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.

In Section 2 we considered partial sums of Fourier series and we asked what happens when . In this entry we take a more general approach by considering sequences of functions and studying their convergence behaviour.

Let be a vector space of functions equipped with a norm . We say that a sequence of functions *converges to in * if and

This means that for every there exists an such that

* Exercise* Consider the vector space of continuous functions equipped with the maximum norm. Show that the functions

converge to

as .

* Example* Consider the vector space of bounded functions equipped with the supremum norm. Show that the functions

converge to

as . Show that does not converge to in the supremum norm however.

Note that the functions , in the above example are all continuous, but the limit is not.

* Note (Interchanging limits)* Limits cannot always be interchanged. For example let

Then

and

Hence, in this example,

In particular, assume that is continuous and converges at every point to a function , we do not necessarily have that is continuous.

For the example above we have

There is, however, a type of convergence which ensures that if the sequence of functions are continuous then the limit is also continuous. This type of convergence is called uniform convergence.

Definition

Let be a sequence of functions and let We say that converges uniformly to if for every there is an integer such that implies thatfor all .

In other words, let be the supremum norm Then uniform convergence just means that

The difference between uniform convergence and pointwise convergence is that for pointwise convergence the choice of depends on and on whereas for uniform convergence is only allowed to depend on (but **not** on ).

Note that uniform convergence implies pointwise convergence ( can you prove this?), whereas pointwise convergence does not imply uniform convergence.

If is a sequence of continuous functions which converge uniformly to a function , then is continuous.Theorem(Uniform convergence theorem)

A consequence of this is that if is continuous for all and as , then if has a discontinuity implies that the convergence of to cannot be uniform.

For Fourier series, this implies the following: Assume that is periodic with period and note that the partial sums are continuous. Then if is not continuous at some point , then cannot converge uniformly to as on any open interval containing .

There is a very convenient test for uniform convergence due to Weierstrass.

Theorem(Weierstrass test)

Suppose is a sequence of functions defined on a set and supposeThen the series

converges uniformly on if converges.

* Example* Let (where ) and

where and . We must choose , such that

Let then

that is, we can choose Hence it remains to show that the sum

converges. This can be done using the ratio test

* Exercise* Show that the functions

converge uniformly to

as .

** Mean square convergence of Fourier series**

An interesting property of the partial sums of a Fourier series is that among all trigonometric polynomials of degree , the partial sum yield the best approximation of in the -norm (or mean square sense). Note that the -norm is given by

Lemma(Best approximation lemma)

Assume is integrable and let denote the th partial sum of the Fourier series, thenfor any real numbers and .

* Exercise * Give a geometrical interpretation of this result.

Theorem

Let be periodic, bounded and integrable on . Then the Fourier series of converges in the mean square sense, that is,

You can find a proof of this result on this post.

Further, Parseval’s identity holds

Hey Josef,

I would like to know which subject(s) from the maths department go into more depth on these types of concepts (convergence of sequences of functions, series, etc.)?

Hi Christian, Math3611 (Higher) Analysis is a third year course (S1) which deals with convergence of sequences of functions and related issues. See here for the course handout.