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This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences—that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions.
The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression.
In organizing their exposition, the authors have carefully balanced an emphasis on key conceptual insights against the need to provide the technical underpinnings of rigorous analysis. Students of mathematics, physics, engineering and other sciences will find the theory and applications covered in this volume to be of real interest.
The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Fourier Analysis is the first, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.
Regarding the researches of d'Alembert and Euler could one not add that if they knew this expansion, they made but a very imperfect use of it. They were both persuaded that an arbitrary and discontinuous function could never be resolved in series of this kind, and it does not even seem that anyone had developed a constant in cosines of multiple arcs, the first problem which I had to solve in the theory of heat. J. Fourier, 1808-9
In the beginning, it was the problem of the vibrating string, and the later investigation of heat flow, that led to the development of Fourier analysis. The laws governing these distinct physical phenomena were expressed by two different partial differential equations, the wave and heat equations, and these were solved in terms of Fourier series.
Here we want to start by describing in some detail the development of these ideas. We will do this initially in the context of the problem of the vibrating string, and we will proceed in three steps. First, we describe several physical (empirical) concepts which motivate corresponding mathematical ideas of importance for our study. These are: the role of the functions cos t, sin t, and [e.sup.it] suggested by simple harmonic motion; the use of separation of variables, derived from the phenomenon of standing waves; and the related concept of linearity, connected to the superposition of tones. Next, we derive the partial differential equation which governs the motion of the vibrating string. Finally, we will use what we learned about the physical nature of the problem (expressed mathematically) to solve the equation. In the last section, we use the same approach to study the problem of heat diffusion.
Given the introductory nature of this chapter and the subject matter covered, our presentation cannot be based on purely mathematical reasoning. Rather, it proceeds by plausibility arguments and aims to provide the motivation for the further rigorous analysis in the succeeding chapters. The impatient reader who wishes to begin immediately with the theorems of the subject may prefer to pass directly to the next chapter.
1 The vibrating string
The problem consists of the study of the motion of a string fixed at its end points and allowed to vibrate freely. We have in mind physical systems such as the strings of a musical instrument. As we mentioned above, we begin with a brief description of several observable physical phenomena on which our study is based. These are:
simple harmonic motion,
standing and traveling waves,
harmonics and superposition of tones.
Understanding the empirical facts behind these phenomena will motivate our mathematical approach to vibrating strings.
Simple harmonic motion
Simple harmonic motion describes the behavior of the most basic oscillatory system (called the simple harmonic oscillator), and is therefore a natural place to start the study of vibrations. Consider a mass {m} attached to a horizontal spring, which itself is attached to a fixed wall, and assume that the system lies on a frictionless surface.
Choose an axis whose origin coincides with the center of the mass when it is at rest (that is, the spring is neither stretched nor compressed), as shown in Figure 1. When the mass is displaced from its initial equilibrium position and then released, it will undergo simple harmonic motion. This motion can be described mathematically once we have found the differential equation that governs the movement of the mass.
Let y(t) denote the displacement of the mass at time t. We assume that the spring is ideal, in the sense that it satisfies Hooke's law: the restoring force F exerted by the spring on the mass is given by F = -ky(t). Here k > 0 is a given physical quantity called the spring constant. Applying Newton's law (force = mass × acceleration), we obtain
-ky(t) = my"(t);
where we use the notation y" to denote the second derivative of y with respect to t. With c = [square root of k/m], this second order ordinary differential equation becomes
(1) y"(t) + [c.sup.2]y(t) = 0.
The general solution of equation (1) is given by
y(t) = a cos ct + b sin ct ;
where a and b are constants. Clearly, all functions of this form solve equation (1), and Exercise 6 outlines a proof that these are the only (twice differentiable) solutions of that differential equation.
In the above expression for y(t), the quantity c is given, but a and b can be any real numbers. In order to determine the particular solution of the equation, we must impose two initial conditions in view of the two unknown constants a and b. For example, if we are given y(0) and y'(0), the initial position and velocity of the mass, then the solution of the physical problem is unique and given by
y(t) = y(0) cos ct + y'(0)/c sin ct.
