Integral Transforms Related to the Fourier Transform

Abstract

Fourier analysis can take different forms as we adapt it to various problems at hand. The main results of the Fourier integral theorem are used to justify continuous partial-wave analyses in terms of functions other than the oscillating exponential ones. Section 8.1 presents the bilateral and the more common unilateral Laplace transform where the expanding functions are the decreasing exponential functions exp(pq). In Section 8.2 we expand functions in terms of powers x ^iq _± ^−1/2 (bilateral Mellin) or of powers x ^q (common Mellin) as a continuous analogue of the ordinary Taylor series expansion. Section 8.3 deals with Fourier transforms of functions of N variables and applies them to the general solution of the N-dimensional elastic-diffusive equation. In particular, the three-dimensional wave and general heat equations are treated. Hankel transforms (Section 8.4) use the Bessel functions as the expanding set and arise out of N-dimensional Fourier transforms of functions of the radius. The elastic¡ªdiffusive equation solutions are completed, and the difference between odd and even dimensions is pointed out. We list, finally, several transform pairs which use cylindrical functions as their expanding set. Under the title of “other” integral transforms, in Section 8.5 we give a rough outline of the Sturm¡ªLiouville approach. This is applied in particular to transforms using Airy functions. Other approaches lead to Hilbert and Stieltjes transforms. All sections are basically independent of one another except for Hankel transforms, which are built out of N-dimensional Fourier transforms. Those transforms which are only briefly mentioned in the text are accompanied by a bibliographical survey.