thenholds uniformly for all x ∈ [- 1,1], where h(x) = 1 +     

$L^p$函数的分布函数在端点处的极限性质

We consider the limiting property of the distribution function of L~p function at endpoints 0 and ∞ and prove that for λ 0 the following two equations limλ→+∞λ~pm({x : |f(x)| λ}) = 0, limλ→0+λ~pm({x : |f(x)| λ}) = 0hold for f ∈ L~p(Rn) with 1 ≤ p ∞. This result is naturally applied to many operators of type(p, q) as well.     

Global Topological Linearization with Unbounded Nonlinear Term

§ 1.Statementof Theorem　 Consider the systemx′=Ax + f (x) ,y′=By +φ(x) +ψ(y) ,(1 )where x∈ Rn1,y∈ Rn2 ,f,φ andψ are locally Lipschitzian.If x is in Rn we denote itsEnclidend norm by| x| and if A is an n×n matrix we denote its operator norm by| A| .Let Reλ(A) be the real partof eigenvalues of A.　 Suppose that Reλ(A) <0 and Reλ(B) >0 .Without loss of generality,we may assamethatd| x| 2dtx′=Ax≤ -α| x| 2 , (2 )| e- Bt|≤ k . e-βt　 (t≥ 0 ) , (3 )whereα,β and k are all…     

RESEARCH ANNOUNCEMENTS Rough Marcinkiewicz Integrals With L(log~ L)~2 Kernels on Product Spaces

Let RN (N 2, N = n or m) be the N-dimensional Euclidean space and SN-1 be the unitsphere in RN. For nonzero point x RN, we denote x' = x/ |x|. E. M. Stein[12] defined a highdimensional analogue of the Marcinkiewicz integral on Rn by (f)(x)= (R|Fs(x)|2ds)1/2,where Fs(x) =dv, is a homogeneous function of degree zero, whoserestriction to Sn-1 is in L1 (Sn-1) and satisfies the cancellation property sn-1 (x')dx'= 0. Itis well-known that the Marcinkiewicz integral is very useful in harmonic…     

On the uniform strong approximation of Marcinkiewicz type for multivariable continuous functions

The rate of uniform strong approximation of Marcinkiewicz type for multivariable continuous func-tions is obtained in this paper as follows:‖1/k 1 k∑j=0|Sj(f)- f|q‖≤C/k 1 k∑j=0 Eqj(f),where Sj (f) denotes the square partial Fourier sum of f and Ej (f) denotes the square best approximation of f by trigonometric polynomials of degree (j, j, … ,j),j = 0,1, 2,….     

PHASE RETRIEVAL OF TIME-LIMITED SIGNALS

The problem of reconstructing a signal ψ(x) from its magnitude |ψ(x)| is of considerable interest to engineers and physicists.This article concerns the problem of determining a time-limited signal f with period 2π when |f(eix)| is known for x ∈ [-π,π].It is shown that the conditions |g(eix)| = |f(eix)| and |g(ei(x+b))-g(eix)| = |f(ei(x+b))-f(eix)|,b = 2π,together imply that either g = wf or g = vf,where both w and v have period b.Furthermore,if 2bπ is irrational then the functions w and v reduce to some constants c1 and c2,respectively;if 2bπ is rational then w takes the form w=eiαB1(eix)B2(eix) and v takes the form ei(x2πN/b+α)B1(eix)B2(eix),where B1 and B2 are Blaschke products.     

A NOTE ON SINGULAR INTEGRALS WITH DOMINATING MIXED SMOOTHNESS IN TRIEBEL-LIZORKIN SPACES

Let h be a measurable function defined on R+×R+. LetΩ∈ L(log L+)νq(Sn1-1×Sn2-1)(1 ≤νq≤ 2) be homogeneous of degree zero and satisfy certain cancellation conditions. We show that the singular integral T f(x1, x2) = p. v.∫ Rn1+n2Ω(y′1, y′2)h(|y1|, |y2|)|y1|n1|y2|n2f(x1- y1, x2- y2)dy1dy2maps from Sα1, α2p, q˙F(Rn1× Rn2) boundedly to itself for 1 p, q ∞, α1, α2 ∈ R.     

Hardy spaces H_L~p(R~n) associated with operators satisfying k-Davies-Gaffney estimates

Let L be a one-to-one operator of type ω having a bounded H∞ functional calculus and satisfying the k-Davies-Gaffney estimates with k ∈ N. In this paper, the authors introduce the Hardy space HLp(Rn) with p ∈ (0, 1] associated with L in terms of square functions defined via {e-t2kL}t>0 and establish their molecular and generalized square function characterizations. Typical examples of such operators include the 2k-order divergence form homogeneous elliptic operator L1 with complex bounded measurable coefficients and the 2k-order Schrdinger type operator L2 := (-Δ)k + Vk, where Δ is the Laplacian and 0≤V∈Llock(Rn). Moreover, as an application, for i ∈ {1, 2}, the authors prove that the associated Riesz transform ▽k(Li-1/2) is bounded from HLip (Rn) to Hp(Rn) for p ∈ (n/(n + k), 1] and establish the Riesz transform characterizations of HL1p (Rn) for p ∈ (rn/(n + kr), 1] if {e-tL1 }t>0 satisfies the Lr-L2 k-off-diagonal estimates with r ∈ (1, 2]. These results when k := 1 and L := L1 are known.     

