History Fourier Series

8/2/2019 History Fourier Series

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.2Introduction to Harmonic Analysis

2.1 How trigonometric series came about:the vibrating string

"The eighteenth century stands out in mathematical history as an era of greatgenius. Through the work of an astonishing array of masters the science was


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"troduction to Hannonic Analysis

of examples for clarification. In most calculus books it is proved that

32

111111111 - - + - - - + - - - + - - - + - = Ig2234 5 6 7 8 9

and also that

11111111 31+ - - - + - + - - - + - + - - - +... = -lg2.3 2 5 7 4 9 11 8 2The second series is obtained by rearranging the terms of the first one:Every two positive terms are followed by a negative one. This seemingly

harmless modification changes the sum of the series. As a second examplewe mention the telescopic series

00

L(Xk+l - xk).k=


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34 AlToduction to Hannonic Analysis

turned out to be so subtle that in their pursuit a final clarification imposed

itself.At the beginning of the eighteenth century, near the close of Isaac New-

ton's life (1643-1727) and with Gottfried Wilhelm von Leibniz (1646-17I 6) just dead, the effectiveness of calculus - their creation - as aninstrument for the treatment of problems in mechanics was generally rec-ognized. Mechanics with its abundance and variety of problems exerted astrong fascination on those interested in attaining analytic mastery over themanifestations of nature.

Problems pertaining to the motions of single mass particles were solvedto a reasonable extent. Beyond them the forefront of advance dealt withmatters of greater complexity: motions of bodies with many degrees offreedom, reactions of flexible continuous mass distributions, vibrations ofelastic bodies. .

In particular the motions of tautly stretched elastic strings of length I,held fixed at its extremes, in response to a displacement from the state ofequilibrium, received a great deal of attention and these investigations areamong the most important in the eighteenth century development of therational mechanics of deformable media. In many cases the response canbe acoustically perceived, ranging from the hum of a heavy structural wireto the notes of musical string instruments. It can also be visually noticeablebeing sometimes marked by features such as the presence of nodal pointsthat maintain a state of rest while the string between them is in agitation


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..1. How trigonometric series came about: tbe vibrating string 35

It was at this juncture that many divergent conceptions, which were toplaya crucial role in the development of mathematics and of mechanics aswell, came in to confrontation.

It was believed that continuous material bodies could be approximatedby a system of discrete mass particles of finite number n. These discrete

systems could be made to merge into the continuous case by letting the sizeof the individual particles diminish indefinitely and by letting their numbergo to infinity.

In the case of the homogeneous string the natural approximation thatpresents itself is that of a string of n beads of equal mass mounted at equally

spaced points along the string, which itself is assumed to be weightless(though strong), perfectly flexible, and elastic. Friction is ignored, and ifM is the total mass of the particles and T is the tension, it is furthermoreassumed that the ratio M / T is negligible for the purpose of taking gravityout of the picture and concentrating attention on tension alone. That is the

case for musical strings. For example each string of a grand piano, evidentlyof small mass, undergoes a tension that corresponds to a force from 75 to90 kilos.

Clearly the original problem has been replaced by an idealized abstractone. To do so always requires deep understanding and sound judgment, for

the subsequent theory stands or falls according to results that at strategicpoints have to agree with the data of observations.


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36 . Introduction to Harmonic Analysisstandard. Then by an ingenions use of familiar fonuulas from calculus he

solved the equation, finding its functional solution

y(x, t) = (x+ ct) + W(x- ct)

where nd Ware arbitrary functions with continuous second-order deriva-tives. The above fonuula has an intuitive interpretation: The tenu x +ct)can be interpreted as a uniform motion of the initial state x) to the leftwith speed c; likewise W(x - ct) can be interpreted as a similar motion ofW(x) to the right.

