THE GLOBAL APPROXIMATION BY LEFT-BERNSTEIN-DURRMEYER QUASI-INTERPOLANTS IN Lp[0, 1]

In this paper,we will use the 2r-th Ditzian-Totik modulus of smoothness wp^2r(f,t)p to discuss the direct and inverse theorem of approximation by Left-Bernstein-Durrmeyer quasi-interpolants Mn^[2r-1]f for functions of the space Lp[0,1](1≤p≤ ∞)。     

正系数多项式倒数逼近的点态和整体估计

利用Ditzian-Totik光滑模对于[0,1]上定义的非角连续函数f(x)，且f(x)≠0，文中证明存在正系数多项式Pn(x)及常数C，使得|f9x)-1/Pn(x)|≤Cωψ^λ（f,n^-1/2(ψ(x) 1/√n)^1-λ)。当λ＝1时，上述结果导出已有的整体估计，而当0≤λ＜1时，得到倒数逼近一个新的点态局部估计。     

Pointwise Approximation by Szàsz-Mirakjan Quasi-Interpolants

Recently some classical operator quasi-interpolants were introduced to obtain much faster convergence. A.T. Diallo investigated some approximation properties of Szàsz-Mirakjan Quasi-Interpolants, but he obtained only direct theorem with Ditzian-Totik modulus ω2rψ(f, t). In this paper, we extend Diallo's result and solve completely the characterization on the rate of approximation by the method of quasi-interpolants to functions f ∈ CB[0, ∞) by making use of the unified modulus ω2rψλ> (f, t) (0 ≤λ≤ 1).     

Entropy numbers of Besov classes of generalized smoothness on the sphere

We investigate the asymptotic behavior of the entropy numbers of Besov classes BBΩp,θ(Sd 1)of generalized smoothness on the sphere inL q(Sd 1)for 1≤p,q,θ≤∞,and get their asymptotic orders.We also obtain the exact orders of entropy numbers of Sobolev classesBWr p(Sd 1)inL q(Sd 1)whenpand/orqis equal to 1 or∞.This provides the last piece as far as exact orders of entropy numbers ofBWr p(Sd 1)inL q(Sd 1)are concerned.     

一类推广的Bernstein-Kantorovich算子的点态逼近

讨论Bernstein-Kantorovich算子的一种推广形式的逼近性质,运用插项的方法证明了逼近正定理,并证明了逆定理,得到了逼近等价定理.完善了算子在逼近性质方面的结果.     

LP[0,1](1＜p＜∞)空间正系数多项式倒数逼近的一个推广

本文推广了LP[0,1](1＜p＜∞)空间函数的正系数多项式的倒数逼近的结论,即证明了:设f(x)∈LP[0,1],1＜p＜∞,且在(0,1)内严格1次变号,则存在一点x0∈(0,1)及一个n次多项式Pn(x)∈∏n(+)使得‖f(x)-x-x0/Pn(x)‖LP[0,1]≤Cpω(f,n-1/2)LP[0,1],其中∏n(+)为次数不超过n的正系数多项式的全体.     

The Jackson Inequality for the Best L2-Approximation of Functions on [0, 1] with the Weight x

Let L^2（[0, 1], x） be the space of the real valued, measurable, square summable functions on [0, 1] with weight x, and let n be the subspace of L2（[0, 1], x） defined by a linear combination of Jo（μkX）, where Jo is the Bessel function of order 0 and {μk} is the strictly increasing sequence of all positive zeros of Jo. For f ∈ L^2（[0, 1], x）, let E（f, n） be the error of the best L2（[0, 1], x）, i.e., approximation of f by elements of n. The shift operator off at point x ∈[0, 1] with step t ∈[0, 1] is defined by T（t）f（x）=1/π∫0^π f（√x^2 ＋t^2-2xtcosO）dθ The differences （I- T（t））^r/2f = ∑j=0^∞（-1）^j（j^r/2）T^j（t）f of order r ∈ （0, ∞） and the L^2（[0, 1],x）- modulus of continuity ωr（f,τ） = sup{｜｜（I- T（t））^r/2f｜｜：0≤ t ≤τ] of order r are defined in the standard way, where T^0（t） = I is the identity operator. In this paper, we establish the sharp Jackson inequality between E（f, n） and ωr（f, τ） for some cases of r and τ. More precisely, we will find the smallest constant n（τ, r） which depends only on n, r, and % such that the inequality E（f, n）≤ n（τ, r）ωr（f, τ） is valid.     

Bernstein-Durrmeyer算子拟中插式在Orlicz空间中的逼近

本文在Orlicz空间中研究了Bernstein-Durrmeyer算子拟中插式B_n~(2r-1)(f,x)逼近性质.利用2r阶Ditzian-Totik模与K-泛函的等价性,Jensen不等式,H?lder不等式,Berens-Lorentz引理得到了逼近的正,逆和等价定理,从而推广了Bernstein-Durrmeyer算子拟中插式B_n~(2r-1)(f,x)在L_P空间的逼近结果.     

