I have a conceptual question concerning the fourier transform. As it has been ex
ID: 3028462 • Letter: I
Question
I have a conceptual question concerning the fourier transform. As it has been explained to me, functions may be represented as an infinite series of sines and cosines. The way the fourier transform essentially "works" is by multiplying the function by sines and cosines of varying frequencies and then integrating across all time. When the function has power at a certain frequency you will get a non- zero result (mathematically known as correlation).
My question is, since we are integrating across all time, wouldn't the result of the integration be infinite for these frequencies? I have been told there is a scaling factor built into the fourier transform, but I need clarification on what it is and how this works. Thank you!
Explanation / Answer
As we are integrating across all time, it wouldn't result in the integration as infinite for these frequencies.
Explanation for why scaling factor into Fourier Transform:
the reason is that the action of "scaling" on the circle, and the action of "scaling" on the real line, are two different things.
On the circle, the mapping a "wraps around", and for 'a' a nonzero integer, the image of the circle under this map wraps around |a| times.
On the real line, the mapping xax is one-to-one.
And this makes a huge difference.
When you do the rescaling coscos2 on the circle, you are not just compressing the function by making the characteristic length-scale smaller, you are also cramming two copies of the rescaled function into the same circle. In fact, this works for any periodic function: the mapping g()g(a) scales spatially and also makes |a| copies of the function.
When you do the rescaling f(x)f(ax) on the real line, you are only compressing the function spatially. There still is only one copy of the function. This difference in the number of copies is, morally speaking, why there is a factor of |a| difference in the two formulae.
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