Then the conditions for the fixed ends at x = 0,and x = 1, the initialdisplacement of the string y = f (x), 0 ::::x :::: 1, at time 1 = 0, and thezero initialvelocity imply ,

f(x + Cl) + f(x - Cl)Y(X,I) =. 2 (2.1)

provided f (x) is extended by the condition that it be odd and periodic of

period 2 (Rogosinski [1959]).Even though (2.1) makes sense under no regularity assumptions what-soever d' Alembert's critical mind saw no reason that all possible motions


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. 2.1. How trigonometric series came about: the vibrating string 37

~//0"" /",/ V3 1"", -',Figure 2,4, The "plucked" stringanime t = 0, for c = 1 and P = (1/3, 1/3).

own. Then he stated that the case of "discontinuous" functions (at that time

"discontinuous" meant nondifferentiable, that is functions with comers)must be encompassed to allow for the above situation. D' Alembert was notconvinced and kept on going his own way in a subsequent paper.3

~' " '"/ ,/ :"'" ""/ '/ , ", ""// 0 / VB V2 '" 1 "'"Figure 2,5. D'Alemhert's solutionat time t = 1/6.


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38 "-Introduction to Harmonic Analysis

/"" //"", " , ',', ,. ", /

0 """ 1/3 /' I",

~" '

", I '" /""" I /' "" /

"" : /' "'" ,/',., ',/

Figure 2.6. D'Alembert's solutionat time t = 2/3.

the string is expressible in the form00

y(x t)=

~ A. nnx nnct

, ~ nsm-cos-n=l I ~

with appropriate coefficients An. This meant a representation in sine seriesfor the initial ordinate f (x). There it was

00

f(x ) = ~ . A . nnx~ nsm-n=l I .

(2.2)

Euler, the most important mathematician of the century, placed the weightof his authority against that formulation.5 The iq.finity of unknown con-

stants in (2.2) might seem to allow sufficient generality for representationin trigonometric series, he argued, but in fact overriding reasons showedhi ld b h d f P i di i ( f h i h h d id f (2 2


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.2.2. Heat diffusion 39

no.,restrictio.ns upo.n the shape o.f the curve marking the initial po.sitio.n.In that, fo.llo.wingEuler, he suppo.rted the functio.nal so.lutio.nand rejectedtrigo.no.metric series. Nevertheless he came clo.se to. Berno.ulli's fo.rmula,witho.ut realizing it.

Also.it has to.be mentio.ned that in a paper7 written in 1777 and publishedo.nlyin 1798 after his death, assuming functio.nskno.wnupo.n so.megro.undo.ro.therto.be representable in terms o.fco.sineseries o.fthe type (2.2), Eulerfo.und the no.w standard fo.rmula fo.r the co.efficients (see (2.3) in Sectio.n2.3).

2.2 Heat diffusion

Then it was Fo.urier's turn. He derived frQm physical fundamentals thepartial differential equatio.n o.fheat diffusio.n in cQntinuo.usbQdies such as

a lamina, a thin bar, an annulus, a sphere. In searching fQr sQlutiQns(seeSectio.n 2.6 fQr a mQdern treatment), he started with the lamina and rightaway trigQnQmetricseries po.pped up again. The particular lamina Qnwhichhe was wQrking was Qffinite width and semi-infinite length. He centered ito.nthe x-axis, starting at the Qrigin. The width ran frQm -1 to. 1 alQng they-axis (Fig. 2.7).

The cho.ice Qf thebQundary cQnditiQns was a separate issue. FQurier


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40 .troduction to HannonicAnalysis

y

Figure 2.7. The thin lamina of Fourier's memoir.

had already appeared in d' A1embert's paper of 1750 mentioned before.Upon substitution into the above equation, it is found that q; and 1/fsatisfyordinary differential equations

q;(x)/q;"(x) = -1/f(y)/1/f"(y) = A

that are easily solved. With A = 1/n2 he found the solutions u(x, y) =enx cos ny Then he claimed that the general solution was given by a com


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82.2. Heat diffusion 41

y

Figure 2.8. The square wave, first to be developed in Fourier series.

iron determination, he found the correct value 4/ JTof the first coefficientand subsequently of all the others.

