Surprisingly, we have not yet exhausted the set of all possible factorizations of the DFT matrix Fn. In fact, a case can be made that we are but halfway along in our survey, for until now we have neglected the *prime factor FFTs*. This framework is based upon a number-theoretic splitting of the DFT matrix and a large fraction of the FFT literature is devoted to its implementation and various properties.

Elsewhere in this final chapter we examine two leading application areas that rely heavily upon FFTs: convolution and the numerical solution of the Poisson equation. Methods for one- and two-dimensional convolution are given in §4.2. The duality between fast convolution and FFTs is established and some new FFT frameworks emerge. Fast methods for the product of a Toeplitz matrix and a vector are also discussed.

A real-data FFT framework is derived in §4.3 and comparisons are made with the rival Hartley transform. By exploiting structure, it is possible to halve the amount of required arithmetic. Fast algorithms for various sine and cosine transforms are given in §4.4 and then used in §4.5 to solve the Poisson equation problem. The trigonometric transforms that we develop have many applications and are a fitting tribute to the FFT and its wide applicability.