One can easily verify that there exist constants A > 0 and [??] [member of] R such that
a cos ct + b sin ct = A cos(ct - [??]).
Because of the physical interpretation given above, one calls A = [square root of [a.sup.2] + [b.sup.2]] the "amplitude" of the motion, c its "natural frequency," [??] its "phase" (uniquely determined up to an integer multiple of 2[pi]), and 2[pi]/c the "period" of the motion.
The typical graph of the function A cos(ct - [??]), illustrated in Figure 2, exhibits a wavelike pattern that is obtained from translating and stretching (or shrinking) the usual graph of cos t.
We make two observations regarding our examination of simple harmonic motion. The first is that the mathematical description of the most elementary oscillatory system, namely simple harmonic motion, involves the most basic trigonometric functions cos t and sin t. It will be important in what follows to recall the connection between these functions and complex numbers, as given in Euler's identity [e.sup.it] = cos t + i sin t. The second observation is that simple harmonic motion is determined as a function of time by two initial conditions, one determining the position, and the other the velocity (specified, for example, at time t = 0). This property is shared by more general oscillatory systems, as we shall see below.
Standing and traveling waves
As it turns out, the vibrating string can be viewed in terms of one-dimensional wave motions. Here we want to describe two kinds of motions that lend themselves to simple graphic representations.
First, we consider standing waves. These are wavelike motions described by the graphs y = u(x; t) developing in time t as shown in Figure 3.
In other words, there is an initial profile y = [??](x) representing the wave at time t = 0, and an amplifying factor [psi] (t), depending on t, so that y = u(x; t) with
u(x; t) = [??](x)[psi](t):
The nature of standing waves suggests the mathematical idea of "separation of variables," to which we will return later.
A second type of wave motion that is often observed in nature is that of a traveling wave. Its description is particularly simple: there is an initial profile F(x) so that u(x,t) equals F(x) when t = 0. As t evolves, this profile is displaced to the right by ct units, where c is a positive constant, namely
u(x; t) = F(x - ct):
Graphically, the situation is depicted in Figure 4.
Since the movement in t is at the rate c, that constant represents the velocity of the wave. The function F(x + ct) is a one-dimensional traveling wave moving to the right. Similarly, u(x,t) = F(x + ct) is a one-dimensional traveling wave moving to the left.
Harmonics and superposition of tones
The final physical observation we want to mention (without going into any details now) is one that musicians have been aware of since time immemorial. It is the existence of harmonics, or overtones. The pure tones are accompanied by combinations of overtones which are primarily responsible for the timbre (or tone color) of the instrument. The idea of combination or superposition of tones is implemented mathematically by the basic concept of linearity, as we shall see below.
We now turn our attention to our main problem, that of describing the motion of a vibrating string. First, we derive the wave equation, that is, the partial differential equation that governs the motion of the string.
1.1 Derivation of the wave equation
Imagine a homogeneous string placed in the (x; y)-plane, and stretched along the x-axis between x = 0 and x = L. If it is set to vibrate, its displacement y = u(x; t) is then a function of x and t, and the goal is to derive the differential equation which governs this function.
For this purpose, we consider the string as being subdivided into a large number N of masses (which we think of as individual particles) distributed uniformly along the x-axis, so that the [n.sup.th] particle has its x-coordinate at [x.sup.n] = nL/N. We shall therefore conceive of the vibrating string as a complex system of N particles, each oscillating in the vertical direction only; however, unlike the simple harmonic oscillator we considered previously, each particle will have its oscillation linked to its immediate neighbor by the tension of the string.
We then set [y.sub.n](t) = u(xn,t), and note that [x.sub.n]+1 - [x.sub.n] = h, with h = L/N. If we assume that the string has constant density p > 0, it is reasonable to assign mass equal to ph to each particle. By Newton's law, ph[y".sub.n](t) equals the force acting on the n-th particle. We now make the simple assumption that this force is due to the effect of the two nearby particles, the ones with x-coordinates at [x.sub.n]-1 and [x.aub.n]+1 (see Figure 5). We further assume that the force (or tension) coming from the right of the [n.sup.th] particle is proportional to (yn+1 - yn)/h, where h is the distance between [x.sub.n+1] and [x.sub.n]; hence we can write the tension as
([tau]/h)(yn+1 - yn),
where [tau] > 0 is a constant equal to the coefficient of tension of the string. There is a similar force coming from the left, and it is
{[tau]/h)(yn-1 - yn).