Next Fourier explained what kind of function his series represented: It isperiodic of period 2; for y between -1 and 1it takes the constant value 1;at

y = :I: 1 it is zero; and for y between 1 and 3 it is equal to -1. Speaking inmodem terms, Fourier found the "Fourier series" of the square wave (Fig.2.8). He computed the Fourier series for other specific periodic functions,such as the sawtooth wave and the triangular wave (Fig. 2.9), as well asthe sine series of a "wave" that is constant, say 1, on the interval (0, ex)

and zero on the interval (ex,JT). Euler had stated8 that functions of thiskind were impossible to represent in trigonometric series. Fourier did not I

1

-1 0 1


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42 8 Introduction to Harmonic Analysis

Fourier's examination of the foundational aspects of his new result con-cerned the generality of the function to which it applied. From physicalconsiderations, he presumed that the result was true for "any" function,even those not differentiable. Indeed his formula involved the computationof an integral which, having the geometrical meaning of an area, does not

require any differentiability at all.Then Fourier showed a much simpler method of obtaining the same for-

mula, that is the now standard one of multiplying (2.2) through by sin 17: xand integrating term by term. The formula and even the method was thatused by Euler in the mentioned paper of 1777, published posthumously in

1798. (Fourier, who was reported to have learned of it later as indicated byLacroix, did not let go without comment the charge of his having failed torefer to earlier works on the subject. For, in a leiter, he wrote: "I am sorrynot to have known the mathematician who first made use of this method

because I would have cited him. Regarding the researches of d' Alembertand Euler could one not add that if they knew this expansion they made buta very imperfect use of it.")

About his last method he remarked that "it is just a useful abbreviation,"but it is totally insufficient to solve all of the difficulties that his theory ofheat presents: One has to be directed by other methods too, "needed by thenovelty and the difficulty of the subject."

Really none of his methods is conclusive; rather it appears that just as


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8 2.3. Fourier coefficients and series 43

linear or nonlinear according to circumstances, but after him an effort wasmade to render a nonlinear physical problem into a linear model in orderto exploit the power and generality of the method Fourier developed in his1807 memoir.

2.3 Fourier coefficientS and series

Under the broad assu"mption of absolute integrability, usually satisfied inpractice, a function f (x) defined on (-7r, 7r) can be developed in Fourierseries

ao .f(x) = -+(alcosx+b1smx)2

+ (a2 cos 2x + b2 sin 2x) + ... + (an cos nx + bn sin nx) +... . (2.3)

The coefficients, named Fourier coefficients, are defined by

1 1o=

;; - f(x) dx

I 1l = ;; - f(x) cos x dx


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44 entroduction to Harmonic Analysis

1t

-1tx

1t

f(x) =2(sin x_.! si~ 2x +.! sin 3x - ...)2 3

4 1 1f(x) =11: (cos x +"9 cos 3x + 25 cos 5x + ...)

Figure 2.9. The sawtooth and the triangular wave with corresponding Fourierseries.

A result that goes under the name of the Riemann-Lebesgue theoremproves that the coefficients an and bn tend to zero as n tends to infinity.


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..3. Fourier coefficients and series 45

The totality of Fourier coefficients is called the spectrum and the indices n

that fabe1 them frequencies. The second one remains among real numbersand it accounts for the name, hannonic analysis, given to this theory

A 00

f(x) = 20 + LAn cos (nx +


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46 . Introductionto HarmonicAnalysis2.4 Dirichlet function and theoremIn a short time the disputes raised by Fourier's work reached a final clari-ficatiou. It was the dawn of modem mathematics when, in 1837, Dirichlet

proposed the by now standard concept of function y = f (x) that associatesa unique y to every x. Dirichlet gave the following striking example, named

Dirichlet function after him:

D (x ={

o if x is irrational,)

