7.4 Fiddly Things

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Remark 7.4.1. This section is about niggling little matters of principle and

detail. Basically, we want to know when we can trust sequences of functions

to converge to something. And it would be a good idea to know what the

words mean.

It makes sense to say that a sequence of functions in C[a, b] converges in the

metric derived from the inner product:

Definition 7.4.1. Natural Numbers: N is the set of natural numbers,

{0, 1, 2, · · · }

Definition 7.4.2. Integers: Z is the set of integers: {· · · ,−1, 0, 1, 2, · · · }.

We write Z+ for the set of positive intgers {1, 2, 3, 4, · · · }.

Definition 7.4.3. Rational Numbers: We write Q for the set of rational

numbers, that is numbers which can be expressed in the form a/b for a, b 2 Z

and b 6= 0.

Remark 7.4.2. You probably think that you know what the natural numbers

and integers are, and who am I to shake your certainties? Actually, you

are merely familiar with them. The above ‘definitions’ are actually nothing

of the sort. If you are the kind of person who wants to know what a natural

number is, or a real number, then I am afraid I shan’t be telling you in this

course. You know enough about their properties to be able to catch me out

in any of the more obvious lies, and this will be good enough to pass the

examination. Whether it will be enough for your intellectual life is entirely

up to you. Mathematicians got on very well for several centuries without

knowing what a real number is, why shouldn’t you? And when some of them

138 CHAPTER 7. FOURIER THEORY

did find out, they got very uncomfortable. You have to decide if you are the

sort of person who has to know or whether you accept whatever you are told

by someone in authority. Crikey, that’s me in this case! Who’d have thought

it.

Definition 7.4.4. If {fn : n 2 N} is a sequence of points in a metric space,

then the limit of the sequence is f iff

8 " 2 R+, 9N 2 N : 8 n 2 N, n > N ) d(f, fn) < "

Remark 7.4.3. I am thinking of the case where the fn and f are functions;

thinking of a function as a point in a space may seem strange, but that was

implied by our taking projections.

Definition 7.4.5. Cauchy sequences A sequence of points fn in a metric

space is a Cauchy sequence iff

8 " 2 R+ 9 N 2 N 8 n,m 2 N, n,m > N ) d(fn, fm) < "

Remark 7.4.4. The sequence of ‘points’ are getting closer together. It is

easy to see that if a sequence fn converges to f (Written fn ! f) then the

sequence is a Cauchy sequence. (Exercise: prove this claim.) We would

expect that the converse is true, if a sequence is cauchy then it converges

to something. After all, we can picture a succession of little balls of radius

" getting progressively smaller. The balls are inside each other if we choose

them sensibly, so they seem to be homing in on something. Unfortunately

the space can have holes in it.

Example 7.4.1. The rational numbers, Q were defined above to be those

numbers which can be written as a/b where a and b are integers. It is well

known that

p

2, e and _ are all irrational. So the sequence

1, 1.4, 1.41, 1.414, · · ·

consisting of the finite approximations to

p

2 of increasing precision is a

Cauchy sequence in Q which does not converge to anything in the space Q.

Definition 7.4.6. A metric space is complete when every Cauchy sequence

in it converges to an element of the space.

Then the problem is that Q is not complete. It has holes in it. (Rather a lot

of holes. In fact more holes than points. But that is another story.)

The same thing can easily happen in function spaces.

7.4. FIDDLY THINGS 139

Figure 7.4: Approximating a discontinuous function

Exercise 7.4.1. Define

tanh(x) , ex − e−x

ex + e−x and fn(x) , tanh(nx)

Show that fn is a cauchy sequence in C(R) but that the limit function is

sign(x) which is not in the space.

Remark 7.4.5. It is worth plotting these function in Mathematica: see

figure 7.4

I have shown f1, f2, f3, f4 and f24. Verifying that the sequence is cauchy

is a useful exercise in getting things clear in your mind. Verifying that it

converges to something not in the space is also good for keeping your ideas

in order.

Remark 7.4.6. The problem is that we started out with a rather limited

class of functions, the continuous ones, and we only need to be able to integrate

them and products of them with other functions. So the first step is

to say that instead of working in C[a, b] we would do well to work in a space

of functions which is large enough to contain discontinuous functions which

can still be integrated.

Which functions can be integrated? You might suppose they all can be;

this is because you have only met nice friendly functions, not the mean, evil

functions which resist integration.

