距离函数小波.第III部分：“奇异”变换和级数
Distance function wavelets - Part III: "Exotic" transforms and series
摘要：
报告的第三部分包括各种非常规的距离函数小波（DFW）。首次将偏微分方程的维数和阶作为DFW变换和级数中尺度参数的替代，特别是在空间和时空位势问题中。注意，递归多重互易公式是DFW级数。格林第二恒等式用于避免零阶基本解在生成DFW级数时的奇异性。各种复合偏微分方程的基本解被发现是非常灵活和有效的，以处理广泛的问题。我们还讨论了潜在的连接之间的关键概念的尺寸，规模和顺序的PDE通过耗散声波传播的分析。势问题的形状参数也被用作“尺度参数”来创建非正交DFW。本文还简要地讨论和证明了各种坐标变量变换和级数的DFW对应。实际上，各向异性和非均匀DFW的开发，通过使用测地距离变量。DFW和相关的基函数也被用于构造核距离S形函数，这在人工神经网络和机器学习中是潜在有用的。与前两份报告一样，甚至更糟的是，这项研究牺牲了数学的严谨性，反过来又解放了想象力。大多数结果都是在没有严格分析的情况下直观获得的。后续研究仍在进行中。本文旨在激发更多的研究进入这一有前途的领域。
英文摘要：
Part III of the reports consists of various unconventional distance function wavelets (DFW). The dimension and the order of partial differential equation (PDE) are first used as a substitute of the scale parameter in the DFW transforms and series, especially with the space and time-space potential problems. It is noted that the recursive multiple reciprocity formulation is the DFW series. The Green second identity is used to avoid the singularity of the zero-order fundamental solution in creating the DFW series. The fundamental solutions of various composite PDEs are found very flexible and efficient to handle a borad range of problems. We also discuss the underlying connections between the crucial concepts of dimension, scale and the order of PDE through the analysis of dissipative acoustic wave propagation. The shape parameter of the potential problems is also employed as the "scale parameter" to create the non-orthogonal DFW. This paper also briefly discusses and conjectures the DFW correspondences of a variety of coordinate variable transforms and series. Practically important, the anisotropic and inhomogeneous DFW's are developed by using the geodesic distance variable. The DFW and the related basis functions are also used in making the kernel distance sigmoidal functions, which are potentially useful in the artificial neural network and machine learning. As or even worse than the preceding two reports, this study scarifies mathematical rigor and in turn unfetter imagination. Most results are intuitively obtained without rigorous analysis. Follow-up research is still under way. The paper is intended to inspire more research into this promising area.
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