【摘要】 狄拉克δ-函数实际上是离散情况下的Kroneckerδ-函数的连续化,它在数学和物理中都有重要的应用.基于广义函数概念引入狄拉克δ-函数的精确定义,证实狄拉克δ-函数不是通常Lebesgue局部可积意义下的普通函数;文中分别以单位矩形脉冲函数、高斯函数、钟形函数和Sinc函数的序列在弱极限意义下来逼近狄拉克δ-函数.另外,验证了狄拉克δ-函数可以作为Heaviside函数的广义导数,以及其高价广义导数,并给出狄拉克δ-函数的卷积性质、伸缩性质、复合变换性质、正交性和狄拉克梳函数,最后引入了狄拉克δ-函数与广义傅里叶变换的关系,以及其在泊松方程Dirichlet边值问题求解中的应用.更多还原

【Abstract】 It is indicated that Dirac δ-function is a continuation of the discrete Kronecker δ-function, which plays an important role in both mathematics and physics. In this paper, the precise definition of Dirac δ-function is introduced based on the concept of generalized functions, and it is proved that the Dirac δ-function is not a usual function in the Lebesgue sense of local integrable one. To this end, the Dirac δ-function is here approximated in the sense of weak limit by making use of the sequences of the unit rectangle impulse functions, Gauss functions, Bell-shaped functions and Sinc-functions, respectively. In addition, it is checked that the Dirac δ-function is obtained as a generalized derivative of the Heaviside function, and its higher derivative is also shown. Moreover, the convolutions, scales, compound transformations, orthogonality and Comb Dirac functions are recalled, respectively. Finally, the relationship between Dirac δ-function and generalized Fourier transform is introduced, and we present an application to solve the Dirichlet boundary value problem of the Poisson equation.更多还原