ABSTRACT
Although the generalized function is a generalization of the concept of ordinary function, the rigorous de”nition of the former is based on the latter. One way to approach the theory of generalized functions is to consider the limit of a sequence of ordinary functions: (1) if all functions of the sequence take a ”xed value at a given point or have a given integral, then the limit function will inherit the same property, regardless of whether the convergence is uniform or not; and (2) if the convergence is nonuniform, then, even if the functions of the sequence are continuous or dižerentiable or analytic, the limit function that is the generalized function may not inherit these properties (Section I.23.4), that is, it could be discontinuous or even singular. For example, the Gaussian functions with unit integral and variance σ are all analytic; the limit σ → 0 of zero variance is the Dirac impulse (Section 1.3) that is a generalized function, which is zero everywhere except at the origin, where it is in”nite. ¡e nature of the singularity at the origin is speci”ed by the integral across it being unity. ¡is suggests that the primitive of the Dirac impulse is the Heaviside unit step that is a generalized function that is zero before the origin and unity a±er (Section 1.2). It can be veri”ed rigorously that this unit step function is the limit as σ → 0 of the integrals of Gaussian functions that specify error functions. If instead the derivatives of the Gaussian functions are considered, the limit as σ → 0 speci”es (Section 1.4) the derivatives of the Dirac impulse; since the Gaussian functions are analytic, they have derivatives of all orders, and thereby derivatives of all orders of the Dirac impulse and Heaviside unit step can be obtained (Section 1.5). ¡e Heaviside unit step speci”es a (Section 1.8) discontinuity with jump unity and can be used to represent any function with isolated, ”nite discontinuities (Section 1.8); taking the corresponding generalized function, the derivatives of all orders can be obtained (Section 1.9), including at the points of discontinuity, in terms of Dirac impulse and its derivatives. ¡is result relies on the properties of the unit step and impulse as generalized functions (Section 1.6); these can be used to construct other generalized functions, for example, the sign and the modulus (Section 1.7).
