$p$ -Adic Haar多分辨率分析
$p$-Adic Haar multiresolution analysis
摘要：
本文引入了{ \em $p$ -adic多分辨分析（MRA）}的概念。我们使用一个“自然”的精化方程，其解（可精化函数）是单位圆盘的特征函数。该方程反映了单位圆盘的特征函数是半径为$p^{-1}$的不相交圆盘的$p$特征函数之和的事实。对案例$p=2$进行了详细的研究。我们的MRA是真实Haar MRA的2-adic模拟。但与真实设置相反，生成我们的Haar MRA的可加细函数是周期性的，周期为1，这对于真实的可加细函数永远不成立。这一事实意味着在同一Haar MRA生成的${\cL}^2(\bQ_2)$中存在无穷多个不同的2-进正交小波基。所有这些基础都是建立起来的。由于$p$ -adic伪微分算子与小波基密切相关，因此我们的基可以被广泛地应用。
英文摘要：
In this paper, the notion of {\em $p$-adic multiresolution analysis (MRA)} is introduced. We use a ``natural'' refinement equation whose solution (a refinable function) is the characteristic function of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is the sum of $p$ characteristic functions of disjoint discs of radius $p^{-1}$. The case $p=2$ is studied in detail. Our MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real setting, the refinable function generating our Haar MRA is periodic with period 1, which never holds for real refinable functions. This fact implies that there exist infinity many different 2-adic orthonormal wavelet bases in ${\cL}^2(\bQ_2)$ generated by the same Haar MRA. All of these bases are constructed. Since $p$-adic pseudo-differential operators are closely related to wavelet-type bases, our bases can be intensively used for applications.
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