On the foundations of transfer function analysis

Abstract

This paper presents an approach to the integral transform analysis of linear differential systems which is based on the expansion theorem of Weyl, Stone, Titchmarsh, and Kodaira (W-S-T-K). The problem is essentially to find an integral transform which is compatible with the canonical operator of a given class of systems. If this can be done, then the differential equation of any member of that class can be converted into an algebraic equation in the transform demain. The relevence and range of applicability of the W-S-T-K theory to the above problem are discussed. The usefulness of this appraoch is emphasized by the fact that, when it is applicable, it provides not only an answer to the question of whether a compatible integral tranform exists, but also provides an explicit procedure for deriving the transform. Two examples illustrating the theory are given. The first is a derivation of a transform which includes the Fourier, two-sided Laplace, and various versions of the Mellin transform as special cases. The second is a derivation of the Hankel transform.