一种分形插值曲线上的调和结构和有效电阻

Harmonic structure and effective resistance on a fractal interpolation curve

为了研究分形插值曲线上的调和分析问题,根据分形曲线的生成特点,从其自相似结构出发,在一种自仿射分形插值曲线上,通过边界点集上自相似拉普拉斯变换的构造,建立了插值曲线上的电阻网络,运用有限网络中拉普拉斯变换限制的方法,给出了相应的调和结构.结果表明该分形集的调和结构是由满足一定条件的比例系数与二阶对称矩阵构成的.同时从电学角度定义了顶点集中任意两点间与拉普拉斯变换相关的有效电阻,这样定义的有效电阻是顶点集上的一个度量,从而得到了此类插值曲线上一个新的度量空间.

In order to study the problem of harmonic analysis on fractal interpolation curve,its self-similar structure was considered according to the characteristics from creating curve.A sequence of resistance networks on a self-affine fractal interpolation curve was established by the construction of similar Laplacian on boundary points,and through limiting the Laplacian of network the corresponding harmonic structure was given.The results indicate that scale factor qualified some conditions and a symmetrical matrix form the fractal harmonic structure.At the same time,from the electrical point of view,the concept of effective resistance associated with the Laplacian between any two points was defined.This effective resis-tance is a metric on boundary points thus there′s a new metric space on this interpolation curve.