Example 7.4.2. Let evil f : I −! R be defined as f(x) = −1 if x 2 Q \ I

and f(x) = +1 if x 2 I \ Q. Now it is easy to see that between every two

distinct rational numbers there is another different rational number. It is

140 CHAPTER 7. FOURIER THEORY

not too hard to prove that between every distinct rational numbers there is

an irrational number. (Given a < b 2 Q add

p

2(b − a)/2 to a. It is easy to

show this must be between a and b and also to show it is irrational.)

It is also easy to show that between any two distinct irrational numbers there

is a rational number. Take the decimal expansions of the two numbers, go

down the line until they differ and one digit is bigger than the corresponding

place in the other number. Now choose the larger digit and follow it by zeros

for ever. The result is clearly between the two numbers and is also obviously

rational.

It follows that evil f as defined above is not Riemann integrable, since any

partition of I will have for each partition interval [a, b] points of both types.

So the supremum of f on the interval will be 1 and the infimum will be −1 and

this won’t get any better as we make the partition finer. Since the integral is

defined only if the limit of the suprema tends to the limit of the infima over

the partition intervals, we have a non (Riemann) integrable function.

Note that f2 is the constant function 1 and is rather easily integrated.

Remark 7.4.7. To get out of the difficulty I shall work with the space of

piecewise continuous functions. These functions are certainly integrable as

are their products and sums, which are also piecewise integrable, so they

form a vector space which includes things like the sign function. They do

not include the evil function defined above.

Definition 7.4.7. A function f : [a, b] −! R is said to be piecewise continuous

iff f is continuous on [a, b] except at a finite set of points and the

limits

lim

h"0

f(x0 + h), lim

h#0

f(x0 + h)

exist for all interior points, and the appropriate limit exists for the end points.

Remark 7.4.8. limh"0 f(x0 +h) means that h approaches 0 from below, i.e.

that h is negative but gets less so. Contrariwise for limh#0 f(x0 + h)

Exercise 7.4.2. Draw the graphs of some functions in the class of piecewise

continuous functions, and some not.

Proposition 7.4.1. The set PC[a, b] of piecewise continuous functions on

[a, b] is a vector space.

Proof We merely have to note that it is closed under addition and scalar

multiplication. The latter is trivial. The former will usually require us to

take the intersection of intervals on which both functions are continuous. _

7.4. FIDDLY THINGS 141

Proposition 7.4.2. The product of two functions in PC[a, b] is also in

PC[a, b].

Proof: This again requires us to find intervals on which both functions are

continuous and rely on the result that the product of continuous functions is

continuous. _

Remark 7.4.9. It is obvious from the definition that any continuous function

is in PC[a, b].

Remark 7.4.10. It is also obvious that all functions in PC[a, b] are Riemann

integrable. After all, continuous functions are, and the piecewise continuous

fail to be continuous at only a finite set of points, and we can forget about

what happens at a finite set of points because they have length zero and will

not affect any integral. So to integrate one of them, calculate the integrals

of the intervals over which they are continuous and add up the answers.

Remark 7.4.11. This sounds like a good space to work in. We can still do

all the projection onto orthogonal functions that we want. There is however

a slight catch:

Example 7.4.3. Not! 0 is the zero function from I to R which sends every

number to 0. I define o : I −! R to be zero except at 1 where it takes the

value 1. Now these are different functions (not very, but that’s the point) but

in the L2 metric, the distance between them is zero. As far as the integral

of the square of the difference is concerned, the difference is actually just

o−0 = o and o2 = o. And the integral of this over I is zero. So d(0, o) = 0

but 0 6= o

Bummer. This can’t be allowed in a metric space. It contradicts the first

axiom.

Remark 7.4.12. We get around this problem by simply declaring that we

shall deal with not the functions, but classes of functions. And two functions

are in the same class precisely when the integral of the square of their difference

is zero. So in particular, 0 and o are equivalent. They differ only on a

finite set of points, so we shall regard them as the same. Then on the classes

we have that the distance between things is zero only if they are the same.

Remark 7.4.13. This looks messy but everything works out. Trust me. Or

better yet, don’t trust me. Check up on everything I do from here on to

make sure it is not a swindle. Even better, go back over everything I have

done and make sure I haven’t pulled some trick on you. I am but indifferent

honest. In particular, if student one uses f1 and g1 where student two uses f2

142 CHAPTER 7. FOURIER THEORY

and g2 and they calculate < f, g > for the inner product between the classes,

do they get the same result when f1, f2 2 f, g1, g2 2 g? If not, all bets are

off.

