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This is a book chapter by Peter J. Olver, dated 12/11/12 and copyrighted 2012, not Phil's own writing. It opens with the history and importance of Fourier series, then in Section 12.1 shows how the heat equation on a periodic bar leads to eigenfunction expansions. It links discrete systems (Ku=f, gradient flow, vibrations) to continuous media and self-adjoint boundary value problems, and later sections review computational techniques and analytical foundations.

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Chapter 12 FourierSeries Just before 1800, the French mathematician/physicist/eng ineer Jean Baptiste Joseph Fourier made an astonishing discovery. As a result of his inv estigations into the partial dif- ferential equations modeling vibration and heat propagati on in bodies, Fourier was led to claimthat“every”function couldberepresented by aninﬁni teseriesofelementary trigono- metric functions — sines and cosines. As an example, conside r the sound produced by a musical instrument, e.g., piano, violin, trumpet, oboe, or drum. Decomposing the signal into its trigonometric constituents reveals the fundament al frequencies (tones, overtones, etc.) that are combined to produce its distinctive timbre. T he Fourier decomposition lies at the heart of modern electronic music; a synthesizer combi nes pure sine and cosine tones to reproduce the diverse sounds of instruments, both natura l and artiﬁcial, according to Fourier’s general prescription. Fourier’s claim was so remarkable and unexpected that most o f the leading mathe- maticians of the time did not believe him. Nevertheless, it w as not long before scientists came to appreciate the power and far-ranging applicability of Fourier’s method, thereby opening up vast new realms of physics, engineering, and else where, to mathematical anal- ysis. Indeed, Fourier’s discovery easily ranks in the “top t en” mathematical advances of all time, a list that would include Newton’s invention of the calculus, and Gauss and Rie- mann’s establishment of diﬀerential geometry that, 70 year s later, became the foundation of Einstein’s general relativity. Fourier analysis is an es sential component of much of mod- ern applied (and pure) mathematics. It forms an exceptional ly powerful analytical tool for solving a broad range of partial diﬀerential equations. App lications in pure mathematics, physics and engineering are almost too numerous to catalogu e — typing in “Fourier” in the subject index of a modern science library will dramatica lly demonstrate just how ubiq- uitous these methods are. Fourier analysis lies at the heart of signal processing, including audio, speech, images, videos, seismic data, radio transmi ssions, and so on. Many modern technological advances, including television, music CD’s and DVD’s, video movies, com- puter graphics, image processing, and ﬁngerprint analysis and storage, are, in one way or another, founded upon the many ramiﬁcations of Fourier’s di scovery. In your career as a mathematician, scientist or engineer, you will ﬁnd that Fou rier theory, like calculus and linear algebra, is one of the most basic and essential tools i n your mathematical arsenal. Mastery of the subject is unavoidable. Furthermore, a remarkably large fraction of modern pure mat hematics is the result of subsequent attempts to place Fourier series on a ﬁrm mathema tical foundation. Thus, all of the student’s “favorite” analytical tools, including th e deﬁnition of a function, the ε–δ deﬁnition of limit and continuity, convergence properties in function space, including uni- 12/11/12 631 c/circleco√yrt2012 Peter J. Olver form convergence, weak convergence, etc., the modern theor y of integration and measure, generalized functions such as the delta function, and many o thers, all owe a profound debt to the prolongedstruggle to establish a rigorousframework for Fourier analysis. Even more remarkably, modern set theory, and, as a result, mathematic al logic and foundations, can be traced directly back to Cantor’s attempts to understand t he sets upon which Fourier series converge! As we will appreciate, Fourier series are, in fact, a very nat ural outgrowth of the basic linear algebra constructions that we have already dev eloped for analyzing discrete dynamical processes. The Fourier representation of a