一类基于给定数据的分形插值算法与多小波尺度函数的构造

分形插值方法为复杂现象的确定性表示提供了一种好的方法,如经济和地震学领域的数据模拟.目前在实际应用中大多基于仿射分形插值(AFIFs),插值函数具有自相似形、连续和处处不可微等特征.该文基于给定的数据类型来考虑分形插值算法,并提供相应的数值例子.特别地,用Hermite分形插值给出了一类L2(R)的紧支撑小波基的尺度函数,不同于用AFIFs建立的多尺度分析,得到的尺度函数具有可微性,能够用来建立微分方程的数值方案.

The Construction of Data-oriented Fractal Interpolation Algorithm and its Multiscale Functions

Fractal interpolation methods provide good deterministic representations of complex phenomena such as economics series,seismic data,etc.To date,many applications are based on affine fractal interpolation functions(AFIFs),which are continuous,nowhere differentiable and self-similar.In this study,we present an algorithm of Hermite fractal interpolation based on given datum,and numerical examples are presented.Especially,we construct a compactly supported wavelet basis of L~2(R) using Hermite fractal interpolation functions which is based on function values and the first derivative values. Differing form wvavelets constructed by AFIFs,scale functions have differentiability and can be used to construct numerical scheme for differential equations.