1.f

' . IX IS raaona ,

for x in the interval (-]f, ]f). That D(x) it is not an "ordinary" functioncan be easily perceived: Between any two rational numbers (where D takesthe value one), as close as they might be, there are always infinitely manyirrational numbers (where D takes the value zero). Similarly, betweeu any

two irrationals there are always infinitely many rationals. This seemingword ganle has an implication: Due to the infinitely many "jumps;' fromzero to I, the graph of D cannot be drawn, not even qualitatively. In spite ofthat the Dirichlet function is well defined: At t = 0.5 it takes the value 1, att = .j2 the value zero, just to give a couple of examples. More generally, forevery x in ( -]f, ]f) it suffices to establish whether x is rational or irrationalto know the value of D(x).


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.2.4. Dirichletfunction and dJeorem 47

x sin 1/x

0.5x

Figure 2.10. Graph of a function that oscillates infinitely many times in proximityof the origin.

Even less eccentric functions have to be ruled out, such as those thatoscillate infinitely many times. An example is in Figure 2.10 (note thatthe graph is only qualitative, for the infinitely many oscillations of y =x sin X-I in proximity of the origin cannot be rendered).

Nevertheless Fourier's intuition was correct overall due to the followingDirichlet theorem9 If a function f has at most a finite number of maxima

and minima and of discontinuities, then its Fourier series at all points xwbere f is continuous converges to f (x); at a point of discontinuity x = Xuit t


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\

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48 .Introduction to Harmonic Analysis

3

-" " x-2

Figure 2.11. At the point of discontinuity Xo = I the Fourier series of f convergest02.

,,~~:{.

~


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.2.5. Lord Kelvin, Michelson, and Gibbs phenomenon 49

grable, then the Fourier series converges almost everywhere, that is exceptpossibly on a set of Lebesgue measure zero.

2.5 Lord Kelvin, Michelson, and Gibbs }:)henomenon

Tides are an oscillatory phenomenon to which it is natura.l to apply Fourieranalysis. Being primarily due to the combined gravitational effects of the

moon and of the sun upon the oceans, the simplest mOdel accounts only

for these two forces, additionally assumed to be periodic. The periods arethat of the rotation of the Earth with respect to the Moon and that of therotation of the Earth with respect to the Sun. Take for insta.nce the solar tide,which is the least complicated of the two. The fundamenta.l harmonic (2.5)has a frequency of one cycle per day and the n = 2 harmonic - whichis stronger in any given month - has a frequency of two cycles per day.(Actually neither the solar tide nor the lunar tide are exactly periodic and sothe corresponding Fourier coefficients al and az are not eXa.ctly independentof time, as periodicity would imply. Coefficients slightly varying over thecourse of the year are used to obtain a better mathematical 11l0del.In Section

4.9 another example can be found.)Lord Kelvin (1824-1907), who began his scientific career with articles 10


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50 9Introduction to Harmonic Analysis

ficients fed in was to move the blips closer to the points of discontinuity, butthey remained there and their height remained about 18% above the correctvalue. After making every effort to remove mechanical defects that couldaccount for the blips, their existence was confirmed by hand calculation(Michelson [1898]).

Josiah Willard Gibbs (1839-1903), one of America's greatest physicists,whose main field of interest was theoretical physics and chemistry (his for-mulation of thermodynamics transformed a large part of physical chemistryfrom an empirical to a deductive science) had spent almost three years inEurope and studied with Karl Weierstrass (1815-1897). In two letters to

Nature - the second, Gibbs [1899], a correction of the first - showingan appreciation of mathematical fine points, he clarified the above phe-nomenon that ever since has gone under his name.

A careful examination of Dirichlet's theorem detects its pointwise na-ture: The convergence of a Fourier series at any point, those of disconti-

nuity included, is described. The procedure is that of fixing a point andthen summing all the infinitely many terms of the series. If only a finitenumber of them is summed, then an approximate value is obtained. Suchan approximation will now be examined in a neighborhood of a point ofdiscontinuity.