funct ion is a continuous counter- part of the eigenvector expansions used to solve linear syst ems of ordinary diﬀerential equations. The fundamental partial diﬀerential equations governing heat propagation and vibrations in continuous media can be viewed as the function space counterparts of such discrete systems. In the continuous realm, solutions are ex pressed as linear combinations of simple “separable” solutions constructed from the eigen values and eigenfunctions of an associated self-adjoint boundary value problem. The eﬃcac y of Fourier analysis rests on the orthogonality properties of the trigonometric functio ns, which is a direct consequence of their status as eigenfunctions. So, Fourier series can be rightly viewed as a function space version of the ﬁnite-dimensional spectral theory of s ymmetric matrices and orthog- onal eigenvector bases. The main complication is that we mus t now deal with inﬁnite series rather than ﬁnite sums, and so convergence issues tha t do not appear in the ﬁnite- dimensional situation become of paramount importance. Oncewehaveestablishedthepropertheoreticalbackground , thetrigonometricFourier serieswillno longerbeaspecial, isolatedphenomenon, but , rather, initsnaturalcontext as thesimplest representative of a broad class of orthogonal eigenfunctio n expansions based on self-adjoint boundary value problems. Modern and classi cal extensions of the Fourier method, including Fourier integrals, discrete Fourier ser ies, wavelets, Bessel functions, spherical harmonics, as well as the entire apparatus of mode rn quantum mechanics, all rest onthe same basictheoretical foundation, andso gainingfam iliaritywiththe general theory and abstract eigenfunction framework will be essential. Ma ny of the most important cases used in modern physical and engineering applications will a ppear in the ensuing chapters. We begin our development of Fourier methods with a section th at will explain why Fourier series naturally appear when we move from discrete s ystems of ordinary diﬀeren- tial equations to the partial diﬀerential equations that go vern the dynamics of continuous media. The reader uninterested in motivations can safely om it this section as the same material reappears in Chapter 14 when we completely analyze the dynamical partial diﬀer- ential equations that lead to Fourier methods. Beginning in Section 12.2, we shall review, omitting proofs, the most basic computational techniques i n Fourier series, for both ordi- naryand generalizedfunctions. Intheﬁnal section, weincl udeanabbreviatedintroduction to the analytical background required to develop a rigorous foundation for Fourier series methods. 12.1. Dynamical Equations of Continuous Media. The purpose of this section is to discover why Fourier series arise naturally when we move from discrete systems of ordinary diﬀerential equat ions to the partial diﬀerential 12/11/12 632 c/circleco√yrt2012 Peter J. Olver equations that govern the dynamics of continuous mechanica l systems. In our reconstrucd- tion of Fourier’s thought processes, let us start by reviewi ng what we have learned. In Chapter 6, we characterized the equilibrium equations of discrete mechanical and electrical systems as a linear algebraic system Ku=f (12.1) with symmetric, positive (semi-)deﬁnite coeﬃcient matrix K. There are two principal types of dynamical systems associated with such equilibriu m equations. Free vibrations aregovernedbyNewton’sLaw, whichleadstoasecondordersy stemofordinarydiﬀerential equations (9.59), of the form d2u dt2=−Ku. (12.2) On the other hand, the gradient ﬂow equations (9.21), namely du dt=−Ku, (12.3) are designed to decrease the quadratic energy function q(u) =1 2uTKuas rapidly as possible. In each case, the solutionto the system was made by imposing a particular ansatz or inspired guess†for the basic solutions. In the case of a gradient ﬂow, the sol utions are of exponential form e−λtv, while for vibrations they are of trigonometric form cos( ωt)v or sin(ωt)vwithω2=λ. In either case, substituting the relevant solution ansatz reduces the dynamical system to the algebraic eigenvalue problem Kv=λv (12.4) for the matrix K. Each eigenvalue and eigenvector creates a particular solu tion or natural mode, and the general solution to the dynamical system can be expressed as a linear super- position of these fundamental modes. The remarkable fact is that the samemathematical framework, suitably reinterpreted, carries over directly to the continuous realm! In Chapter 11, we developed the equilibrium equations gover ning one-dimensional continuous media — bars, beams, etc. The solution is now a fun ctionu(x) representing, say, displacement of the bar, while the positive (semi-)deﬁ nite matrix is replaced by a certin positive (semi-)deﬁnite linear operator K[u]. Formally, then, under an external forcing function f(x), the equilibrium system can be written the abstract form K[u] =f. (12.5) By analogy with (12.1,3), the corresponding gradient ﬂow sy stem will thus be of the form‡ ∂u ∂t=−K[u]. (12.6) †See the footnote in Example 7.32 for an explanation of this term. ‡Sinceu(t,x) now depends upon time as well as position, we switch from ordinary to p artial derivative notation. 12/11/12 633 c/circleco√yrt2012 Peter J. Olver Such partial diﬀerential equations model diﬀusion process es in which a quadratic energy functional is decreasing as rapidly as possible. A good phys ical example is the ﬂow of heat in a body; the heat disperses throughout the body so as to decrease the thermal energy as quickly as it can, tending (in the absence of extern al heat sources) to thermal equilibrium. Other physical processes modeled by (12.6) in clude diﬀusion of chemicals (solvents, pollutants, etc.), and of populations (animals , bacteria, people, etc.). The simplest and most instructive example is a uniform perio dic (or circular) bar of length 2π. As we saw in Chapter 11, the equilibrium equation for the tem perature u(x) takes the form K[u] =−u′′=f, u (−π) =u(π), u′(−π) =u′(π), (12.7) associated with the positive semi-deﬁnite diﬀerential ope rator K=D∗◦D= (−D)D=−D2(12.8) acting on the space of 2 πperiodic functions. The corresponding gradient ﬂow (12.6) is the partial diﬀerential equation ∂u ∂t=∂2u ∂x2, u (t,−π) =u(t,π),∂u ∂x(t,−π) =∂u ∂x(t,π),(12.9) known as the heat equation since it models (among other diﬀusion processes) heat ﬂow. The solution u(t,x) represents the temperature at position xand time t, and turns out to be uniquely prescribed by the initial distribution: u(0,x) =f(x),−π≤x≤π. (12.10) Heat naturally ﬂows from hot to cold, and so the fact that it ca n be described by a gradient ﬂow should not be surprising; a derivation of (12.9) from phy sical principles will appear in Chapter 14. Solving the periodic heat equation was the semin al problem that led Fourier to develop the profound theory that now bears his name. As in the discrete version, the elemental solutions to a diﬀu sion equation (12.6) are found by introducing an exponential ansatz: u(t,x) =e−λtv(x), (12.11) in which we replace the eigenvector vby a function v(x). These are often referred to as separable solutions to indicate that they are the product of a function of talone times a function of xalone. We substitute the solution formula (12.11) into the d ynamical equations (12.6). We compute ∂u ∂t=∂ ∂t/bracketleftbig e−λtv(x)/bracketrightbig =−λe−λtv(x),while−K[u] =−K/bracketleftbig e−λtv(x)/bracketrightbig =−e−λtK[v], since the exponential factor is a function of t, whileKonly involves diﬀerentiation with respect to x. Equatingthese two expressions and canceling thecommonex ponential factor, we conclude that v(x) must solve a boundary value problem of the form K[v] =λv. (12.12) 12/11/12 634 c/circleco√yrt2012 Peter J. Olver We interpret λas theeigenvalue andv(x) as the corresponding eigenfunction for the operator Ksubjecttotherelevantboundaryconditions. Eacheigenval ueandeigenfunction pair will produce a solution (12.11) to the partial diﬀerent ial equation, and the general solution can be built up through linear superposition. For example, substitution of the exponential ansatz (12.11 ) into the periodic heat equation (12.9) leads to the eigenvalue problem v′′+λv= 0, v (−π) =v(π), v′(−π) =v′(π),(12.13) for theeigenfunction v(x). Now, it is not hard to show that if λ <0 orλis complex, then the only periodic solution to (12.13) is the trivial solutio nv(x)≡0. Thus, all eigenvalues must be real and non-negative: λ≥0. This is not an accident — as we will discuss in detail in Section 14.7, it is a direct consequence of the posi tive semi-deﬁniteness of the underlying diﬀerential operator (12.8). When λ= 0, periodicity singles out the nonzero constants v(x)≡c/negationslash= 0 as the associated eigenfunctions. If λ=ω2>0, then the general solution to the diﬀerential equation (12.13) is a linear com bination v(x) =acosωx+bsinωx ofthebasis solutions. Anonzero function ofthisform wills atisfy the2 πperiodicboundary conditions if and only if ω=kis an integer. Therefore, the eigenvalues λ=k2, 0≤k∈N, are the squares of positive integers. Each positive eigenva lueλ=k2>0 admits two linearly independent eigenfunctions, namely sin kxand coskx, while the zero eigenvalue λ= 0 has only one, the constant function 1. We conclude that the basic trigonometric functions 1,cosx,sinx,cos2x,sin2x,cos3x, ... (12.14) form a complete system of independent eigenfunctions for th e periodic boundary value problem (12.13). By construction, each eigenfunction gives rise to a particu lar solution (12.11) to the periodic heat equation (12.9). We have therefore discovere d an inﬁnite collection of inde- pendent solutions: uk(x) =e−k2tcoskx,/tildewideuk(x) =e−k2tsinkx, k = 0,1,2,3,.... Linear superposition tells us that ﬁnite linear combinatio ns of solutions are also solutions. However, these will notsuﬃce to describe the general solution to the problem, and so we are led to propose an inﬁnite series† u(t,x) =a0 2+∞/summationdisplay k=1/bracketleftBig ake−k2tcoskx+bke−k2tsinkx/bracketrightBig (12.15) †For technical reasons, one takes the basic null eigenfunction to be1 2instead of 1. The explanation will be revealed in the following section. 12/11/12 635 c/circleco√yrt2012 Peter J. Olver to represent the general solution to the periodic heat equat ion. As in the discrete ver- sion, the coeﬃcients ak,bkare found by substituting the solution formula into the init ial condition (12.10), whence u(0,x) =a0 2+∞/summationdisplay k=1/bracketleftbig akcoskx+bksinkx/bracketrightbig =f(x). (12.16) Theresult isthe Fourier series representation ofthe initialtemperaturedistribution. O nce we have prescribed the Fourier coeﬃcients ak,bk, (12.15) provides an explicit formula for the solution to the periodic initial-boundary value proble m for the heat equation. However, since we are dealing with inﬁnite series, the prece ding is purely a formal construction, and requires some serious mathematical anal ysis to place it on a ﬁrm footing. The key questions are •First, when does such an inﬁnite trigonometric series conve rge? •Second, what kinds of functions f(x) can be represented by a convergent Fourier series? •Third, if we have such an f, how do we determine its Fourier coeﬃcients ak,bk? •And lastly, since we are trying to solve diﬀerential equatio ns, can we safely diﬀerentiate a Fourier series? These are the fundamental questions of Fourier analysis, an d must be properly dealt with before we can make any serious progress towards solving the h eat equation. A similar analysis applies to a second order dynamical syste m of the Newtonian form ∂2u ∂t2=−K[u]. (12.17) Such diﬀerentialequationsareused todescribethefreevib rationsofcontinuousmechanical systems, such as bars, strings, and, in higher dimensions, m embranes, solid bodies, ﬂuids, etc. For example, the vibration system (12.17) correspondi ng to the diﬀerential operator (12.8) is the wave equation ∂2u ∂t2=∂2u ∂x2. (12.18) The wave equation models stretching vibrations of a bar, sou nd vibrations in a column of air, e.g., inside a wind instrument, transverse vibrations of a string, e.g., a violin string, surfaces waves on a ﬂuid, electromagnetic waves, and a wide v ariety of other vibrational and wave phenomena. As always, we need to impose suitable boundary conditions in order to proceed. Con- sider, for example, the wave equation with homogeneous Diri chlet boundary conditions ∂2u ∂t2=∂2u ∂x2, u (t,0) = 0, u (t,ℓ) = 0, (12.19) that models, for instance, the vibrations of a uniform violi n string whose ends are tied down. Adapting our discrete trigonometric ansatz, we are na turally led to look for a separable solution of the form u(t,x) = cos(ωt)v(x) (12 .20) 12/11/12 636 c/circleco√yrt2012 Peter J. Olver in which ωrepresents the vibrational frequency. Substituting into t he wave equation and the associated boundary conditions, we deduce that v(x) must be a solution to the eigenvalue problem d2v dx2+ω2v= 0, v (0) = 0 = v(ℓ), (12.21) in which ω2=λplays the role of the eigenvalue. We require a nonzero solution to this linear boundary value problem, and this requires ω2to be strictly positive. As above, this can be checked by directly solving the boundary value pr oblem, but is, in fact, a consequence of positive deﬁniteness; see Section 14.7 for d etails. Assuming ω2>0, the general solution to the diﬀerential equation is a trigonome tric function v(x) =acosωx+bsinωx. The boundary condition at x= 0 requires a= 0, and so v(x) =bsinωx. The second boundary condition requires v(ℓ) =bsinωℓ= 0. Assuming b/negationslash= 0, as otherwise the solution is trivial, ωℓmust be an integer multiple of π. Thus, the natural frequencies of vibration are ωk=kπ ℓ, k = 1,2,3,... . The corresponding eigenfunctions are vk(x) = sinkπx ℓ, k = 1,2,3,.... (12.22) Thus, we ﬁnd the following natural modes of vibration of the w ave equation: uk(t,x) = coskπt ℓsinkπx ℓ,/tildewideuk(t,x) = sinkπt ℓsinkπx ℓ. Each solution represents a spatially periodic standing wav e form. We expect to write the general solution to the boundary value problem as an inﬁnite series u(t,x) =∞/summationdisplay k=1/parenleftbigg bkcoskπt ℓsinkπx ℓ+dksinkπt ℓsinkπx ℓ/parenrightbigg (12.23) in the natural modes. Interestingly, in this case at each ﬁxe dt, there are no cosine terms, and so we have a more specialized type of Fourier series. The s ame convergence issues for suchFourier sine series arise. It turns out that the general theory of Fourier series will also cover Fourier sine series. We have now completed our brief introduction to the dynamica l equations of continu- ousmediaandtheFourierseriesmethodofsolution. Thestud ent shouldnowbesuﬃciently motivated, and it is time to delve into the underlying Fourie r theory. In Chapter 14 we will return to the applications to the one-dimensional heat and wave equations. 12/11/12 637 c/circleco√yrt2012 Peter J. Olver 12.2. Fourier Series. While the need to solve physically interesting partial diﬀe rential equations served as our (and Fourier’s) initial motivation, the remarkable r ange of applications qualiﬁes Fourier’s discovery as one of the most important in all of mat hematics. We therefore take some time to properly develop the basic theory of Fourier ser ies and, in the following chapter, a number of important extensions. Then, properly e quipped, we will be in a position to return to the source — solving partial diﬀerenti al equations. The starting point is the need to represent a given function f(x), deﬁned for −π≤ x≤π, as a convergent series in the elementary trigonometric fun ctions: f(x) =a0 2+∞/summationdisplay k=1[akcoskx+bksinkx]. (12.24) The ﬁrst order of business is to determine the formulae for th e Fourier coeﬃcients ak,bk. The key is orthogonality. We already observed, in Example 5. 12, that the trigonometric functions are orthogonal with respect to the rescaled L2inner product /angbracketleftf;g/angbracketright=1 π/integraldisplayπ −πf(x)g(x)dx (12.25) on the interval†[−π,π]. The explicit orthogonality relations are /angbracketleftcoskx;coslx/angbracketright=/angbracketleftsinkx;sinlx/angbracketright= 0, /angbracketleftcoskx;sinlx/angbracketright= 0, /bardbl1/bardbl=√ 2,/bardblcoskx/bardbl=/bardblsinkx/bardbl= 1,fork/negationslash=l, for allk,l, fork/negationslash= 0,(12.26) wherekandlindicate non-negative integers. Remark: If we were to replace the constant function 1 by1√ 2, then the resulting functions would form an orthonormal system. However, this e xtra√ 2 turns out to be utterly annoying, and is best omitted from the outset. Remark: Orthogonality of the trigonometric functions is not an acc ident, but follows from their status as eigenfunctions for the self-adjoint bo undary value problem (12.13). The general result, to be presented in Section 14.7, is the fu nction space analog of the orthogonality of eigenvectors of symmetric matrices, cf. T heorem 8.20. If we ignore convergence issues for the moment, then the orth ogonality relations (12.26) serve to prescribe the Fourier coeﬃcients: Taking t he inner product of both sides †We have chosen the interval [ −π,π] for convenience. A common alternative is the interval [0,2π]. In fact, since the trigonometric functions are 2 πperiodic, any interval of length 2 π will serve equally well. Adapting Fourier series to intervals of ot her lengths will be discussed in Section 12.4. 