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/