Appendix

Theory of Distributions

The Theory of Distributions is the mathematical framework which underlies the generalized Fourier Transforms.

Definition 41

Given a test function $x$, a distribution $T$ operates on $x$ to produce a number $<T,x>$.

Definition 42

The distrbution induced by a function $g$ is defined as

$<T_g, x> = \int_{-\infty}^{\infty}{g(t)^*x(t)dt}$

Notice two things:

• $<T_g, x>$ is linear

• $<\alpha T_g, x> = \alpha^*<T_g, x>$

With these definitions, we can now define the Dirac delta in terms of distrubions. Let $g$ be any function such that

$\int_{\infty}^{\infty}{g(t)dt} = 1$

Define $g_\epsilon$ to be

$g_\epsilon = \frac{1}{\epsilon}g(\frac{t}{\epsilon})$

Now we can defined $\delta(t) = \lim_{\epsilon \rightarrow 0}{T_{g_\epsilon}}$

$<\delta, x> = \int_{-\infty}^{\infty}{\delta(t)x(t)dt} = x(0)$

which is the property of the Dirac Delta we want. Now we can define the generalized Fourier Transform in terms of distributions.

Definition 43

The generalized Continuous Time Fourier Transform of a distrubtion $T$ is

$<FT, X> = 2\pi<T, x>$

for test function $x$ whose Fourier Transform is $X$