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The central limit theorem. To prove the central limit theorem we make use of the Fourier transform which is one of the most useful
 

Summary: The central limit theorem.
To prove the central limit theorem we make use of the Fourier transform which is one of the most useful
tools in pure and applied analysis and is therefore interesting in its own right.
We say a f : R C is summable if
|f(x)| dx < .
For any such function we define its Fourier transform
^f : R C
by setting
^f(t) = e-itx
f(x) dx for t R.
Note that f ^f is linear.
For a R set a(x) = x + a. Then
f a(t) = eita ^f(t).
Indeed,
f a(t) = e-itx
f(x + a) dx
= e-it(y-a)
f(y) dy substitute y - a for x
= eita
e-ity

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics