Generalized transforms and their asymptotic behaviour

Abstract

Certain properties of generalized transforms of the type Pco g(x) h(oc, x) dx Jo are derived when g is a generalized function in the terminology of Lighthill (1958) and Jones (1966 b). The kernel function h is assumed to be smooth and of sufficiently slow growth at infinity for the generalized transform to exist for any generalized function g. Nevertheless, the class of kernel functions is wide and includes functions such as eiaa;2 and the Bessel function J n(ccx). Theorems concerning the derivative and the limit (in the generalized sense) of the generalized transform are established. The problem of the inversion of generalized transforms is also discussed. The analogue of the Riemann-Lebesgue lemma for generalized transforms is obtained when g is a conventional function and the restrictions on h are relaxed so that it need only be the derivative of a function with suitable properties. The asymptotic behaviour as cc -» + 00 of the generalized transform is examined under the condition that g is infinitely differentiable (in the ordinary sense) at all but a finite number of points. It is shown that the main contribution to the asymptotic development comes from intervals near these points and the point at infinity. Criteria are provided which demonstrate that in many important practical cases the contribution from the point at infinity is essentially exponentially small and therefore negligible. The contributions from the other critical points are determined under a variety of circumstances. In all cases the aim has been to consider conditions which are likely to be of practical value, to be capable of relatively straightforward verification and yet yield theorems of reasonable utility and wide applicability. Some illustrations of the applications of the theorems are given; they include Bessel functions,