12/11/12 638 c/circleco√yrt2012 Peter J. Olver with cos lxforl >0, and invoking the underlying linearity‡of the inner product, yields /angbracketleftf;coslx/angbracketright=a0 2/angbracketleft1;coslx/angbracketright+∞/summationdisplay k=1[ak/angbracketleftcoskx;coslx/angbracketright+bk/angbracketleftsinkx;coslx/angbracketright] =al/angbracketleftcoslx;coslx/angbracketright=al, since, bytheorthogonalityrelations(12.26),alltermsbu tthelthvanish. Thisservestopre- scribe the Fourier coeﬃcient al. A similar manipulation with sin lxﬁxesbl=/angbracketleftf;sinlx/angbracketright, while taking the inner product with the constant function 1 g ives /angbracketleftf;1/angbracketright=a0 2/angbracketleft1;1/angbracketright+∞/summationdisplay k=1[ak/angbracketleftcoskx;1/angbracketright+bk/angbracketleftsinkx;1/angbracketright] =a0 2/bardbl1/bardbl2=a0, which agrees with the preceding formula for alwhenl= 0, and explains why we include the extra factor of1 2in the constant term. Thus, if the Fourier series converges to the function f(x), then its coeﬃcients are prescribed by taking inner product s with the basic trigonometric functions . The alert reader may recognize the preceding argument — it i s thefunctionspaceversionofourderivationorthefundamen talorthonormalandorthogonal basis formulae (5.4,7), which are valid in any inner product space. The key diﬀerence here is that we are dealing with inﬁnite series instead of ﬁnite su ms, and convergence issues must be properly addressed. However, we defer these more del icate considerations until after we have gained some basic familiarity with how Fourier series work in practice. Let us summarize where we are with the following fundamental deﬁnition. Deﬁnition 12.1. TheFourier series of a function f(x) deﬁned on −π≤x≤πis the inﬁnite trigonometric series f(x)∼a0 2+∞/summationdisplay k=1[akcoskx+bksinkx], (12.27) whose coeﬃcients are given by the inner product formulae ak=1 π/integraldisplayπ −πf(x)coskxdx, k = 0,1,2,3,..., bk=1 π/integraldisplayπ −πf(x)sinkxdx, k = 1,2,3,....(12.28) Notethatthefunction f(x)cannotbecompletelyarbitrary,since, attheveryleast, t he integrals in the coeﬃcient formulae must be well deﬁned and ﬁ nite. Even if the coeﬃcients (12.28) are ﬁnite, there is no guarantee that the resulting F ourier series converges, and, even if it converges, no guarantee that it converges to the or iginal function f(x). For these reasons, we use the ∼symbol instead of an equals sign when writing down a Fourier s eries. Before tackling these key issues, let us look at an elementar y example. ‡More rigorously, linearity only applies to ﬁnite linear combinations, n ot inﬁnite series. Here, thought, we are just trying to establish and motivate the basic formulae , and can safely defer such technical complications until the ﬁnal section. 12/11/12 639 c/circleco√yrt2012 Peter J. Olver Example 12.2. Consider the function f(x) =x. We may compute its Fourier coeﬃcients directly, employing integration by parts to eva luate the integrals: a0=1 π/integraldisplayπ −πxdx= 0, ak=1 π/integraldisplayπ −πxcoskxdx=1 π/bracketleftbiggxsinkx k+coskx k2/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleπ x=−π= 0, bk=1 π/integraldisplayπ −πxsinkxdx=1 π/bracketleftbigg −xcoskx k+sinkx k2/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleπ x=−π=2 k(−1)k+1. (12.29) Therefore, the Fourier cosine coeﬃcients of the function xall vanish, ak= 0, and its Fourier series is x∼2/parenleftbigg sinx−sin2x 2+sin3x 3−sin4x 4+···/parenrightbigg . (12.30) Convergence of this series is notan elementary matter. Standard tests, including the ratio and root tests, fail to apply. Even if we know that the series c onverges (which it does — for all x), it is certainly not obvious what function it converges to. Indeed, it cannot converge to the function f(x) =xfor all values of x. If we substitute x=π, then every term in the series is zero, and so the Fourier series converge s to 0 — which is not the same asf(π) =π. Thenthpartial sum of a Fourier series is the trigonometric polynomial† sn(x) =a0 2+n/summationdisplay k=1[akcoskx+bksinkx]. (12.31) By deﬁnition, the Fourier series converges at a point xif and only if the partial sums have a limit: lim n→∞sn(x) =/tildewidef(x), (12.32) which may or may not equal the value of the original function f(x). Thus, a key require- ment is to formulate conditions on the function f(x) that guarantee that the Fourier series converges, and, even more importantly, the limiting sum rep roduces the original function: /tildewidef(x) =f(x). This will all be done in detail below. Remark: The passage from trigonometric polynomials to Fourier ser ies is similar to the passage from polynomials to power series. A power series f(x)∼c0+c1x+···+cnxn+···=∞/summationdisplay k=0ckxk can be viewed as an inﬁnite linear combination of the basic mo nomials 1 ,x,x2,x3,.... According to Taylor’s formula, (C.3), the coeﬃcients ck=f(k)(0) k!are given in terms of †The reason for the term “trigonometric polynomial” was discussed at length in Exam- ple 2.17( c). 