Over the last few sections we’ve spent a fair amount of time
to computing Fourier series, but we’ve avoided discussing the topic of
convergence of the series. In other
words, will the Fourier series converge to the function on the given interval?
In this section we’re going to address this issue as well as
a couple of other issues about Fourier series.
We’ll be giving a fair number of theorems in this section but are not
going to be proving any of them. We’ll
also not be doing a whole lot of in the way of examples in this section.
Before we get into the topic of convergence we need to first
define a couple of terms that we’ll run into in the rest of the section. First, we say that has a jump
discontinuity at if the limit of the function from the left,
denoted ,
and the limit of the function from the right, denoted ,
both exist and .
Next, we say that is piecewise
smooth if the function can be broken into distinct pieces and on each piece
both the function and its derivative, ,
are continuous. A piecewise smooth
function may not be continuous everywhere however the only discontinuities that
are allowed are a finite number of jump discontinuities.
Let’s consider the function,
We found the Fourier series for this function in Example 2 of the previous
section. Here is a sketch of this
function on the interval on which it is defined, i.e. .
This function has a jump discontinuity at because and note that on the intervals and both the function and its derivative are
continuous. This is therefore an example
of a piecewise smooth function. Note
that the function itself is not continuous at but because this point of discontinuity is a
jump discontinuity the function is still piecewise smooth.
The last term we need to define is that of periodic extension. Given a function, ,
defined on some interval, we’ll be using exclusively here, the periodic extension of
this function is the new function we get by taking the graph of the function on
the given interval and then repeating that graph to the right and left of the
graph of the original function on the given interval.
It is probably best to see an example of a periodic
extension at this point to help make the words above a little clearer. Here is a sketch of the period extension of
the function we looked at above,
The original function is the solid line in the range . We then got the periodic extension of this by
picking this piece up and copying it every interval of length 2L to the right and left of the original
graph. This is shown with the two sets
of dashed lines to either side of the original graph.
Note that the resulting function that we get from defining
the periodic extension is in fact a new periodic function that is equal to the
original function on .
With these definitions out of the way we can now proceed to
talk a little bit about the convergence of Fourier series. We will start off
with the convergence of a Fourier series and once we have that taken care of
the convergence of Fourier Sine/Cosine series will follow as a direct
consequence. Here then is the theorem giving
the convergence of a Fourier series.
Convergence of
Fourier series
The first thing to note about this is that on the interval both the function and the periodic extension
are equal and so where the function is continuous on the periodic extension will also be continuous
and hence at these points the Fourier series will in fact converge to the
function. The only points in the
interval where the Fourier series will not converge to
the function is where the function has a jump discontinuity.
Let’s again consider Example 2 of the previous
section. In that section we found that
the Fourier series of,
on to be,
We now know that in the intervals and the function and hence the periodic extension
are both continuous and so on these two intervals the Fourier series will
converge to the periodic extension and hence will converge to the function
itself.
At the point the function has a jump discontinuity and so
the periodic extension will also have a jump discontinuity at this point. That means that at the Fourier series will converge to,
At the two endpoints of the interval, and ,
we can see from the sketch of the periodic extension above that the periodic
extension has a jump discontinuity here and so the Fourier series will not
converge to the function there but instead the averages of the limits.
So, at the Fourier series will converge to,
and at the Fourier series will converge to,
Now that we have addressed the convergence of a Fourier
series we can briefly turn our attention to the convergence of Fourier
sine/cosine series. First, as noted in
the previous section the Fourier sine series of an odd function on and the Fourier cosine series of an even
function on are both just special cases of a Fourier
series we now know that both of these will have the same convergence as a
Fourier series.
Next, if we look at the Fourier sine series of any function,
,
on then we know that this is just the Fourier
series of the odd extension of restricted down to the interval . Therefore we know that the Fourier series
will converge to the odd extension on where it is continuous and the average of the
limits where the odd extension has a jump discontinuity. However, on we know that and the odd extension are equal and so we can
again see that the Fourier sine series will have the same convergence as the
Fourier series.
