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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.

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    To start with, you need some conditions on $f$ and $g$, for example their product $f\cdot g$ should be integrable - can you find such conditions? Also, what integral do you use in the course?2010-11-29
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    f and g don't represent any specific functions. If there are restraints determining whether the transform can be performed, I would appreciate mention of those in the answer. I'm not sure what you mean by "what integral", I was unaware that there are more than one kind of "integral".2010-11-29
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    @nomoreink: All this is mentioned in the article linked by George S. and also in the main article in wikipedia.2010-11-29
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    The integral you know is called the Riemann integral. If you continue to study mathematics you will learn about the Lebesgue integral which is a kind of extension of the Riemann integral (not really but almost). Fourier analysis is best studied in terms of the Lebesgue integral, basically due to convergence issues and translation invariance.2010-11-29

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