A First Course in Wavelets with Fourier Analysis, 2nd Edition

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 ...