Fourier TransformYou are currently
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Suppose you have a vector $A$ in 2-D space, and suppose you have another vector $B$. Suppose you want to know how much of $A$ is pointing in the direction of $B$. When you ask that question it doesn't matter if $B$ is long or short, we we may as well assume that $B$ has length 1. It's of unit length. This concept of "how much of $A$ is in the direction of $B$" is called the "dot product", and there are standard ways to compute it.
So now suppose you also have two vectors $u$ and $v$ of unit length, and which are at right angles to each other. Then we can ask how much of $A$ is in the direction of $u$ and how much is in the direction of $v$, and the answers are (unsurprisingly) the dot products, $A.u$ and $A.v$
Interestingly, we can then write $A=(A.v)v+(A.u)u$ and we have a way of expressing $A$ in terms of $u$ and $v.$ So given a vector we can work out how to express it as the sum of other vectors, although it only works if the vectors are of unit length and at right angles.
So how does it do this?
What we need is a concept of the dot product in a vector space. In short, we need the basic properties of vectors (adding, multiplying by a constant, distributivity of these) and a concept of angle between two vectors as provided by the dot product and its properties.
Well, functions can be thought of as a vector space. Give two functions $f(x)$ and $g(x)$ we can define their sum $(f+g)(x)$ as: $(f+g)(x)=f(x)+g(x).$ We can multiply by a constant in the obvious way. So we just need a concept of the dot product. That we get by analogy with the dot product of two vectors.
We take the dot product by multiplying point wise and adding up. So we do the same. Given $f(x)$ and $g(x)$ we multiply pointwise and sum:
Now, of course, things are more complex when we want functions that are more general than just the linear combination of sine waves with integer frequencies, but that is the underlying idea of what a Fourier Transform really is. We are finding out how much of each frequency is needed to construct a function we're interesting in.
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