多尺度分析生成元的刻画

本文将给出多尺度分析生成元的一种完全刻画.将证明:函数φ∈L~2(R)是二进多尺度分析生成元的充要条件是(1)存在{a_k}∈l~2,φ(x)=∑_(k∈Z)a_kφ(2x-k);(2)存在正数A

Characterization of Generators for Multiresolution Analysis

We give a complete characterization of generators for multiresolution analysis. Precisely, we prove the following results: $\phi\in L^2(\mathbb{R})$ is a genarator of a dyadic multireso-lution if and only if (1) there exists $\{a_k\}\in l^2$ such that $\phi(x)=\sum_{k\in \mathbb{Z}}{a_k \phi(2x-k)};$ (2) there exists positive numbers $A$ and $B$ such that $A \leq \Phi(\omega)\leq B$, a.e., where $\Phi(\omega)=\sum_{l\in \mathbb{Z}}{|\hat{\phi}(\omega+2l\pi)|}^2;$ (3) the function $F(x,y)=\frac{1}{y-x}\int_x^y{|\hat{\phi}(\omega)|}^2d\omega$ is dyadicly away from zero at the origin.