# Appendix

## Theory of Distributions

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

{% hint style="info" %}

### Definition 41

Given a test function $$x$$, a distribution $$T$$ operates on $$x$$ to produce a number $$\<T,x>$$.
{% endhint %}

{% hint style="info" %}

### Definition 42

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

$$\<T\_g, x> = \int\_{-\infty}^{\infty}{g(t)^\*x(t)dt}$$
{% endhint %}

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.

{% hint style="info" %}

### 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 $$X$$
{% endhint %}


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