二维分形插值函数及其小波类型级数表示

本文对一类很广泛的二维分形插值函数进行了研究，给出了分形插值函数连续的充分必要条件和一种构造迭代函数系统使其吸引子是连续函数的方法，并将分形插值函数表示为一个二维小波类型级数，其“母函数”是由迭代函数系统中的位移函数所决定，这种表示方式不仅提供了一种生成分形插值函数有效方法，而且对研究分形插值函数的性质及所描述的物理对象的特性也是十分有意义。     

向量递归迭代函数系统及其不变测度

本文阐述了RIFS和IFS之间的关系,并对定义在一组紧空间上的向量递归迭代函数系统(VRIFS),定义了Markov算子,给出Markov算子是压缩算子的条件,描述了其不动点即广义递归迭代函数系统的不变测度.文中还给出了相应的拼贴定理.     

Iterated function systems on multifunctions and inverse problems

In this paper, we first consider the problem of defining IFS operators on the space of non-empty compact and convex subsets of . After defining a complete metric on , we construct an IFS operator and show some properties. A notable feature is the definition of a type of weak inner product on . We then define a family of complete metrics on the space of all measurable set-valued functions (with values in ), and extend the weak inner product to this space. Following this, we construct IFS operators on these spaces. We close with a brief discussion of the inverse problem of approximating an arbitrary multifunction by the attractor of an IFS.     

Non-spectral self-affine measure problem on the plane domain

The self-affine measure μM,D corresponding to an expanding integer matrix

     

IFS on boundaries of homogeneous trees: WQB condition and ultifractal analysis

We study, from the point of view of the multifractal analysis, iterated function systems on totally disconnected spaces, namely, the boundaries of homogeneous trees. In particular, we study in this setting the “weak quasi-Bernoulli” property introduced by Testud [3, 4]. After projection on ? or ?², we get new examples of self-similar measures which, being WQB, obey the multifractal formalism for positive q's.     

Frames built on fractal sets

Frames of finite dimensional Hilbert spaces have recently been of great interest in applications to modern communication networks transport packets. In this note, continuous and discrete frames, living on fractal sets, of both finite and infinite dimensional separable abstract Hilbert spaces are found. In particular, we find discrete frames, robust to erasures, of finite dimensional Hilbert spaces using iterated function systems.     

基于一类新的胞腔排除遗传算法求解迭代函数系逆问题

提出求解迭代函数系（IFS）逆问题的一类有效遗传算法,该算法基于新发展的可拼接/可分解编码,并结合使用胞腔排除技巧,对于典型图像的应用表明;该方法可有效应用于基于矩匹配表示的IFS逆向题求解,从而为IFS逆问题的数值方法研究提供了一条新颖途径。     

Reproducing Kernels and Coherent States on Julia Sets

We construct classes of coherent states on domains arising from dynamical systems. An orthonormal family of vectors associated to the generating transformation of a Julia set is found as a family of square integrable vectors, and, thereby, reproducing kernels and reproducing kernel Hilbert spaces are associated to Julia sets. We also present analogous results on domains arising from iterated function systems. The research of the first two authors was supported by Natural Sciences and Engineering Research Council of Canada.     

Hyperbolic dynamics in graph-directed IFS

A cone space is a complete metric space (X,d) with a pair of functions cs,cu:X×X→R, such that there exists K>0 satisfying