What is the Fourier transform of the product of two functions?

Given $x(t) = f(t) \cdot g(t)$, what is the Fourier transform of $x(t)$? If possible, please explain your answer.

The motivation behind the question is homework, but this is a basic principle in the class that I never quite grasped properly. My current homework builds upon the principle of the question. So answering this question by no means will be doing my homework for me. This is why I am asking the general case.

Solution 1:

Transforms such as Fourier transform or Laplace transform, takes a product of two functions to the convolution of the integral transforms, and vice versa.

This is called the Convolution Theorem, and is available with proof at wikipedia.

Solution 2:

Fourier transform of a product is the convolution of the corresponding transforms. For details on conditions on the functions refer links below

http://en.wikipedia.org/wiki/Fourier_transform

http://en.wikipedia.org/wiki/Fourier_transform#Convolution_theorem