Altogether, these forces which act in opposite directions give us the desired relation between the oscillators [y.sub.n](t), namely
(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
On the one hand, with the notation chosen above, we see that
[y.sub.n]+1(t) + [y.sub.n]-1(t) - [2y.sub.n](t) = u([x.sub.n] + h,t) + u([x.sub.n] - h,t) - 2u([x.sub.n],t).
On the other hand, for any reasonable function F(x) (that is, one that has continuous second derivatives) we have
F(x + h) + F(x - h) - 2F(x)/[h.sup.2] [arrow right] F"(x) as h [arrow right] 0.
Thus we may conclude, after dividing by h in (2) and letting h tend to zero (that is, N goes to infinity), that
p [[??].sup.2]u/[[??]t.sup.2] = [tau] [[??].sup.2]u/[[??]x.sup.2]
or
1/[[??].sup.2] [[??].sup/2]u/[[??]t.sup.2], with c = [square root of [tau]/p.
This relation is known as the one-dimensional wave equation, or more simply as the wave equation. For reasons that will be apparent later, the coefficient c > 0 is called the velocity of the motion.
In connection with this partial differential equation, we make an important simplifying mathematical remark. This has to do with scaling, or in the language of physics, a "change of units." That is, we can think of the coordinate x as x = aX where a is an appropriate positive constant. Now, in terms of the new coordinate X, the interval 0 [less than or equal to] x [less than or equal to] L becomes 0 [less than or equal to] X [less than or equal to] L/a. Similarly, we can replace the time coordinate t by t = bT, where b is another positive constant. If we set U(X,T) = u(x,t), then
[??]U/[??] = a [??]u/[??]x, [[??].sup.2]/[[??]X.sup.2] = [a.sup.2] [[??].sup.2]u/[[??]x.sup.2],
and similarly for the derivatives in t. So if we choose a and b appropriately, we can transform the one-dimensional wave equation into
[[??].sup.2]U/[[??]T.sup.2] = [[??].sup.2]U/[[??]X.sup.2],
which has the effect of setting the velocity c equal to 1. Moreover, we have the freedom to transform the interval 0 [less than or equal to] x [less than or equal to] L to 0 [less than or equal to] X [less than or equal to] [pi]. (We shall see that the choice of [pi] is convenient in many circumstances.) All this is accomplished by taking a = L/[pi] and b = L/(c[pi]). Once we solve the new equation, we can of course return to the original equation by making the inverse change of variables. Hence, we do not sacrifice generality by thinking of the wave equation as given on the interval [0,[pi]] with velocity c = 1.
1.2 Solution to the wave equation
Having derived the equation for the vibrating string, we now explain two methods to solve it:
using traveling waves,
using the superposition of standing waves.
While the first approach is very simple and elegant, it does not directly give full insight into the problem; the second method accomplishes that, and moreover is of wide applicability. It was first believed that the second method applied only in the simple cases where the initial position and velocity of the string were themselves given as a superposition of standing waves. However, as a consequence of Fourier's ideas, it became clear that the problem could be worked either way for all initial conditions.
Traveling waves
To simplify matters as before, we assume that c = 1 and L = [pi] so that the equation we wish to solve becomes
[[??].sup.2]u/[[??]t.sup.2] = [[??].sup.2]/[[??]x.sup.2] on 0 [less than or equal to] x [less than or equal to] [pi], with t [greater than or equal to] 0.
Continues...
Excerpted from Fourier Analysis by Elias M. Stein Rami Shakarchi Copyright © 2003 by Princeton University Press. Excerpted by permission.
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Overview
This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences—that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions.
The first part implements this ...