$p$ -Adic Haar多分辨分析与伪微分算子
$p$-Adic Haar multiresolution analysis and pseudo-differential operators
摘要：
引入了{ \em $p$ -adic多分辨分析（MRA）}的概念。我们讨论一个“自然”的加细方程，其解（一个加细函数）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$p^{-1}$的互不相交的圆盘的$p$特征函数之和的事实。该细化方程生成MRA。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1-周期的，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。所有这些基地都有描述。利用一维MRA的张量积构造了${\cL}^2(\bQ_2^n)$的多维2-adic Haar正交基。给出了多维$p$ -进小波是伪微分算子本征函数的一个判据。我们还证明了这些小波是Taibleson多维分数算子的本征函数。这些事实为在应用中密集使用我们的基础创造了必要的先决条件。
英文摘要：
The notion of {\em $p$-adic multiresolution analysis (MRA)} is introduced. We discuss 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 a sum of $p$ characteristic functions of mutually disjoint discs of radius $p^{-1}$. This refinement equation generates a MRA. 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 1-periodic, 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 described. We also constructed multidimensional 2-adic Haar orthonormal bases for ${\cL}^2(\bQ_2^n)$ by means of the tensor product of one-dimensional MRAs. A criterion for a multidimensional $p$-adic wavelet to be an eigenfunction for a pseudo-differential operator is derived. We proved also that these wavelets are eigenfunctions of the Taibleson multidimensional fractional operator. These facts create the necessary prerequisites for intensive using our bases in applications.
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