Two-scale difference equations I: existence and global regularity of solutions
Two-scale difference equations I: existence and global regularity of solutions
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DOI：
10.1137/0522089
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发表时间：
1991-09
影响因子：
2
通讯作者：
I. Daubechies;J. Lagarias
中科院分区：
文献类型：
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作者：
I. Daubechies;J. Lagarias
双尺度差分方程是形式为$f(X)=\sum_{n=0}^N c_n f(αx-β_n)$的函数方程，其中$α和$β_0是实常数，$c_n是复常数。这些方程的解出现在样条论中，出现在构造曲线的插值法中，出现在构造紧支集的小波中，出现在构造分形图中，出现在概率论中。本文研究了这类方程$L^1$-解的存在唯一性。特别地，它刻画了$L^1$-具有紧支集的解。介绍了一种研究这类方程特例的时间域方法，其中α，β0，cdots，βn为整数，我们称之为格型双尺度差分方程组。证明了如果格型双尺度差分方程解在$C^m(\mathbb{R})$中有紧支集，则$m<；{{(\beta_n-\beta_0)}/{(\α-1)}}-1$.
A two-scale difference equation is a functional equation of the form $f(x) = \sum _{n = 0}^N c_n f(\alpha x - \beta _n )$, where $\alpha > 1$ and $\beta _0 < \beta _1 <\cdots <\beta _n $, are real constants, and $c_n $ are complex constants. Solutions of such equations arise in spline theory, in interpolation schemes for constructing curves, in constructing wavelets of compact support, in constructing fractals, and in probability theory. This paper studies the existence and uniqueness of $L^1 $-solutions to such equations. In particular, it characterizes $L^1 $-solutions having compact support. A time-domain method is introduced for studying the special case of such equations where $\{ {\alpha ,\beta _0 , \cdots ,\beta _n } \}$ are integers, which are called lattice two-scale difference equations. It is shown that if a lattice two-scale difference equation has a compactly supported solution in $C^m (\mathbb{R})$, then $m < {{(\beta _n - \beta _0 )} / {(\alpha - 1)}} - 1$.