距离函数小波——第一部分：亥姆霍兹与对流-扩散变换及级数
Distance function wavelets - Part I: Helmholtz and convection-diffusion transforms and series
AI总结：
本报告介绍了基于亥姆霍兹和对流-扩散方程基本解的距离函数小波（DFW）变换与级数，包括HF、HL变换及各向异性小波，用于处理不连续和跟踪数据问题，并评估任意维度线性扩散问题的解析解。
摘要：
本报告旨在介绍基于亥姆霍兹方程、修正亥姆霍兹方程和对流-扩散方程的基本解与通解的距离函数小波（DFW）的研究进展，其中包括各向同性的亥姆霍兹-傅里叶（HF）变换与级数、亥姆霍兹-拉普拉斯（HL）变换，以及各向异性的对流-扩散小波和脊波。后者旨在处理不连续和跟踪数据问题。解决了HF级数的边缘效应问题。提出并讨论了DFW变换的替代存在条件。为了简化和精简HF和HL变换的表达式，引入了一种新的依赖于维度的函数记号。HF级数还用于评估任意维度和几何形状的线性扩散问题的解析解。本报告的不足之处在于，由于作者数学知识有限，缺乏严谨的数学分析。
英文摘要：
This report aims to present my research updates on distance function wavelets (DFW) based on the fundamental solutions and the general solutions of the Helmholtz, modified Helmholtz, and convection-diffusion equations, which include the isotropic Helmholtz-Fourier (HF) transform and series, the Helmholtz-Laplace (HL) transform, and the anisotropic convection-diffusion wavelets and ridgelets. The latter is set to handle discontinuous and track data problems. The edge effect of the HF series is addressed. Alternative existence conditions for the DFW transforms are proposed and discussed. To simplify and streamline the expression of the HF and HL transforms, a new dimension-dependent function notation is introduced. The HF series is also used to evaluate the analytical solutions of linear diffusion problems of arbitrary dimensionality and geometry. The weakness of this report is lacking of rigorous mathematical analysis due to the author's limited mathematical knowledge.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
