ABSTRACT
Fourier analysis has been employed with great success in a wide range of applications. The underlying theory is based on a small set of linear transforms on particular linear spaces. It may, in fact, be best to refer to two parallel theories of Fourier analysis—the function/functional theory and the discrete theory—according to whether these linear spaces are infinite or finite dimensional. The function/functional theory involves infinite dimensional spaces of functions on ℝ n and can be further divided into two “subtheories”—one concerned with functions that are periodic on ℝ n (or can be treated as periodic extensions of functions on finite subregions) and one concerned with more general functions and functionals on ℝ n . This theory played the major role in applications up to half a century or so ago. However, the tools from the discrete theory (involving vectors in ℂ N instead of functions on ℝ n ) are much more easily implemented on digital computers. This became of interest both because much of the function/functional theory analysis can be closely approximated within the discrete theory and because the discrete theory is a natural setting for functions known only by samplings. In addition, “fast” algorithms for computing the discrete transforms were developed, allowing discrete analysis to be done very quickly even with very large data sets. For these reasons the discrete theory has become extremely important in modern applications, and its utility, in turn, has greatly extended the applicability of Fourier analysis.
