Despite a fear of oil slicks on the beach in Encinitas, we want to sketch the theory of the fast Fourier transform or FFT. The simplest version of this transform allows one to compute the DFT on a finite cyclic group of order n = 2k elements in n log w operations rather than n2 operations. Other algorithms exist if n is not a power of 2 but we will not discuss them here. The FFT has revolutionized digital signal processing.

We are mostly following the discussion in G. Strang [1986]. The FFT is usually attributed to Cooley and Tukey [1965]. However, Heidemann et al. [1984] note that the Cooley-Tukey algorithm was essentially known to Gauss in 1805, which is two years before Fourier and 160 years before Cooley and Tukey.

Brigham [1974] says that R. L. Garwin asked Tukey to give him a rapid way to compute the Fourier transform during a meeting of the President's Scientific Advisory Committee.

Selberg's trace formula was formulated in 1956 by Selberg [1989, Vol. I, pp. 423–63] for arithmetic groups Г like SL(2, ℤ) acting on the Poincaré upper half plane H defined in (2) of Chapter 23. Despite the slightly fearsome quotes above, it has proved to be extremely useful. Applications include:

• a derivation of Weyl's law on the distribution of eigenvalues of the Laplacian on Г\H,

• an answer to the question: Can you hear the shape of Г\H?,

• the discussion of the analytic properties of Selberg's zeta function (which are essentially equivalent to the trace formula).

We will summarize some of this in Chapter 23. Our main goal here is to consider finite analogues of some of these results. At this point the reader might wish to review the discussion “beginnings of a trace formula” at the end of Chapter 3 as well as Chapter 12 (Poisson's summation formula).

Finite versions of the trace formula do not seem to have been much studied. We first began to think about these things after reading Arthur's derivation of the Frobenius reciprocity law from the trace formula in Aubert et al. [1989, pp. 11–27].

The classical uncertainty principle says that a function and its Fourier transform cannot both be highly localized or concentrated. In quantum mechanics, this becomes the statement that it is impossible to find a particle's position and momentum simultaneously, as Penrose makes quite explicit in the quote above. But, as Donoho and Stark remark, the uncertainty principle shows its (ugly/beautiful) face in many other areas such as signal processing and medical imaging. References for this chapter are Donoho and Stark [1989], Dym and McKean [1972], Grünbaum [1990], Smith [1990], Terras [1985, Vol. I], Elinor Velasquez [1998], and Wolf [1994].

In the first part of this chapter we obtain limit theorems for the simplest random walks on ℤ/nℤ, for n odd, using the DFT and Markov chains. In the second part we redo some of the first part, replacing Markov chains with sums of random variables. We begin with the example of random number generators.

References for this chapter include: Fan Chung [1996], F. Chung, P. Diaconis, and R. Graham [1987], P. Diaconis [1988], P. Diaconis and M. Shashahani [1986], P. Diaconis and D. Stroock [1987], P. Doyle and J. L. Snell [1984], W. Feller [1968], R. Guy [1984, Vol. 3, Section K45], J. G. Kemeny and J. L. Snell [1960], W. LeVeque [1974, Vol. 3, Section K45], K. Rosen [1993, Section 8.7], J. T. Sandefur [1990], J. L. Snell [1975], and M. Schroeder [1986, Chapter 27]. See also Chapters 17 and 18 for more information on random number generators.

Random Number Generators

There are many reasons why programs such as Mathematica and Matlab are capable of giving us a random number at the drop of a hat or the push of a key. Applications include computer simulations, sampling, testing of computer algorithms, decision making, Monte Carlo methods for numerical integration, and fault detection.

In the “good” old days, people obtained random numbers from tables.

In this age of computers, it is very natural to replace the continuous with the finite. One thinks nothing about replacing the real line ℝ with a finite circle (i.e., a finite ring ℤ/nℤ) and similarly one replaces the real Fourier transform with the fast Fourier transform (FFT). Then computers become happier about our computations. Moreover, the prerequisites for our book become much less formidable than they were for our earlier volumes on harmonic analysis on symmetric spaces (Terras [1985, 1988]). In fact, some (such as Greenspan [1973]) might arguethat, since the universe is finite, it is more appropriate to use finite models than infinite ones. Greenspan decides to “deny the concept of infinity” despite feeling “unmathematical” and “un-American” in doing so. Physicists have also begun considering dynamicalsystems over finite fields (see Nambu [1987]).

Here our goal is to consider finite analogues of the symmetric spaces such as ℝn and the Poincaré upper half plane, which were studied in Terras [1985, 1988]. We will discover finite analogues of all the basic theorems in Fourier analysis, both commutative and noncommutative, including the Poisson summation formula and the Selberg trace formula. One motivation of this study is to develop an understanding of the continuous theory by developing the finite model.

First we consider the easiest kind of Fourier analysis – that on the additive group ℤ/nℤ, the integers modulo n. This is an abelian group of order n and it is cyclic (generated by the congruence class 1 mod n). Thus it is the simplest possible group for Fourier analysis. Yet it seems to have the most applications. As we saw in the last chapter, it may be viewed as the multiplicative group of nth roots of unity. This can be drawn as n equally spaced points on a circle of radius 1. Thus ℤ/nℤ is a finite analogue of the circle (or even of the real line).

The discrete Fourier transform on ℤ/nℤ, or DFT, arises whenever anyone needs to compute the classical Fourier series and integrals of sines and cosines. In fact, the first application of the discrete Fourier transform was perhaps A.-C. Clairaut's use of it in 1754 to compute an orbit, which can be considered as a finite Fourier series of cosines.