Let us consider the simplest example, the (asymmetrical) square wavedefined by


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&.5. Lord Kelvin, Michelson, and Gibbs phenomenon 51

-z, -, , x2,

b,

4,

2 3 4 5 6 7n

Figure 2.13. Above. tbe square wave. Below, its Fourier coefficients bn.

52 t d ti t H i A l i


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52 .troduction to HarmonicAnalysis

x

x

Figure 2.15. The partial sum S, (x). of the Fourier series of the square wave. forn = 15 above aud n = 22 below.

the square wave. If many more terms are added, then the approximation

improves (Fig. 2.15). Nevertheless, independeutly of how many terms onemight sum, it is impossible to obtain a good approximation over an entire

1

1

86 Th t ti f ll 53


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86. The construction of a cellar 53

2.6 The construction of a cellar

The problem of heat diffusion attracted Fourier's attention as well as thatof other contemporary mathematicians. For instance Poisson published alarge part of his results, which originally appeared in the Journal de I'EcolePolytechnique, in the book Theorie- mathematique de la chaleur (1835)(Mathematical Theory of Heat). The topic was interesting not only froma theoretical viewpoint, relative to the detenmnation of the temperature inthe interior of the Earth, but also from a practical viewpoint, snch as metalworking.

To illustrate the principle in a simple way, suppose a cellar has to be built.It is natural to ask what its depth should be in order for it to be cool andwith only small variations of temperature as the seasons change (ideally itought to keep the same temperature at all times).

As time goes by, heat propagates through the earth's crust as a waveso that the hottest day of the year at the surface is not the hottest day inthe interior because the peak of temperature there will occur with a delay(change of phase). Think for instance of springs in the mountains: Duringsummer their water is cool, often so cool indeed that it can be drunk only insmall sips, whereas during winter it is found at an ideal temperature. Alsoit can be expected that both heat and cold, when they reach deep inside, areattenuated (damping).

If th bl i f l t d d l d th ti ll b l th


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54 8ntroduction to Hannonic Analysis

FouR I ~R'S C Ut4 R

Figure 2.16. Fourierdescendinginto a cellar. (Drawingby EnricoBombieri.)


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..6. The construction of a cellar 55

Note that the time variable t and the space variable x are now separated

and that the dependency upon t is explicit. Differentiating term by term -assuming that is petmissible - and upon substitution, (2.8) becomes

00 00d 2'" . '" Un'

L...J iwnun(x) e''''nr = K L...J dx2 e''''nr.n~-oo n=-oo

By the uniqueness of Fourier series (that can be proved), the coefficients onboth sides have to be equal. Thus the heat equation gets transformed intoinfinitely many equations, one for every coefficient

d2un .K~=lwnUn.dx (2.10)

Time t has disappeared: (2.10) is an ordinary differential equation in the xvariable, easy to solve, being linear with constant coefficients. The solutionis

Un(x) = Cnea"x + Dne-a"x (2.11)where Cn and Dn are arbitrary constants and an = (1:!: i).Jlnlw/2K,depending upon n > 0 or n < O. Because the real part of an is positive,Cn = O. Otherwise the solution in (2.11) would go to infinity as x tends toinfinity, whereas it has to be bounded for lack of interior sources (boundarycondition at infinity). Therefore Un(x) = Dne-a"x. Finally back to (2.9):When the boundary condition U(0, t) = fo (t) is imposed and developed in

56 .. I t d ti t H i A l i


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56 .. Introduction to Harmonic Analysisrnw


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.. The Fouriertransfonn 57

I

-I

Figure 2.17.The rectangularpulse r(t).

integration to be equal to 2uJ-l sin w (Fig. 2.18). The geometric meaning ofR (w) = J~ 1cos wt dt for a fixed w is that of an area that varies continuouslywith w. So R(w) is continuouseven though r(t) is not. Continuity is ageneral property of the Fourier transfonn F, defined in (2.13), under thebasic assumptionthat f is absolutelycontinuous.

