**APPENDIX A**

**TECHNICAL MATTERS**

**A.1 PROOF OF THE FOURIER INVERSION FORMULA**

In this section we give a rigorous proof of Theorem 2.1 (the Fourier Inversion Theorem), which states that for an integrable function *f* we have

Inserting the definitions of *F* and *F* ^{-1}, we must show

We restrict our attention to functions, *f*, which are nonzero only on some finite interval to avoid the technicalities of dealing with convergent integrals over infinite intervals [for details, see Tolstov (1962)]. If *f* is nonzero only on a finite interval, then the *t* integral occurs only on this finite interval (instead of – ∞ < *t* < ∞ as it appears). The λ integral still involves an infinite interval and so this must be handled by integrating over a finite interval of the form –*l* ≤ λ ≤ *l* and then letting *l* → ∞. So we must show

Using the definition of the complex exponential (*e*^{iu} = cos *u* + *i* sin *u*), the preceding limit is equivalent to showing

Since sine is an odd function, the λ integral involving sin((*t* – *x*)λ) is zero. Together with the fact that cosine is an even function, the preceding limit is equivalent to

Now ...