STRONG CONVERSE INEQUALITY FOR SZASZ-DURRMEYER OPERATORS

For Szasz-Durrmeyer operators Ln (f,z), 1＜ p≤∞, we prove that, forsome m, w^2φ(f,1/√n)p ≤(≤M(││Ln,f,x) - ,f││p ││Lmn(f, x) -f││p),where φ(x)^2 =x, M ＞0,w^2φ(f,t)p is Ditzian-Totik modulus of smoothness.     

Lp approximation by Bernstein-Kantorovich quasi-interpolants

Bernstein-Kantorovich quasi-interpolants K(2r-1) n (f, x) are considered and direct,inverse and equivalence theorems with Ditzian-Totik modulus of smoothness ω2rψ(f, t)p (1 ≤ p≤+∞) are obtained.     

Global W2,p Solutions of GBBM Equations on Unbounded Domain

In this paper we study the initial boundary value problem of GBBM equations on unbounded domain u_t - Δu_t = div f(u) u(x,0) = u_0(x) u|_{∂Ω} = 0 and corresponding Cauchy problem. Under the conditions: f( s) ∈ C^sup1 and satisfies (H)\qquad |f'(s)| ≤ C|s|^ϒ, 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3; 0 ≤ ϒ < ∞ if n = 2 u_0(x) ∈ W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω)(W^{2,p}(R^n) ∩ W^{2,2}(R^n) for Cauchy problem), 2 ≤ p < ∞, we obtain the existence and uniqueness of global solution u(x, t) ∈ W^{1,∞}(0, T; W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω))(W^{1,∞}(0, T; W^{2,p}(R^n) ∩ W^{2,2} (R^n)) for Cauchy problem), so the results of [1] and [2] are generalized and improved in essential.     

Lp[0,1]空间Müntz有理逼近的Jackson型估计

设Λ={λn}n∞=1为正的实数数列,且当n→∞时,有λn↘0.本文给出了当λn≤Mn-1/2,n=1,2,…,(其中M＞0为一正常数)时Müntz系统{xλn}的有理函数在Lp[0,1]空间的逼近速度,主要结论为Rn(f,Λ) Lp≤CMω(f,n-1/2)Lp,1≤p≤∞.     

Mixed modulus of smoothness with Muckenhoupt weights and approximation by angle

Mixed modulus of smoothness in weighted Lebesgue spaces with Muckenhoupt weights are investigated. Using mixed modulus of smoothness we obtain Potapov type direct and inverse estimates of angular trigonometric approximation of functions in these spaces. Also we obtain equivalences between mixed modulus of smoothness and K-functional and realization functional. Fractional order modulus of smoothness is considered as well.     

关于Wiener过程泛函连续模的精确收敛速度

设{W（t）,t≥0}是一标准Wiener过程,记S是Strassen重对数律的紧集类·本文中我们讨论了两个变量sup_0≤t≤1-h inf_f∈S sup_0≤x≤1 |(W(t+hx)-W(t))(2h log h^-1)^-1/2 - f(x)|及inf_0≤t≤1-h sup_0≤x≤1 |(W(t + hx) - W(t))(2hlogh^-1)^-1/2- f(x)|（对任何f∈S）趋于零的精确的收敛速度.作为一个推广,我们建立了Wiener过程的不可微模与泛函的连续模之间的一种关系.     

Bergman空间B_q~p(01)中的Hardy-Littlewood逆定理

本文在Bergman空间Bqp(01)中研究关于旋转连续模的Hardy Littlewood逆定理,在通常条件下,得到了与在空间Hp(0相似文献   

二阶微分系统奇异正定超线性周期边值问题的多重正解

主要研究了二阶微分系统具有奇异正定超线性周期边值问题多重正解的存在性问题,利用Leray-Schauder抉择定理和锥不动点定理给出了奇异正定超线性周期边值问题-(p(t)x′)′+q1(t)x=f1(t,x,y),t∈I=[0,1]-(p(t)y′)′+q2(t)y=f2(t,x,y)x(0)=x(1),x[1](0)=x[1](1)y(0)=y(1),y[1](0)=y[1](1)(1.1)的多重正解的存在性,其中非线性项fi(t,x,y)(i=1,2)在x=∞,y=∞点处超线性,在(x,y)=(0,0)处具有奇性.这里定义x[1](t)=p(t)x′(t),y[1](t)=p(t)y′(t)为准导数,其中系数p(t),qi(t)(i=1,2)是定义在[0,1]上的可测函数,且p(t)>0,qi(t)>0(i=1,2),a.e[0,1],fi(t,x,y)∈C(I×R×R,R+),R+=(0,+∞).     

Pointwise weighted approximation by Bernstein operators

We consider the pointwise weighted approximation by Bernstein operators. The related weight functions are Jacobi weights . We obtain approximation equivalence theorems, using a weighted modulus of smoothness , which is an extension of~[5].     

关于Bernstein多项式的一个特征

本文建立Ｂｅｒｎｓｔｅｉｎ多项式Ｂｎ（ｆ，ｘ）的阶是的一个特征定理。