Similarly the Fourier seriescorresponding to the coefficientsin (2.12)leads to the so-called Fourierintegraltheorem or inversionformula


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58 . Introductionto HarmonicAnalysis1

100

f(t) = - F(w) ei." dw,21f -00

which holds under the assumption that f, F are absolutely integrable andcontinuous. A more general statement that takes into account functions f

showing a finite number of "jumps" discontinuities holds (Komer [1988]).Thus functions satisfying the stated assumptions can be expressed as a"sum" of infinitely many harmonics F(w) ei"" , where w is any real number.The above (2.14) first appeared in a paperl1 by Fourier that was a summaryof his book yet to be published. It is common to find (2.13) and (2.14)

written in two other equivalent ways, one being

(2.14)

100 '

F(w) = -00 f(t) e-i2x"" dt, (2.15)

f(t) = i: F(w)ei2."" dwand the other

F(w) = ~ 100

v"ii -00 f(t) e-i"" dt,(2.16)

1 00

F( ) i""d


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60 Introduction to HannonicAnalysis


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a)

I

1"8

c)

d)

. Introduction to HannonicAnalysisb)

I 1

I

8

v.

8"0)

16

~.9. 'lhmslatlonsandJ/J/crJ:'Cllcc MIINCM IIII


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I

0>

Figure 2.20. The Fourier transform of eiW01(I).

A A'1

Figure 2.21. Tbe rectangular pulse translated to A and - A.

4 4.

Ismo>

/'

m

'" ,---, ', '

/ ""


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62 ~ Introduction to Harmonic Analysis

the fonnula r(t - A) + r(t + A). The corresponding Fourier transfonn isr(w)(eiAw + e-iAw) = 2 r(w) cos Aw, where the oscillating factor cos Awaccounts for the so-called interference fringes (Fig. 2.22). If A is very big,

then cos Aw oscillates sharply making the fringes very dense and difficultto distinguish one' from the other; if on the contrary A is very small, then

the fringes are very sparse. In Figure 2.22, A is of the order of magnitudeof 1- so A is comparableto the pulse duration - and the fringes standout nicely (see also Figures 5.7 and 5.8 pertaining to the same phenomenonin two dimensions).

The phenomenon of interference is relevant to many different fields. Oneexample is the principle of interferometry (Chapter 9), which greatly im-proved the resolving power of radiotelescopes, so allowing the developmentof radioastronomy. This in turn chmged our conception of the universe.

2.10 Waves, a unifying concept in science

Since Fourier's time the theory has been greatly extended in dimension1 (Zygmund~[1968], Edwards [1967]), greater than 1 (Stein and Weiss[1971]), andlnore generally on groups and even on discrete structures. Thedecompositidn in waves typical of Fourier analysis finds applications topartial differential equations which lie at the origin of the theory The well


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_.10. Waves, a unifying concept in science 63

The role of hannonic analysis in quantum mechanics has been hinted atin Section 2.8. Moreover the Fourier transfo~ is one of the most widelyused techniques in quantum physics (Abrikosov et al. [1975], Dirac [1947],Walls and Milburn [1994]) from solid state physics to quantum optics andquantum field theory.

This book takes off in yet another direction that relies upon the wave-form model of the electromagnetic spectrum. This leads to major fields ofcontemporary research that make use in a fundamental way of hannonicanalysis in one dimension, such as electrical signals and computer music, intwo dimensions, such as Fourier optics, computerized tomography, and ra-dioastronomy, and in three dimensions, such as crystallography and nuclearmagnetic resonance.

In this we might be following Fourier's own inclination, as reported ina letter by Karl Gustav Jacob Jacobi (1804-1851) to Legendre. Jacobi-

himself thinking that "the only goal of science is the honor of the humanspirit and in this respect a question in the theory of numbers is as valuableas a problem in physics" - refers in the same letter to a rather differentbelief, when he writes: "It is true that Fourier was of the opinion that thechief end of mathematics is the public good and the explanation of the

natural phenomena" (Jacobi [1846]).

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History Fourier Series