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### Questions about Fourier Series

```
Date: 11/18/2000 at 03:16:40
From: Anwar Saleh
Subject: Definition of Fourier Series

Thanks for the time you are putting here to encourage the others on
the other side!

I really have a simple question regarding the correct definition of
Fourier series as accepted by the majority of mathematicians. I was
reading a book about Fourier series and the theory involved (Carslaw's
book) and the author mentioned at the beginning that Fourier series
are special cases of the more general trigonometric series. He said
the following:

"A trigonometric series

a_0 + (a_1 cos(x) + b_1 sin(x)) + (a_2 cos(x) + b_2 sin(x)) + ...

is said to be a Fourier's series, if constants a_0, a_1, b_1, ...
satisfy the equations

a_0 = (1/2pi) * INT[f(x) dx]          from -pi to pi
a_n = (1/pi)  * INT[f(x)*cos(nx) dx]  from -pi to pi
b_n = (1/pi)  * INT[f(x)*sin(nx) dx]  from -pi to pi

where n >= 1, and the Fourier's Series is said to correspond to the
function f(x)."

The author did not mention anything about the convergence (existence)
of the integrals representing the Fourier's coefficients of the
expansion. Therefore, my question really has two parts:

1. Does the accepted definition assume that these integrals must
converge or exist for a series to be accepted as a Fourier series? Or
does convergence not really matter and therefore all mathematical
functions that can live in the mathematician's mind, no matter how
wild they are, really be used to construct series that can be called
"Fourier's series"?

2. If the definition demands the convergence of these integrals for it
to be a Fourier series, then in what sense does this convergence occur
(Riemann, Lebesgue, or what)?

I admit the question is a strange one but since I live far away from
mathematics, I need your help to come closer to mathland in the
shortest possible time. It is really amazing to realize that the
shortest and most profitable way to acquire a piece of knowledge is
not through books, notes, tutorials, papers, or even videotapes, but
really through people who have already paid the price in terms of
time, money, pain, patience, and sleepless nights and know exactly
what the requestor is going through. This is not to say that the
requestor should stay idle and expect everything will go smoothly. He
must pay the price by exhausting all his efforts before he can make an
emergency call (similar to when you get a flat tire!). Thanks again
for your time and interest.
```

```
Date: 11/18/2000 at 10:39:48
From: Doctor Fenton
Subject: Re: Definition of Fourier Series

Dear Anwar,

You have asked some penetrating and very interesting questions. The
way I think of the difference between trigonometric and Fourier series
is that you can take any sequence of coefficients a_n and b_n, put
them in front of trigonometric functions cos(nx) and sin(nx), and you
have a trigonometric series:

a_0+(a_1 cos(x) + b_1 sin(x))+(a_2 cos(2x) + b_2 sin(2x))+...
^             ^
|             |

(Incidentally, your formula, which I copied here, should have 2x in
the argument as indicated, and in general, a_n is the coefficient of
cos(nx) and b_n is the coefficient of sin(nx).)

A Fourier series, on the other hand, must have coefficients that are
generated from a single function f(x) by the formulas you gave, so the
coefficients cannot be arbitrarily chosen.

For your first question, I would say that all the integrals of the
coefficients must exist. (Incidentally, you use "exist" and "converge"
interchangeably, but I don't recall seeing the term "converge" used to
describe the existence of a specific integral. It's a minor point of
terminology, since the existence of an integral depends upon the
convergence of some approximating sequence such as the Riemann sums or
integrals of simple functions, but the standard terminology is that
the approximations "converge" and the integral of f "exists.") If any
integral doesn't exist, then some terms in the series are undefined,
and I don't think anyone would regard such a series as a Fourier
series.

A series is usually defined to be its sequence of partial sums, and
the series is meaningful only if all the terms and partial sums are
meaningful. For example, I don't think anyone would regard

ln(-1), ln(-2), ln(-3), ...

sqrt(-1), sqrt(-2), ...
or
1/0, 2/0, 3/0, ...

as sequences of real numbers. Given a sequence, you can consider
whether it converges, and it is meaningful to talk about sequences
which diverge, such as

1, 2, 3, 4, 5, ...
or
1, -1, 1, -1, 1, -1, ...

but in all these cases, all the terms of the sequence make sense.

As for your second question, as to what type of integral is
considered, I would say that there's no clear answer. An author
usually considers the integrals needed for a particular investigation.
If Riemann integrals are enough, only Riemann integrals will be
discussed.

The point of Fourier series is somewhat different: given a function
f(x) on [-pi,pi], if all the coefficients are defined, then you have a
Fourier series S[f](x) associated with f. For what values of x does
the Fourier series converge? That is, if S_n[f](x) is the nth partial
sum of S[f], then for which values of x does

lim   S_n[f](x)
n->oo

exist? Next, if the series does converge, what is the relation between
f(x) and S[f](x)? The convergence of the partial sums can also be
taken in other senses, such as Cesaro summability, in which you look
at the averages of the S_n[f] to see if they converge:

(S_1[f](x)+S_2[f](x)+S_3[f](x)+...+S_n[f](x))
lim   ----------------------------------------------
n->oo                    n

You can also consider the convergence of S_n[f] to f in some "average
sense," such as

pi
lim   INT |f(x)-S_n[f](x)|^2 dx ,
n->oo -pi

which is called "mean square convergence."

The types of functions you consider, Riemann-integrable or Lebesgue-
integrable, etc., generally depend upon exactly which questions you
want to consider, and what "tools" are available for analysis.

I hope this has answered your questions, at least to some degree.
Please write again if you have further questions.

- Doctor Fenton, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Calculus

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