Questions About Fourier SeriesDate: 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/ |
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