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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
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
Last modified 1yr ago