General interpolation on the lattice h{\Bbb Z}^s: Compactly supported fundamental solutions

Summary. In this paper we combine an earlier method developed with K. Jetter on general cardinal interpolation with constructions of compactly supported solutions for cardinal interpolation to gain compactly supported fundamental solutions for the general interpolation problem. The general interpolation problem admits the interpolation of the functional and derivative values under very weak restrictions on the derivatives to be interpolated. In the univariate case, some known general constructions of compactly supported fundamental solutions for cardinal interpolation are discussed together with algorithms for their construction that make use of MAPLE. Another construction based on finite decomposition and reconstruction for spline spaces is also provided. Ideas used in the latter construction are lifted to provide a general construction of compactly supported fundamental solutions for cardinal interpolation in the multivariate case. Examples are provided, several in the context of some general interpolation problem to illustrate how easy is the transition from cardinal interpolation to general interpolation. Received May 11, 1993 / Revised version received August 16, 1994     

Description of the interpolation spaces for a pair of local Morrey-type spaces and their generalizations

The real interpolation method is considered and it is proved that for general local Morrey-type spaces, in the case in which they have the same integrability parameter, the interpolation spaces are again general local Morrey-type spaces with appropriately chosen parameters. This result is a particular case of the interpolation theorem for much more general spaces defined with the help of an operator acting from some function space to the cone of nonnegative nondecreasing functions on (0, ∞). It is also shown how the classical interpolation theorems due to Stein-Weiss, Peetre, Calderón, Gilbert, Lizorkin, Freitag and some of their new variants can be derived from this theorem.     

三元重心Thiele型混合有理插值

有理插值比多项式插值有更好的近似,但有理插值一般很难控制极点的产生.基于Thiele型连分式插值与重心有理插值,构造三元重心Thiele型混合有理插值,当选取适当的权后能避免部分极点的产生.文章最后通过数值例子验证了这种方法的正确性和有效性.     

Band-Limited Wavelets

Using the harmonic method,we get a class of more general band-limited wavelet,which was got by complicate operator interpolation method before.Our result is alittle better than the result by operator interpolation method.The fast band-limitedwavelet transform shall be given in another paper.     

三角形REISSNER-MINDLIN板元

本文提出构造无自锁现象的Reissuer－Mindlin板元的一个一般性方法.此方法将剪切应变用它的适当的插值多项式代替,当板厚趋于零时这对应于插值点的Kirchhoff条件,因而单元无自锁现象.根据这种方法我们构造两个三角形元──一个3节点元和一个6节点元,并给出数值结果.     

Krattenthaler反演公式的简短证明

为了提出研究Lagrange反演关系的统一方法,Krattenthaler建立了算子方法,并由此得到一个广泛的反演关系:Gould-Hsu-Carliz-Krattenthaler反演公式.本文利用Lagrange插值公式给出它的一个简短证明,进一步地证明在Lagrange插值的意义下,该反演关系是唯一性的.     

A Short Proof of Krattenthaler Formulas

With an effort to investigate a unified approach to the Lagrange inverse Krattenthaler established operator method we finally found a general pair of inverse relations, called the Krattenthaler forumlas. The present paper presents a very short proof of this formula via Lagrange interpolation. Further, our method of proof declares that the Krattenthaler result is unique in the light of Lagrange interpolation. Received April 9, 1999, Accepted December 7, 2000     

High order compact schemes for gradient approximation

In this paper, we propose three gradient recovery schemes of higher order for the linear interpolation. The first one is a weighted averaging method based on the gradients of the linear interpolation on the uniform mesh, the second is a geometric averaging method constructed from the gradients of two cubic interpolation on macro element, and the last one is a local least square method on the nodal patch with cubic polynomials. We prove that these schemes can approximate the gradient of the exact solution on the symmetry points with fourth order. In particular, for the uniform mesh, we show that these three schemes are the same on the considered points. The last scheme is more robust in general meshes. Consequently, we obtain the superconvergence results of the recovered gradient by using the aforementioned results and the supercloseness between the finite element solution and the linear interpolation of the exact solution. Finally, we provide several numerical experiments to illustrate the theoretical results.     

A basis for the operational method of finding interpolation polynomials

A rigorous basis is obtained for the operational method of obtaining interpolation polynomials. A general method is described for deriving forms of interpolation polynomials involving given specific differences.Translated from Matematicheskie Zametki, Vol. 10, No. 2, pp. 243–247, August, 1971.The author wishes to thank A. D. Gorbunov for suggesting this problem.     

Example of an interpolation domain

Cahen et al. (J. Algebra 225 (2000) 794), have defined a domain to be an interpolation domain if, essentially, Lagrange interpolation can be done using integer-valued polynomials. They prove results indicating that in some cases every overring of an interpolation domain is again an interpolation domain, and they ask whether the statement holds in general. In the present note, we provide a counterexample to the general statement.     

Harnack Estimates for Weak Solutions to a Singular Parabolic Equation

By an interpolation method, an intrinsic Harnack estimate and some supestimates are established for nonnegative solutions to a general singular parabolic equation.     

