上密度无限制的指数多项式逼近

We take a new approach to obtaining necessary and sufficient conditions for the incompleteness of exponential polynomials in L p α ,where L p α is the weighted Banach space of complex continuous functions f defined on the real axis R satisfying ＋∞ ∞ ｜f（t）｜ p e -α（t） dt） 1/p ,1 p ∞,and α（t） is a nonnegative continuous function defined on the real axis R.In this paper,the upper density of the sequence which forms the exponential polynomials is not required to be finite.In the study of weighted polynomial approximation,consideration of the case is new.     

Regional, Single Point, and Global Blow-Up for the Fourth-Order Porous Medium Type Equation with Source

Blow-up behaviour for the fourth-order quasilinear porous medium equation with source,ut=-（｜u｜^nu）xxxx＋｜u｜^p-1u in R×R＋,where n 〉 0, p 〉 1, is studied. Countable and finite families of similarity blow-up patterns of the form us（x,t）=（T-t）^-1/p-1f（y）,where y=x/（T-t）^β＇ β=p-（n＋1）/4（p-1）,which blow-up as t → T^- 〈∞ are described. These solutions explain key features of regional （for p = n＋1）, single point （for p 〉 n＋1）, and global （for p ∈ （1,n＋1））blowup. The concepts and various variational, bifurcation, and numerical approaches for revealing the structure and multiplicities of such blow-up patterns are presented.     

Existence of Renormalized Solutions for Nonlinear Parabolic Equations

We give an existence result of a renormalized solution for a class of nonlin- ear parabolic equations b（x,u）/ t-div （a（x,t,u, u））＋g（x,t,u,u ）＋H（x,t, u）=f,in QT, where the right side belongs to LP＇ （0,T;W-1,p＇（Ω）） and where b（x,u） is unbounded function of u and where - div （ a （ x, t, u, u） ） is a Leray-Lions type operator with growth ｜u ｜p- 1 in V u. The critical growth condition on g is with respect to u and no growth condition with re sp ect to u, while the function H （x, t, u） grows as｜ u ｜p - 1.     

Nonradial Entire Large Solutions of Semilinear Elliptic Equations

We consider the problem of whether the equation △u = p（x）f（u） on RN, N ≥ 3, has a positive solution for which lim ｜x｜→∞（x） = ∞ where f is locally Lipschitz continuous, positive, and nondecreasing on （0,oo） and satisfies ∫1∞[F （t）]^- 1/2dt = ∞ where F（t） = ∫0^tf（s）ds. The nonnegative function p is assumed to be asymptotically radial in a certain sense. We show that a sufficient condition to ensure such a solution u exists is that p satisfies ∫0∞ r min｜x｜=r P （x） dr = ∞. Conversely, we show that a necessary condition for the solution to exist is that p satisfies ∫0∞r1＋ε min ｜x｜=rp（x）dr =∞ for all ε〉0.     

OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR DYNAMIC EQUATIONS WITH p-LAPLACIAN AND DAMPING

This paper concerns the oscillation of solutions of the second order nonlinear dynamic equation with p-Laplacian and damping（r（t）φ（x^△（t））^△＋p（t）φα（x^△α（t）＋q（t）f（xδ（t））=0on a time scale T which is unbounded above. Sign changes are allowed for the coefficient functions r, p and q. Several examples are given to illustrate the main results.     

Global Poincaré Inequalities on the Heisenberg Group and Applications

Let f be in the localized nonisotropic Sobolev space Wloc^1,p （H^n） on the n-dimensional Heisenberg group H^n = C^n ×R, where 1≤ p ≤ Q and Q = 2n ＋ 2 is the homogeneous dimension of H^n. Suppose that the subelliptic gradient is gloablly L^p integrable, i.e., fH^n ｜△H^n f｜^p du is finite. We prove a Poincaré inequality for f on the entire space H^n. Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of C0^∞（H^n） under the norm of （∫H^n ｜f｜ Qp/Q-p）^Q-p/Qp ＋ （∫ H^n ｜△H^n f｜^p）^1/p. We will also prove that the best constants and extremals for such Poincaré inequalities on H^n are the same as those for Sobolev inequalities on H^n. Using the results of Jerison and Lee on the sharp constant and extremals for L^2 to L（2Q/Q-2） Sobolev inequality on the Heisenberg group, we thus arrive at the explicit best constant for the aforementioned Poincaré inequality on H^n when p=2. We also derive the lower bound of the best constants for local Poincaré inequalities over metric balls on the Heisenberg group H^n.     

某个用Carlson-Shaffer算子定义的解析函数子类的性质(英文)

Let Hn（p）be the class of functions of the form f（z）=z p＋ ＋∞ Σ k=n akzk＋p,which are analytic in the open unit disk U={z：｜z｜1}.In the paper,we introduce a new subclass Bn（μ,a,c,α,p;φ）of Hn（p）and investigate its subordination relations,inclusion relations and distortion theorems.The results obtained include the related results of some authors as their special case.     

Boundedness of solutions for a class of oscillating potentials without the twist condition

In this paper, we prove that the second order differential equation d^2x/dt^2＋x^2n_1f（x）＋p（t）=0with p（t ＋ 1） = p（t）, f（x ＋ T） = f（x） smooth and f（x） 〉 0, possesses Lagrangian stability despite of the fact that the monotone twist condition is violated.     

