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# 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
$$
.

### Definition 42

The distrbution induced by a function
$g$
is defined as
$ = \int_{-\infty}^{\infty}{g(t)^*x(t)dt}$
Notice two things:
• $$
is linear
• $<\alpha T_g, x> = \alpha^*$
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
$ = 2\pi$
for test function
$x$
whose Fourier Transform is
$X$