In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

We consider now the Fourier coefficients of functions and discuss the convergence behaviour of Fourier series. We will see that the convergence behaviour depends on the smoothness of the function . In the following we explain what we mean by smoothness of the function.

**Piecewise continuity**

Let and let . We define the one-sided limits

and

Assume that the one-sided limits and both exist and are finite.

- If then we say that is continuous at the point
- If , or if but is not defined, then has a
*removable discontinuity*at - If , then has a
*jump discontinuity*at

Definition(piecewise continuous)

A function ispiecewise continuousif it is continuous on except at most at a finite number of points where there exist jump discontinuities.

**Example**

The function

is piecewise continuous on , while the function

is *not* piecewise continuous since there exists an infinite number of jump discontinuities at for

Theorem

If is piecewise continuous on a closed, bounded interval , then exists.

We consider several examples of functions some of which are piecewise continuous and some are not.

* Example* Consider the function

Show that is piecewise continuous on every finite interval.

* Example* Consider the function

Show that is *not* piecewise continuous on since does not exist.

* Example* Consider the function

Show that is *not* piecewise continuous on since it has infinitely many points of discontinuity on .

We have already seen how to calculate the Fourier coefficients of a trigonometric polynomial of finite degree. In the following we define Fourier series.

Definition(Fourier series)

Let be -periodic, bounded and integrable on Let theFourier coefficientsbe defined by theEuler formulaeThe (formal) series

is called theFourier series of

In the following we study under which assumptions on the function the Fourier series converges. In particular, we are interested in pointwise convergence.

* Note* If the function is piecewise continuous then it is integrable, but the

*partial sums*

do **not** necessarily converge for every point as In fact, there is a continuous function and an for which the partial sums do not converge as ( For such a function see for instance E.M. Stein and R. Shakarchi, Princeton lectures in analysis I, Fourier analysis. Princeton University Press, Princeton, 2003. Section 2.2)

Hence we need stronger assumptions on the functions to ensure that the Fourier series converges at every point.

**Piecewise differentiable**

Consider a function and a point . We write

if this one-sided limit exists and analogously we write

A function is differentiable at if and only if and

* Note* is not necessarily the same as For instance, let

Then and

but

has no limit as . ( See also here.)

Definition

A function ispiecewise differentiableif the one-sided limits exist everywhere and is differentiable on except at (at most) a finite number of points.

Lemma

If is piecewise differentiable, then is piecewise continuous.

**Example**

The function is continuous on but is **not** piecewise differentiable because and do not exist.

**Example**

The function

is piecewise differentiable on .

**Pointwise convergence**

The following result gives conditions on the function under which the Fourier series converges and under which as

Theorem

Let and suppose that the function has the following properties:

- is -periodic,
- is piecewise continuous on
- and exist.
If is continuous at , then

whereas if has a jump discontinuity at , then

What is remarkable about this result is that the convergence of the Fourier series at a point depends only on the behaviour of the function near This is not so obvious, since the Fourier coefficients are defined by integrating the function multiplied by or over the interval

For further results on pointwise convergence of Fourier series see here and here.

so removable discontinuities have no effect on the piecewise continuity of the graph on some interval??

Hi, good question. First notice that removable discontinuities are different from jump discontinuities. A discontinuity at a point is removable if the left-hand and right-hand limits coincide, whereas it is a jump discontinuity if the left-hand and right-hand limits exist (and are finite) but are different. The definition of piecewise continuity assumes that the function does NOT have any removable discontinuities (i.e. it assumes that all removable discontinuities have been removed already). If a function has a removable discontinuity, then it is NOT piecewise continuous.

Note that, in view of the pointwise convergence theorem of Fourier series, Fourier series do not have removable discontinuities. That is why it is convenient to assume that piecewise continuous functions do not have removable discontinuities, since the Fourier series converges to the function which has all its removable discontinuities removed already.

How do we prove pointwise convergence? Reading around it says that pointiwse convergence is prooved using an episilon-delta (i.e. for (x-a) < delta, f(x) – f(a) < epsilon), pointwise convergence is proven by finding an epsilon that is a function of both delta and x.

However, I can't seem to find any worked examples, or myself work out how to apply it to a case like tute question 5 in the tutorial for week 8.

Furthermore, if we are to ask to prove 'pointwise continuity' and on a given domain, do we have to prove pointiwse convergence across all points within the domain? If so it seems as if we have proven uniform convergence on the domain…

For example, take a function f_n(x) on [0,1] and another function g(x). If we are asked to prove that f_n(x) converges pointwise to g(x), then do we prove the pointwise convergence for all points in the interval [0,1]? I'm just a bit hazy on the procedure we should follow to prove pointwise convergence…

Hi Thien, as you said, you prove pointwise convergence by proving convergence at each point.

For instance, to prove that a sequence of functions converges pointwise to you have to show that for all you have .

For example for we have for all and hence . If , then

Uniform convergence is different, here you need to prove that In the example above, we have for each that hence