Potential space estimates in local Hardy spaces for Green potentials in convex domains

Let Ω be a bounded co.nvex domain in Rn(n≥3) and G(x,y) be the Green function of the Laplace operator -△ on Ω. Let hrp(Ω) = {f ∈ D'(Ω) :(E)F∈hp(Rn), s.t. F|Ω = f}, by the atom characterization of Local Hardy spaces in a bounded Lipschitz domain, the bound of f→(△)2(Gf) for every f ∈ hrp(Ω) is obtained, where n/(n 1)＜p≤1.     

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 (…     

全纯函数的加权积分

§ 1　 IntroductionLet D be the unit disc in the complex plane C,dm be the Lebesgue area measure onD.We denote H (D) the setof all holomorphic functions on D.For 0 相似文献   

Convergence to Diffusion Waves for Nonlinear Evolution Equations with Ellipticity and Damping, and with Different End States

In this paper, we consider the global existence and the asymptotic behavior of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects： {ψt=-（1-α）ψ-θx＋αψxx, θt=-（1-α）θ＋νψx＋（ψθ）x＋αθxx（E） with initial data （ψ,θ）（x,0）=（ψ0（x）,θ0（x））→（ψ±,θ±）as x→±∞ where α and ν are positive constants such that α 〈 1, ν 〈 4α（1 - α）. Under the assumption that ｜ψ＋ - ψ-｜ ＋ ｜θ＋ - θ-｜ is sufficiently small, we show the global existence of the solutions to Cauchy problem （E） and （I） if the initial data is a small perturbation. And the decay rates of the solutions with exponential rates also are obtained. The analysis is based on the energy method.     

Infinitely many solutions of dirichlet problem for <Emphasis Type="Italic">p</Emphasis>-mean curvature operator

The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator: is considered, where Θ is a bounded domain in R ⁿ (n>p>1) with smooth boundary ∂Θ. Under some natural conditions together with some conditions weaker than (AR) condition, we prove that the above problem has infinitely many solutions by a symmetric version of the Mountain Pass Theorem if . Supported by the National Natural Science Foundation of China (10171032) and the Guangdong Provincial Natural Science Foundation (011606).     

A SIMPLIFIED BRAUER＇S THEOREM ON MATRIX EIGENVALUES

Let A=(a _ij)∈C ^n×n and . Suppose that for each row of A there is at least one nonzero off-diagonal entry. It is proved that all eigenvalues of A are contained in . The result reduces the number of ovals in original Brauer’s theorem in many cases. Eigenvalues (and associated eigenvectors) that locate in the boundary of are discussed. The project is supported in part by Natural Science Foundation of Guangdong.     

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(…     

On the Littlewood-Paley theorem for arbitrary intervals

We extend the results of Rubio de Francia and Bourgain by showing that, for arbitrary mutually disjoint intervals Δ_k ⊂ ℤ₊, arbitrary p ∈, (0, 2], and arbitrary trigonometric polynomials f _k with supp , we have

MODIFIED APPROXIMATE PROXIMAL POINT ALGORITHMS FOR FINDING ROOTS OF MAXIMAL MONOTONE OPERATORS

§1　Introduction and preliminariesA set T Rn×Rnis called a monotone operator on Rn,if T has the property(x,y) ,(x′,y′)∈T 〈x -x′,y -y′〉≥0 ,where〈·,·〉denotes the inner product on Rn.T is maximal if(considered as a graph) itis not strictly contained in any other monotone operator on Rn.It is well known that thetheory of maximal monotone operators plays an important role in the study of convexprogramming and variational inequalities since itcan provide a powerful general framework…     

A note on regularity for a class of quasilinear elliptic equations

The following regularity of weak solutions of a class of elliptic equations of the form are investigated.     

Characterization of Compact Support of Fourier Transform for Orthonormal Wavelets of L^2（R^d）

Let{ψμ} be an orthonormal wavelet of L^2（R^d） and the support of a whole of its Fourier transform be Uμsupp{ψμ}=Пi=1^d[Ai, Di]-Пi=1^d（Bi, Ci）, Ai≤Bi≤Ci≤Di. Under the weakest condition that each │ψμ│, is continuous for ω ∈ δ（Пi=1^d[Ai, Di]）, a characterization of the above support of a whole is given.