Appendix

Theory of Distributions

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

Definition 41

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

Definition 42

The distrbution induced by a function gg is defined as

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

Notice two things:

  • <Tg,x><T_g, x> is linear

  • <αTg,x>=α<Tg,x><\alpha T_g, x> = \alpha^*<T_g, x>

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

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

Define gϵg_\epsilon to be

gϵ=1ϵg(tϵ)g_\epsilon = \frac{1}{\epsilon}g(\frac{t}{\epsilon})

Now we can defined δ(t)=limϵ0Tgϵ\delta(t) = \lim_{\epsilon \rightarrow 0}{T_{g_\epsilon}}

<δ,x>=δ(t)x(t)dt=x(0)<\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 TT is

<FT,X>=2π<T,x><FT, X> = 2\pi<T, x>

for test function xx whose Fourier Transform is XX

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