Diagonal Forms of Translation Operators for Helmholtz Equation in Three Dimensions

Fourier techniques have been a popular analytical tool in the study of physics and engineering for more than two centuries. A reason for the usefulness of such techniques is that certain trigonometric functions are eigenfunctions of the differentiation operator and can be effectively used to model solutions of differential equations which arise in the fields mentioned above. More recently, the arrival of digital computers and the development of the Fast Fourier Transform FFT algorithm in the 1960s have established Fourier analysis as a powerful and practical numerical tool. The FFT, which computes discrete Fourier transforms DFTs, is now central to many areas, most notably spectral analysis and signal processing. In some applications, however, the input data is not uniformly spaced, a condition which is required for the FFT. In this paper we present a set of algorithms for computing more efficiently some generalizations of the DFT, namely the forward and inverse transformations described by certain equations.