连续与离散域中的广义差分理论
A Theory of General Difference in Continuous and Discrete Domain
AI总结：
本文提出Tao广义差分（TGD），通过三个关键约束将微分推广到有限区间，并给出旋转与正交两种算子构造方法，在连续和离散域中实现统一卷积模式下的抗噪声精确差分。
摘要：
尽管数值差分算法是数字时代的核心要素，但其容易受到噪声影响。这一问题源于连续微分中的无穷小量与离散对应物中的有限区间之间的关键脱节。这种脱节违背了微分的基本定义（莱布尼茨与柯西）。为了弥合这一差距，我们构建了一种新颖的广义差分（Tao广义差分，TGD）。TGD不再采用通过积分求导的方式，而是通过三个关键约束将微分推广到连续域中的有限区间。这使得我们能够通过由序列构造的连续阶跃函数来计算离散域中序列的广义差分。我们提出了两种构造方法，即旋转构造和正交构造，来构造TGD的算子。所构造的TGD算子在计算任意维度的连续函数、离散序列和数组时采用相同的卷积模式。我们通过示例运算进行的分析展示了TGD在连续域和离散域中的能力，为数字时代实现精确且抗噪声的微分铺平了道路。
英文摘要：
Though a core element of the digital age, numerical difference algorithms struggle with noise susceptibility. This stems from a key disconnect between the infinitesimal quantities in continuous differentiation and the finite intervals in its discrete counterpart. This disconnect violates the fundamental definition of differentiation (Leibniz and Cauchy). To bridge this gap, we build a novel general difference (Tao General Difference, TGD). Departing from derivative-by-integration, TGD generalizes differentiation to finite intervals in continuous domains through three key constraints. This allows us to calculate the general difference of a sequence in discrete domain via the continuous step function constructed from the sequence. Two construction methods, the rotational construction and the orthogonal construction, are proposed to construct the operators of TGD. The construction TGD operators take same convolution mode in calculation for continuous functions, discrete sequences, and arrays across any dimension. Our analysis with example operations showcases TGD's capability in both continuous and discrete domains, paving the way for accurate and noise-resistant differentiation in the digital era.
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。