SOME POLYNOMIAL INEQUALITIES IN THE COMPLEX DOMAIN

P（z）=∑v=0^n cvz^v
be a polynomial of degree n and let M（f, r） = max｜z｜=r ｜f（z） ｜ for an arbitrary entire function f（z）. If P（z） has no zeros in ｜z｜ 〈 1 with M（P,1） = 1, then for ｜α｜ 〈 1, it is proved by Jain[Glasnik Matematicki, 32（52） （1997）, 45-51] that
｜P（Rz）＋α（R＋1/2）^nP（z）｜≤1/2{｜1＋α（R＋1/2）^n｜＋｜R^n＋α（（R＋1/2）^n｜},R≥1,｜z｜=1.
In this paper, we shall first obtain a result concerning minimum modulus of polynomials and next improve the above inequality for polynomials with restricted zeros. Our result improves the well known inequality due to Ankeny and Rivlin and besides generalizes some well known polynomial inequalities proved by Aziz and Dawood.     

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.     

复指数函数系在$L_{\alpha}^{p}$空间中的完备性

设函数 $\alpha(t)$在$\bf R$上非负连续 和 $1\le{p}<+{\infty}$, 则 $L_{\alpha}^p=\{f: \int_{-{\infty}}^{\infty}|f(t)e^{-\alpha(t)}|^p\mathrm{d}t<{\infty}\}$ 是Banach空间. 本文中我们得到了一个复指数函数系在$L_{\alpha}^{p}$ 空间中稠密的充分必要条件.     

从$A^{p}_{\alpha}$到$A^{\infty}(\varphi)$的加权复合算子

设$u \in H(D), \ \phi$为$D$上的解析自映射,定义$H(D)$上的加权复合算子为$u C_{\phi}(f)=$$uf\circ\phi$, \ $f\in H(D)$.本文得到了从$A^{p}_{\alpha}$到$A^{\infty}(\varphi)\ (A_{0}^{\infty}(\varphi))$的加权复合算子$u C_{\phi}$的有界性和紧性的充要条件.     

加权Dirichlet 空间上的一般Ces$\grave{a}$ro算子

对加权Dirichlet空间${\cal D}_{\alpha}=\left\{f\in H(D) ; ||f||_{{\cal D}_{\alpha}}^{2}=|f(0)|^{2}+\int_{D}|f'(z)|^{2}(1-|z|)^{\alpha}\d m(z)<+\infty \right\},~~-1<\alpha<+\infty,$我们研究了其上一般Ces$\grave{a}$ro算子的有界性. 此处$H(D)$表示复平面单位圆盘$D$上全纯函数的全体.     

$L^{p}_{[0,1]}$空间M\"{u}ntz有理逼近的Jackson型估计 (英)

设$\Lambda=\{\lambda_{n}\}_{n=1}^{\infty}$为正的实数数列, 且当$n\rightarrow\infty$时, 有$\lambda_{n}\searrow 0$.本文给出了当 $\lambda_{n}\leq Mn^{-\frac{1}{2}},\;n=1,2, \cdots ,$(其中$M>0$为一正常数)时M\"{u}ntz系统$\{x^{\lambda_n}\}$的有理函数在$ L_{[0,1]} ^{p}$空间的逼近速度,主要结论为$R_{n} (f, \Lambda )_{L^{p}}\leq C_M \omega (f, n^{-\frac{1}{2}})_{L^{p}},\;1 \leq p \leq \infty.$     

半拟线性p-Laplacian问题结点解的存在性

In this paper, we study the existence of nodal solutions for the following problem:-(φ_p(x′))′= α(t)φ_p(x~+) + β(t)φ_p(x~-) + ra(t)f(x), 0 t 1,x(0) = x(1) = 0,where φ_p(s) = |s|~(p-2)s, a ∈ C([0, 1],(0, ∞)), x~+= max{x, 0}, x~-=- min{x, 0}, α(t), β(t) ∈C[0, 1]; f ∈ C(R, R), sf(s) 0 for s ≠ 0, and f_0, f_∞∈(0, ∞), where f_0 = lim_|s|→0f(s)/φ_p(s), f_∞ = lim|s|→+∞f(s)/φ_p(s).We use bifurcation techniques and the approximation of connected components to prove our main results.     

BERGMAN TYPE OPERATOR ON MIXED NORM SPACES WITH APPLICATIONS

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沿平坦凸曲线Hilbert变换的L2有界性

考虑一类平坦凸曲线的Hilbert变换■.对于平坦凸曲线,给出H_γ是L~2有界的一些必要条件.     

在Freud正交多项式零点处的Gr\"{u}nwald插值算子的收敛性

设$W_{\beta}(x)=\exp(-\frac{1}{2}|x|^{\beta})~(\beta > 7/6)$ 为Freud权, Freud正交多项式定义为满足下式$\int_{- \infty}^{\infty}p_{n}(x)p_{m}(x)W_{\beta}^{2}(x)\rd x=\left \{ \begin{array}{ll} 0 & \hspace{3mm} n \neq m , \\ 1 & \hspace{3mm}n = m \end{array} \right.$的     

THE ASYMPTOTIC BEHAVIOUR OF ANALYTIC FUNCTIONS

AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2