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In the last video, we talked about how the Fourier series can be used to estimate a lot
of weird functions on an interval by just using functions of this form:
Which we called the complex exponentials. Remember if we wanted to estimate some function,
we multiplying each of the exponentials functions by a number, called its weight, and then added
the functions together. We found that for a lot of functions, this worked really well.
The next natural question is whether we can use the same trick to estimate a function
on the whole real line. It's tempting to say, if a certain combination estimates the function
well on the interval, surely the same combination must estimate the function well on the whole
line. That's not the case at all though, as you can see from these examples.
That's where the Fourier Transform comes in. It comes up with a way to estimate a function
on the whole real line using complex exponentials. Before we talk about how, let's think about
the complex function e to the power of I k x. The real part of this is this cos function
and the imaginary part is a sin function and both bits have the same wavelength. How does
that wavelength relate to k? Well k is in fact 2 pi divided by the wavelength. We call
k the wave number. If we think back to the fourier series, we only needed to use exponentials
with certain values of k. Unfortunately, that's not the case if we want our estimates to work
for the whole line. We need to consider the exponentials with all possible k values. Now
say we have a function f that we want to approximate on the real line. We have to figure out exactly
how much of each one of these exponentials we need to add to get a good approximation.
So in other words, we're looking for some function of k, something that takes a value
of k and tells you how much of the corresponding exponential you need to add. This is pretty
much what the Fourier transform is. Let f hat be the fourier transform of a function.
Then this equation happens.
Ok, don't freak out. I know there's an integral there, and so it seems really hard to understand
what's actually going on but -its really not. If you've seen my calculus video, or you know
something about calculus already, you know that the integral itself is about an approximation,
so we'll come at this from that angle.
First we pick a small number, let's start with half. What we're going to do is get the
best approximation we can to our function using only the complex exponentials with k
values that are multiples of half. Our task is to find the weights to put in front of
these functions. OK, suppose we know the fourier transform of our function already. We're going
to restrict our attention to the points on the graph where k is a multiple of half. It's
tempting to think the weights to put in front of each of the exponentials should just be
given by the value of the fourier transform at that k, but that's not quite right. Actually,
the appropriate weight is the area of each of these sections, in other words, the fourier
transform multiplied by half. Let's see just how good or bad this approximation is to our
actual function. Not awful actually. The right shape at least. But we can do better. Let's
try using delta x equals one quarter. Now, we've got more exponentials to work with.
We'll use a similar process to find the weights. We'll look at the relevant parts of the fourier
transform. Again the weights are given by these areas, which in this case is the height
multiplied by one quarter. Now let's see how good this is. A bit better than last time.
So we can go ahead and do the same process with delta x equaling a one tenth and then
by the time delta x equals one one hundredth, you can hardly see the difference between
the two functions. If we wanted to summaries what we just did
in equation form, we'd say that our function approximately equals the sum of f hat of k
times dx times e to the power of (ikx), where each k a multiple of delta x. The approximation
gets better, the smaller we make k, so we can say that f equals the limit of delta x
goes to 0 of all that stuff on the right hand side. But this is, by definition, is the integral
of f hat k times e to the ikx dx. This, if you remember, is the integral we were trying
to explain. So you can see that the fourier transform function tells us which of the exponential
functions to give the most and the least weight to when we approximate a function.
With that in mind, it's hard to imagine what the relationship between a function and a
fourier transform is. The surprising fact is though that there is some interesting links
between the two. Next episode, we'll explore this relationship. In particular, I'll explain
why the fourier transform makes the Heisenberg uncertainty principle happen.