Functional calculus under the Tadmor-Ritt condition, and free interpolation by polynomials of a given degree

For Banach space operators T satisfying the Tadmor-Ritt condition ||(zI−T)⁻¹||?C|z−1|⁻¹, |z|>1, we prove that the best-possible constant C_T(n) bounding the polynomial calculus for T, ||p(T)||?C_T(n)||p||_∞, deg(p)?n, behaves (in the worst case) as as n→∞. This result is based on a new free (Carleson type) interpolation theorem for polynomials of a given degree.     

带扩散项四阶抛物方程的一类本性并行差分格式

给出逼近带扩散项四阶抛物方程一组非对称差分格式,对此组非对称格式重新组合,得到了一类新的具有并行本性的算法.随后,利用矩阵法证明了算法的绝对稳定性.最后给出数值实验.     

On -norms of meromorphic functions with fixed poles

We study boundedness of the differentiation and embedding operators in the shift-coinvariant subspaces generated by Blaschke products with sparse zeros, that is, in the spaces of meromorphic functions with fixed poles in the lower half-plane endowed with -norm. We answer negatively the question of K.M. Dyakonov about the necessity of the condition for the boundedness of the differentiation on . Our main tool is a construction of an unconditional basis of rational fractions in .

     

Perturbation of the trigonometric system in L 1(0, π)

In this paper, we study a special perturbation of the function system obtained from the Fejér kernel. It is shown how this relates to the stability of bases and complete systems as well to the stability of the trigonometric system. An approximation algorithm in systems resulting from a perturbation of the original system is given.     

A high‐order finite difference method for 1D nonhomogeneous heat equations

In this article a sixth‐order approximation method (in both temporal and spatial variables) for solving nonhomogeneous heat equations is proposed. We first develop a sixth‐order finite difference approximation scheme for a two‐point boundary value problem, and then heat equation is approximated by a system of ODEs defined on spatial grid points. The ODE system is discretized to a Sylvester matrix equation via boundary value method. The obtained algebraic system is solved by a modified Bartels‐Stewart method. The proposed approach is unconditionally stable. Numerical results are provided to illustrate the accuracy and efficiency of our approximation method along with comparisons with those generated by the standard second‐order Crank‐Nicolson scheme as well as Sun‐Zhang's recent fourth‐order method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

整函数插值序列的稳定性

本文证明了指数π型整函数Bπ，p，1＜p＜∞中某些实插值序列的稳定性，从而证明了存在一个和p有关的正常数δp，使得当sup|uj－j＜δp，u0＝0时，｛Gj（x）｝j∈Z为Bπ，p，1＜p＜∞的一组无条基，其中并且我们还证得是Bπ，p，1＜p＜∞的一组稳定的无条件基．     

High‐order difference scheme for the solution of linear time fractional klein–gordon equations

In this article, we apply a high‐order difference scheme for the solution of some time fractional partial differential equations (PDEs). The time fractional Cattaneo equation and the linear time fractional Klein–Gordon and dissipative Klein–Gordon equations will be investigated. The time fractional derivative which has been described in the Caputo's sense is approximated by a scheme of order , and the space derivative is discretized with a fourth‐order compact procedure. We will prove the solvability of the proposed method by coefficient matrix property and the unconditional stability and ‐convergence with the energy method. Numerical examples demonstrate the theoretical results and the high accuracy of the proposed scheme. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1234–1253, 2014     

A nonuniform L2 formula of Caputo derivative and its application to a fractional Benjamin–Bona–Mahony-type equation with nonsmooth solutions

To recover the full accuracy of discretized fractional derivatives, nonuniform mesh technique is a natural and simple approach to efficiently resolve the initial singularities that always appear in the solutions of time-fractional linear and nonlinear differential equations. We first construct a nonuniform L2 approximation for the fractional Caputo's derivative of order 1 < α < 2 and present a global consistency analysis under some reasonable regularity assumptions. The temporal nonuniform L2 formula is then utilized to develop a linearized difference scheme for a time-fractional Benjamin–Bona–Mahony-type equation. The unconditional convergence of our scheme on both uniform and nonuniform (graded) time meshes are proven with respect to the discrete H¹-norm. Numerical examples are provided to justify the accuracy.     

对我国期货市场波动性的分阶段实证研究

波动性是经济和金融研究的热点问题。本文首先采用无条件波动度量方法对我国三大期货市场1997年—2004年的波动性进行了估计,发现第一阶段97年—02年期货价格总体呈下跌趋势,三大市场整体波动性不大,较高的波动性都出现在期货价格下跌时期,较低的波动性都出现在期货价格上涨时期;第二阶段03年—04年三大市场波动性显著提高,总体价格呈上升趋势,较高的波动性都出现在期货价格上涨时期,而较低的波动性都出现在期货价格下跌时期;本文进一步采用条件波动模型对我国三大期货市场两个阶段收益率与波动性的相关关系及波动性的杠杆效应进行了研究,结果表明铜期货收益率与波动性显著相关,大豆期货收益率与波动性不显著相关;我国三大期货市场均存在杠杆效应,并且两个阶段波动性的杠杆效应相反,其中铜期货市场的杠杆效应更显著。