Mathematical Methods for Physicists: A concise introduction - Site Map

THE CONVOLUTION THEOREM FOR FOURIER TRANSFORMS
Thus Eq. (4.57) becomes
g…!†G…!† ˆ 1
2
ˆ 1
2
ˆ F
From this we have equivalently
Z 1 Z 1
1
Z 1
1
Z 1
1
1
f …x†H…u x†e i!u dxdu
Z 1
e i!u f …x†H…u x†du dx
1
f H ˆ F 1 fg…!†G…!†
g ˆ …1=2†
which is Eq. (4.56).
Equation (4.58) can be rewritten as
f …x†H…u x†du ˆ Fff Hg: …4:58†
Z 1
1
e i!x g…!†G…!†;
FffgFfHg ˆ Fff Hg …g ˆ Fffg; G ˆ FfHg†;
which states that the Fourier transform of the convolution of f(x) and H(x) is equal
to the product of the Fourier transforms of f(x) and H(x). This statement is often
taken as the convolution theorem.
The convolution obeys the commutative, associative and distributive laws of
algebra that is, if we have functions f 1 ; f 2 ; f 3 then
9
f 1 f 2 ˆ f 2 f 1
commutative; >=
f 1 …f 2 f 3 †ˆ…f 1 f 2 †f 3 associative;
…4:59†
>;
f 1 …f 2 ‡ f 3 †ˆf 1 f 2 ‡ f 1 f 3 distributive:
It is not dicult to prove these relations. For example, to prove the commutative
law, we ®rst have
Now let x u ˆ v, then
f 1 f 2
f 1 f 2
ˆ
Z 1
1
Z 1
1
Z 1
1
f 1 …u† f 2 …x u†du:
f 1 …u† f 2 …x u†du
f 1 …x v† f 2 …v†dv ˆ f 2 f 1 :
Example 4.11
Solve the integral equation y…x† ˆf …x†‡ R 1
1
y…u†r…x u†du, where f …x† and
r…x† are given, and the Fourier transforms of y…x†; f …x† and r…x† exist.
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