基于积分中值定理对信号系统中冲击函数做进一步推广

Further Generalization of the Impulse Function in Signal Systems Based on the Integral Mean Value Theorem

摘要: 冲击函数(δ函数)作为广义函数理论的核心概念，其传统构造方法主要依赖于极限过程。本文创新性地从积分中值定理的视角出发，首先通过积分第一中值定理的特殊形式，系统阐述了面积为1的矩形函数在宽度趋近于零时自然诱导出冲击函数的数学机制；进而将这一方法推广至分析“数除以零”的数学禁忌，揭示了该运算在冲击函数框架下必然导致矛盾的本质原因。研究表明，积分中值定理不仅为冲击函数提供了一种新的严格化途径，更能为理解其他奇异积分问题提供统一的分析框架。本文的结论为广义函数理论的基础研究提供了新的思路，并对相关数学物理问题的研究具有启示意义。

Abstract: The delta function, as a fundamental concept in the theory of generalized functions, has traditionally been constructed through limiting processes. This paper innovatively examines the delta function from the perspective of the integral mean value theorem. We first systematically demonstrate how a rectangular pulse function with unit area naturally induces the delta function as its width approaches zero, using a specialized form of the first integral mean value theorem. This approach is then extended to analyze the mathematical prohibition of “division by zero”, revealing how such operations inherently lead to contradictions within the delta function framework. Our research shows that the integral mean value theorem not only provides a novel rigorous approach to the delta function but also offers a unified analytical framework for understanding other singular integral problems. The conclusions present fresh insights for foundational studies in generalized function theory and carry significant implications for research in related mathematical physics problems.