Likewise, we can go through a similar argument for the
Fourier cosine series using even extensions to see that Fourier cosine series
for a function on will also have the same convergence as a
Fourier series.
The next topic that we want to briefly discuss here is when
will a Fourier series be continuous.
From the theorem on the convergence of Fourier series we know that where
the function is continuous the Fourier series will converge to the function and
hence be continuous at these points. The
only places where the Fourier series may not be continuous is if there is a
jump discontinuity on the interval and potentially at the endpoints as we saw
that the periodic extension may introduce a jump discontinuity there.
So, if we’re going to want the Fourier series to be
continuous everywhere we’ll need to make sure that the function does not have
any discontinuities in . Also, in order to avoid having the periodic
extension introduce a jump discontinuity we’ll need to require that . By doing this the two ends of the graph will
match up when we form the periodic extension and hence we will avoid a jump
discontinuity at the end points.
Here is a summary of these ideas for a Fourier series.
Now, how can we use this to get similar statements about
Fourier sine/cosine series on ? Let’s start with a Fourier cosine
series. The first thing that we do is
form the even extension of on . For the purposes of this discussion let’s
call the even extension As we saw when we
sketched several even extensions in the Fourier cosine series section that in
order for the sketch to be the even extension of the function we must have
both,
If one or both of these aren’t true then will not be an even extension of .
So, in forming the even extension we do not introduce any
jump discontinuities at and we get for free that . If we now apply the above theorem to the even
extension we see that the Fourier series of the even extension is continuous on
. However, because the even extension and the
function itself are the same on then the Fourier cosine series of must also be continuous on .
Here is a summary of this discussion for the Fourier cosine
series.
Note that we don’t need any requirements on the end points
here because they are trivially satisfied when we convert over to the even
extension.
For a Fourier sine series we need to be a little more
careful. Again, the first thing that we
need to do is form the odd extension on and let’s call it . We know that in order for it to be the odd
extension then we know that at all points in it must satisfy and that is what can lead to problems.
As we saw in the Fourier
sine series section it is very easy to introduce a jump discontinuity at when we form the odd extension. In fact, the only way to avoid forming a jump
discontinuity at this point is to require that .
Next, the requirement that at the endpoints we must have will practically guarantee that we’ll
introduce a jump discontinuity here as well when we form the odd
extension. Again, the only way to avoid
doing this is to require .
So, with these two requirements we will get an odd extension
that is continuous and so we know that the Fourier series of the odd extension
on will be continuous and hence the Fourier sine
series will be continuous on .
Here is a summary of all this for the Fourier sine series.
The next topic of discussion here is differentiation and
integration of Fourier series. In
particular we want to know if we can differentiate a Fourier series term by
term and have the result be the Fourier series of the derivative of the
function. Likewise we want to know if we
can integrate a Fourier series term by term and arrive at the Fourier series of
the integral of the function.
Note that we’ll not be doing much discussion of the details
here. All we’re really going to be doing
is giving the theorems that govern the ideas here so that we can say we’ve
given them.
Let’s start off with the theorem for term by term
differentiation of a Fourier series.
One of the main condition of this theorem is that the
Fourier series be continuous and from above we also know the conditions on the
function that will give this. So, if we
add this into the theorem to get this form of the theorem,
For Fourier cosine/sine series the basic theorem is the same
as for Fourier series. All that’s
required is that the Fourier cosine/sine series be continuous and then you can
differentiate term by term. The theorems
that we’ll give here will merge the conditions for the Fourier cosine/sine
series to be continuous into the theorem.
Let’s start with the Fourier cosine series.
Next the theorem for Fourier sine series.
The theorem for integration of Fourier series term by term
is simple so there it is.
Note however that the new series that results from term by
term integration may not be the Fourier series for the integral of the
function.