The Fourier Transform
This post is way overdue, so let’s get right to it.
Say we have some function, g(x); thus, this function maps x to some value g(x). You can think of the variable x to represent time. Now we want to be able to approximate g(x) to its sum of trigonometric oscillations i.e.; some sine waves of a certain frequency, ω. There will obviously be some frequencies that fit well, and some that won’t. Therefore we need some value ĝ(ω) that tells us just how much of a given oscillation with frequency ω is present in the approximation for g(x)
Now, let me illustrate this point further with a graph I made, taking into account the blue oscillation, which we’ll call g(x)
As you can see, it’s defined as g(x) = sin(x) + 0.13sin(3x).
The red oscillation and yellow oscillations can be considered approximations to g(x). Let’s take a look at the red one, which is just sin(x): it’s obviously the closest approximation of the two to the original function, and will therefore have a bigger impact on the results. It has a frequency, ω = 1, so let’s say that
ĝ(1) = 1
Now taking a look at the other, less impactful approximation of the yellow oscillation, 0.13sin(3x), we see that it has a frequency ω = 3. Thus,
ĝ(3) = 0.13
Since there are no other functions apparent in the graph (and thus, no more frequencies, ω, used to approximate g(x)) we can write for these sort of non-present frequencies,
ĝ(ω) = 0
If we knew all ĝ(ω) values for all possible frequencies, ω, we could PERFECTLY approximate g(x). This is what the continuous Fourier Transform does: it takes some function, g(x), and returns some other function, ĝ(ω) = Ƒ(g), where Ƒ(g) is the Fourier Transform of the function g(x). This describes how much of a given frequency is present in g(x); it’s just another way of representing the information of g(x), but in a different domain.
This is generally easier to solve than with the function g(x) because sometimes g(x) can be really really complicated — the Fourier Transform has a lot of interesting mathematical properties that make computation much simpler. It can change convolutions to multiplication, and it can turn unsolvable PDEs into ODEs (which we generally have techniques to solve) as well as turn ugly, computation-intensive ODEs into linear equations which are soooo much easier.
I might make a post later on a less applied way of describing the Fourier Transform, but I think relating it to applied mathematics is the easiest way to explain it.