Error estimations for some meshless boundary interpolation methods

Meshless methods have become quite popular in numerical treatment of partial differential equations because of their simplicity and the fact that they require neither domain nor boundary mesh. In general, however, they convert the original problem to a highly ill‐conditioned linear system of algebraic equations with a dense matrix. Recently, a special technique has been proposed which circumvents this computational difficulty. This method, called Direct Multi‐Elliptic Interpolation Method, is based on a scattered data interpolation which defines the interpolation function as a solution of a higher order multi‐elliptic equation. Here the boundary version of this meshless method which is based on a multi‐elliptic boundary interpolation is considered. Error estimations are derived justifying the interpolation function to be a good approximation of the solution of the original boundary value problem as well. At the same time, the problem of large, dense and ill‐conditioned matrices as well as the mesh generation are completely avoided. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

有限元超收敛新论

本文从三个方面讨论二阶椭圆问题有限元超收敛．1．一致网格上的新超收敛结果．利用新的“投影型插值”,我们解决了高次三角形元的超收敛问题．2．一般网格的超收敛性．利用局部插值处理和局部磨光处理我们获得了整体超收敛性结果．3．关于当前的两种超收敛技巧．Cornell学派利用一个精致的内估计和网格的点对称性,获得了一个“普遍”的结果,中国学派利用两个基本估计和离散Green函数理论获得了令人满意的结果,两者均很复杂．本文综合了两个学派的方法,简洁地证得上述普遍结果．     

On interpolation properties of compact bilinear operators

We investigate interpolation properties of compact bilinear operators under the general real method. Results apply in particular to the real method with a function parameter and also to the complex method in the special case when the couple in the target reduces to a single Banach space or the source is a product of two fixed Banach spaces.     

An Improved Linearity-Preserving Cell-Centered Scheme for Nonlinear Diffusion Problems on General Meshes

In this paper, we suggest a new vertex interpolation algorithm to improve an existing cell-centered finite volume scheme for nonlinear diffusion problems on general meshes. The new vertex interpolation algorithm is derived by applying a special limit procedure to the well-known MPFA-O method. Since the MPFA-O method for 3D cases has been addressed in some studies, the new vertex interpolation algorithm can be extended to 3D cases naturally. More interesting is that the solvability of the corresponding local system is proved under some assumptions. Additionally, we modify the edge flux approximation by an edge-based discretization of diffusion coefficient, and thus the improved scheme is free of the so-called numerical heat-barrier issue suffered by many existing cell-centered or hybrid schemes. The final scheme allows arbitrary continuous or discontinuous diffusion coefficients and can be applicable to arbitrary star-shaped polygonal meshes. A second-order convergence rate for the approximate solution and a first-order accuracy for the flux are observed in numerical experiments. In the comparative experiments with some existing vertex interpolation algorithms, the new algorithm shows obvious improvement on highly distorted meshes.     

Stokes问题Q_2-P_1混合元外推方法

考虑Stokes问题的有限元解与精确解插值的Q2-P1混合元的渐近误差展开和分裂外推.首先利用积分恒等式技巧确定了微分方程精确解与有限元插值之间积分式的主项,其次再借助插值后处理和分裂外推技术,得到了比通常的误差估计提高两阶的收敛速度.     

基于有理函数模型的一维最优化方法

在本文中提出了基于有理函数模型的一维最优化方法。这些方法比二次模型方法有较好的数值性态和适应性。我们给出了有理反差商方法和Ｎｅｖｉｌｅ型方法，并将其与二次插值方法进行了数值比较。     

距离空间中的神经网络插值与逼近

已有的关于插值神经网络的研究大多是在欧氏空间中进行的,但实际应用中的许多问题往往需要用非欧氏尺度进行度量.本文研究一般距离空间中的神经网络插值与逼近问题,即先在距离空间中构造新的插值网络,然后在此基础上构造近似插值网络,最后研究近似插值网络对连续泛函的逼近.     

About measures and nodal systems for which the Hermite interpolants uniformly converge to continuous functions on the circle and interval

We obtain the Laurent polynomial of Hermite interpolation on the unit circle for nodal systems more general than those formed by the n-roots of complex numbers with modulus one. Under suitable assumptions for the nodal system, that is, when it is constituted by the zeros of para-orthogonal polynomials with respect to appropriate measures or when it satisfies certain properties, we prove the convergence of the polynomial of Hermite-Fejér interpolation for continuous functions. Moreover, we also study the general Hermite interpolation problem on the unit circle and we obtain a sufficient condition on the interpolation conditions for the derivatives, in order to have uniform convergence for continuous functions.Finally, we obtain some improvements on the Hermite interpolation problems on the interval and for the Hermite trigonometric interpolation.     

On general Hermite trigonometric interpolation

Summary A sequence of general Hermite trigonometric interpolation polynomials with equidistant interpolation points is given. Integrating these interpolation formulae a sequence of quadrature formulae for the integration of periodic functions is obtained. Derivative-free remainders are stated for these interpolation and quadrature formulae.This work was done at the Max-Planck-Institut für Physik und Astrophysik, München.