The solution of linear integral equations of the first kind with convolution kernel is discussed. An interesting alternative solution approach aimed at overcoming the difficulties of the traditional Fourier transform methods is given. The so-called method of differential inversion presented in this paper may be applicable to inverse problems of potential theory, radiative transfer and signal processing.
The authors characterize their new method by the fact that the unknown function is expressed as a series of successive derivatives of the known function. There is a relation to Gauss’s multipole expansion and to the Taylor expansion. However, in differential inversion the concept of distributions is used in a generalized form leading to “hyperdistributions” as fundamental mathematical objects of this theory.
The method may be applied to integral equations of convolution type with kernels possessing finite moments. Relations of the method to Mikusiński fields are mentioned. In particular, the cases of exponential kernel and Gaussian kernel are verified in detail. There are also given references where the method has been successfully implemented in practice.