Node insertion in Coalescence Fractal Interpolation Function

The Iterated Function System (IFS) used in the construction of Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) depends on the interpolation data. The insertion of a new point in a given set of interpolation data is called the problem of node insertion. In this paper, the effect of insertion of new point on the related IFS and the Coalescence Fractal Interpolation Function is studied. Smoothness and Fractal Dimension of a CHFIF obtained with a node are also discussed.     

Fractal Interpolation on a Torus

A very general method of fractal interpolation on T ¹ is proposed in the first place. The approach includes the classical cases using trigonometric functions, periodic splines, etc. but, at the same time, adds a diversity of fractal elements which may be more appropriate to model the complexity of some variables. Upper bounds of the committed error are provided. The arguments avoid the use of derivatives in order to handle a wider framework. The Lebesgue constant of the associated partition plays a key role. The procedure is proved convergent for the interpolation of specific functions with respect to some nodal bases. In a second part, the approximation is then extended to bidimensional tori via tensor product of interpolation spaces. Some sufficient conditions for the convergence of the process in the Fourier case are deduced.      

Box-counting dimensions of fractal interpolation surfaces derived from fractal interpolation functions

A construction method of Fractal Interpolation Surfaces on a rectangular domain with arbitrary interpolation nodes is introduced. The variation properties of the binary functions corresponding to this type of fractal interpolation surfaces are discussed. Based on the relationship between Box-counting dimension and variation, some results about Box-counting dimension of the fractal interpolation surfaces are given.     

分形插值函数图象的Hausdorff维数

本文给出了插值点为的分形插值函数图象的Hausdorff维数的下界估计     

Interpolation for Function Spaces Related to Mixed Boundary Value Problems

Interpolation theorems are proved for Sobolev spaces of functions on nonsmooth domains with vanishing trace on a part of the boundary.     

三角剖分上的仿射迭代函数系统分形插值曲面

对二维平面上三角形区域进行三角剖分,构造仿射变换,由二元分形插值函数引入第三维的值,构成迭代函数系统(IFS).利用此IFS构造了一类山状分形插值曲面.通过数值实验对比分析表明:它比人们以往用矩形剖分得出的分形图形效果更逼真,更接近于自然.     

二元线性样条函数插值

二元样条函数插值在计算几何与计算机辅助几何设计中有着重要的作用.本文给出了一种矩形剖分上二元线性样条函数进行Lagrange插值时插值适定结点组所满足的拓扑与几何性质,这种性质依赖于二元线性样条函数所决定的分片线性代数曲线.     

A constructive approach to cubic Hermite Fractal Interpolation Function and its constrained aspects

The theory of splines is a well studied topic, but the kinship of splines with fractals is novel. We introduce a simple explicit construction for a -cubic Hermite Fractal Interpolation Function (FIF). Under some suitable hypotheses on the original function, we establish a priori estimates (with respect to the L _p -norm, 1≤p≤∞) for the interpolation error of the -cubic Hermite FIF and its first derivative. Treating the first derivatives at the knots as free parameters, we derive suitable values for these parameters so that the resulting cubic FIF enjoys global smoothness. Consequently, our method offers an alternative to the standard moment construction of -cubic spline FIFs. Furthermore, we identify appropriate values for the scaling factors in each subinterval and the derivatives at the knots so that the graph of the resulting -cubic FIF lies within a prescribed rectangle. These parameters include, in particular, conditions for the positivity of the cubic FIF. Thus, in the current article, we initiate the study of the shape preserving aspects of fractal interpolation polynomials. We also provide numerical examples to corroborate our results.     

基于分形法的岩石断裂面粗糙度研究

从分形几何角度详细综述了近年来国内外在研究岩石断裂面粗糙度方面的成果.分别阐述了分形中的盒维数法、小岛法、分形插值法和多重分形法,并且剖析了每种方法在理论与试验方面的优势与不足.在此基础上进行总结与归纳,并提出了对今后岩石断裂面形貌学研究的三点展望.     

Rearrangement-Function Inequalities and Interpolation Theory

Using the tools of Real Interpolation Theory, we develop a general method for proving rearrangement-function inequalities for important classes of operators.     

矩形域上分形插值研究

该文给出了矩形域上分形插值数学模型, 分形插值曲面的计算公式, 证明了分形插值曲面迭代函数系唯一性定理, 导出了分形插值曲面的维数定理,并应用实际数据进行了分形插值曲面的实例研究. 为工程中长期寻求的粗糙表面模拟提供了理论基础和实用方法.     