Exercise 7.4.3. Confirm that the inner product between classes of functions

obtained by choosing any member of one class and any member of the second

class and calculating Z b

a

fi(t)gi(t)

for choices fi, gi is well defined, that is, it does not depend on the choices.

Remark 7.4.14. In view of the last exercise, it is reasonable to talk about

PC[a, b] as if the elements are functions, even though they are not. When I

say something involving a function, you can mentally replace it by the class

of functions which differ from the one I mentioned only on finitely many

points. Technically however:

Definition 7.4.8. PC[a, b] is the set of equivalence classes of piecewise continuous

functions from [a, b] to R, where two functions are equivalent iff they

differ only on a finite set of points.

Proposition 7.4.3. With addition of classes defined by addition of their

elements, scaling defined likewise, the set PC[a, b] is a vector space. With

< [f], [g] > defined on the classes [f], [g] by

< [f], [g] >=

Z b

a

f(t)g(t)dt

where f 2 [f] is an element of the equivalence class and similarly for g,

PC[a, b] is an inner product space.

Proof:

Exercise _

Remark 7.4.15. While it is necessary to ensure that we are not gibbering

when we do things like this, and that treating equivalence classes of functions

as though they are just functions is logically OK, one should not make too

much of it. Either using equivalence classes is going to work because it

doesn’t really matter in any serious application whether we use a function f

or another g which is different from the first but not enough to change any

outcomes, or it will turn out that we get into terrible trouble pulling swifties

like this one. Verifying that we don’t in this case is good for your intellectual

integrity and also very easy. To some people, alas, all of Mathematics is just

inscrutable bafflgab anyway. Let’s be kind to them, but not too kind.

7.4. FIDDLY THINGS 143

Remark 7.4.16. If you have verified the last proposition, you will feel comfortable

about sloppy usage like ‘taking a function f in PC[a, b]’. You will

note that it is sloppy and that it really means we take a function and use

it to specify the equivalence class in PC[a, b]. After a while you may find

yourself slipping into this no doubt deplorable usage yourself. Shortly after

that you will find yourself dismissive of people who insist on using the terms

‘equivalence class of functions’, classifying them as finicky pedants.

It happens to the best of us.

Remark 7.4.17. Now we ask the obvious question: Is PC[a, b] complete? Or

does it still have holes in it? I hate to say this after all the fuss about going

to PC spaces which if fashion were the arbiter would certainly be politically

correct, but the answer is still a resounding NO! The space PC[a, b] is still

shot full of holes.

Remember evil f? We can get a sequence of functions in PC[a, b] which

converges to evil f. This is true because the rational numbers can be counted,

that is, put into 1-1 correspondence with the natural numbers. So although

each member of the sequence is a bona fide member of PC[a, b], the limit

is not. Basically, the nth term in the sequence fails to be continuous at n

points and the limit is not continuous at any of them. So every term in the

sequence is actually in the same equivalence class, but the limiting function

is not!

Bummer squared.

Remark 7.4.18. There is a way of coping with this; we define a new integral

called the Lebesgue integral. If we can express a function as a limit of

Riemann integrable functions, then we can define the Lebesgue integral of

the function as the limit of the Riemann integrals. This is not the usual

definition of the Lebesgue integral but is equivalent to it. So the function

evil f which was +1 except on the rational numbers, is the limit of functions

fn which are +1 except on n distinct rationals, and each of which therefore

has integral Z 1

0

fn = 1

So the Lebesgue integral of evil f is also 1.

From the definition you can see that if a function is Riemann integrable then

it is Lebesgue integrable and the integrals are the same.

The space of equivalence classes of functions from [a, b] to R which have

Lebesgue integrable squares, with the inner product defined via the integral,

144 CHAPTER 7. FOURIER THEORY

is a complete inner product space. It contains evil f in the same class as the

constant function 1.

Remark 7.4.19. Since I don’t wish to get involved with any more niceties

and I do not want to prove the last claim, I shall stick in practice to the

Piecewise Continuous functions PC[a, b] and forget about the possibility of

taking sequences of functions that converge to things like evil f.

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