一类分形插值函数的分数阶微积分

介绍了分形插值函数和迭代函数系统以及v阶黎曼-刘维尔分数阶积分、微分的概念和相关定理.由于分形插值函数满足应用分数阶微积分处理问题的条件,所以利用这些概念及分步积分的方法讨论了折线段分形插值函数的分数阶积分的连续性,可微性及哪些点是不可微的,进一步说明了该插值函数分数阶微分的连续性并指出其不连续点,用黎曼-刘维尔分数阶微积分与分形插值函数结合起来研究,目的是想设法跟经典微积分一样,能找出函数上在该点的微积分的具体的实际应用意义.这些理论为研究分形插值函数的分数阶微积分的实际应用意义提供了一些理论基础.     

分形插值函数的分数阶微积分的分形维数

证明了线性分形插值函数的Riemann-Liouville分数阶微积分仍然是线性分形插值函数.在基于线性分形插值函数有关讨论的基础上,证明了线性分形插值函数的Box维数与Riemann-.Liouville分数阶微积分的阶之间成立着线性关系.文中给出的例子的图像和数值结果更进一步说明了这个结论.     

Quantum logics with the Riesz Interpolation Property

We study the quantum logics which satisfy the Riesz Interpolation Property. We call them the RIP logics. We observe that the class of RIP logics is considerable large—it contains all lattice quantum logics and, also, many (infinite) non‐lattice ones. We then find out that each RIP logic can be enlarged to an RIP logic with a preassigned centre. We continue, showing that the “nearly” Boolean RIP logics must be Boolean algebras. In a somewhat surprising contrast to this, we finally show that the attempt for the σ‐complete formulation of this result fails: We show by constructing an example that there is a non‐Boolean nearly Boolean σ‐RIP logic. As a result, there are interesting σ‐RIP logics which are intrinsically close to Boolean σ‐algebras. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fundamental Sets of Fractal Functions

Fractal interpolants constructed through iterated function systems prove more general than classical interpolants. In this paper, we assign a family of fractal functions to several classes of real mappings like, for instance, maps defined on sets that are not intervals, maps integrable but not continuous and may be defined on unbounded domains. In particular, based on fractal interpolation functions, we construct fractal Müntz polynomials that successfully generalize classical Müntz polynomials. The parameters of the fractal Müntz system enable the control and modification of the properties of original functions. Furthermore, we deduce fractal versions of classical Müntz theorems. In this way, the fractal methodology generalizes the fundamental sets of the classical approximation theory and we construct complete systems of fractal functions in spaces of continuous and p-integrable mappings on bounded domains. This work is supported by the project No: SB 2005-0199, Spain.     

插值法在数据修正中的应用

为了使评估的结果达到某种规定的水平,本文研究了运用线性插值、拉格朗日插值以及牛顿插值方法对某公司员工考核数据按照一定的规则进行了修正,同时,对各种方法的修正前、后的结果做了比较．结果表明拉格朗日插值法效果最好,但是计算量偏大;线性插值法虽然效果一般,但是计算复杂度却较低;而牛顿插值法达不到我们预期的效果．     

Fractal Haar system

The Haar system is an alternative to the classical Fourier bases, being particularly useful for the approximation of discontinuities. The article tackles the construction of a set of fractal functions close to the Haar set. The new system holds the property of constitution of bases of the Lebesgue spaces of p-integrable functions on compact intervals. Likewise, the associated fractal series of a continuous function is uniformly convergent. The case p=2 owns some peculiarities and is studied separately.     

On the Parameter Identification Problem in the Plane and the Polar Fractal Interpolation Functions

Fractal interpolation functions provide a new means for fitting experimental data and their graphs can be used to approximate natural scenes. We first determine the conditions that a vertical scaling factor must obey to model effectively an arbitrary function. We then introduce polar fractal interpolation functions as one fractal interpolation method of a non-affine character. Thus, this method may be suitable for a wider range of applications than that of the affine case. The interpolation takes place in polar coordinates and then with an inverse non-affine transformation a simple closed curve arises as an attractor which interpolates the data in the usual plane coordinates. Finally, we prove that this attractor has the same Hausdorff dimension as the polar one.     

最简型的Hermite插指

本文提出了Hermite插值问题的一种新形式，幂指数形式，简称Hermite插指。     

Interpolation on lattices generated by cubic pencils

Principal lattices are distributions of points in the plane obtained from a triangle by drawing equidistant parallel lines to the sides and taking the intersection points as nodes. Interpolation on principal lattices leads to particularly simple formulae. These sets were generalized by Lee and Phillips considering three-pencil lattices, generated by three linear pencils. Inspired by the addition of points on cubic curves and using duality, we introduce an addition of lines as a way of constructing lattices generated by cubic pencils. They include three-pencil lattices and then principal lattices. Interpolation on lattices generated by cubic pencils has the same good properties and simple formulae as on principal lattices. Dedicated to C.A. Micchelli for his mathematical contributions and friendship on occasion of his sixtieth birthday Mathematics subject classifications (2000) 41A05, 41A63, 65D05. J.M. Carnicer: Partially supported by the Spanish Research Grant BFM2003-03510, by Gobierno de Aragón and Fondo Social Europeo.