12/11/12 640 c/circleco√yrt2012 Peter J. Olver the derivatives of the function at the origin. The partial su ms sn(x) =c0+c1x+···+cnxn=n/summationdisplay k=0ckxk of a power series are ordinary polynomials, and the same conv ergence issues arise. Although superﬁcially similar, in actuality the two theori es are profoundly diﬀerent. Indeed, while the theory of power series was well establishe d in the early days of the cal- culus, there remain, to this day, unresolved foundational i ssues in Fourier theory. A power series either converges everywhere, or on an interval cente red at 0, or nowhere except at 0. (See Section 16.2 for additional details.) On the other hand , a Fourier series can converge on quite bizarre sets. In fact, the detailed analysis of the c onvergence properties of Fourier series led the nineteenth century German mathematician Geo rg Cantor to formulate mod- ern set theory, and, thus, played a seminal role in the establ ishment of the foundations of modern mathematics. Secondly, when a power series converge s, it converges to an analytic function, which is inﬁnitely diﬀerentiable, and whose deri vatives are represented by the power series obtained by termwise diﬀerentiation. Fourier series may converge, not only to periodic continuous functions, but also to a wide variety of discontinuous functions and, even, when suitably interpreted, to generalized functions likethe delta function! Therefore, the termwise diﬀerentiation of a Fourier series is a nontriv ial issue. Once one comprehends how diﬀerent the two subjects are, one b egins to understand why Fourier’s astonishing claims were initially widely dis believed. Before the advent of Fourier, mathematicians only accepted analytic functions as the genuine article. The fact thatFourierseriescanconvergetononanalytic,evendisco ntinuousfunctionswasextremely disconcerting, and resulted in a complete re-evaluation of function theory, culminating in the modern deﬁnition of function that you now learn in ﬁrst ye ar calculus. Only through the combined eﬀorts of many of the leading mathematicians of the nineteenth century was a rigorous theory of Fourier series ﬁrmly established; see S ection 12.5 for the main details and the advanced text [ 199] for a comprehensive treatment. Periodic Extensions The trigonometric constituents (12.14) of a Fourier series are all periodic functions of period 2 π. Therefore, if the series converges, the limiting function /tildewidef(x) must also be periodic of period 2 π: /tildewidef(x+2π) =/tildewidef(x) for all x∈R. A Fourier series can only converge to a 2πperiodic function . So it was unreasonable to expect the Fourier series (12.30) to converge to the non-per iodic tof(x) =xeverywhere. Rather, it should converge to its periodic extension, as we n ow deﬁne. Lemma 12.3. Iff(x)is any function deﬁned for −π < x≤π, then there is a unique 2πperiodic function /tildewidef, known as the 2πperiodic extension off, that satisﬁes /tildewidef(x) =f(x) for all−π < x≤π. 12/11/12 641 c/circleco√yrt2012 Peter J. Olver -5 5 10 15 -3-2-1123 Figure 12.1. Periodic extension of x. Proof: Pictorially, the graph of the periodic extension of a funct ionf(x) is obtained by repeatedly copying that part of the graph of fbetween −πandπto adjacent intervals of length 2 π; Figure 12.1 shows a simple example. More formally, given x∈R, there is a unique integer mso that (2 m−1)π < x≤(2m+1)π. Periodicity of /tildewidefleads us to deﬁne /tildewidef(x) =/tildewidef(x−2mπ) =f(x−2mπ), (12.33) noting that if −π < x≤π, thenm= 0 and hence /tildewidef(x) =f(x) for such x. The proof that the resulting function /tildewidefis 2πperiodic is left as Exercise . Q.E.D. Remark: The construction of the periodic extension of Lemma 12.3 us es the value f(π) at the right endpoint and requires /tildewidef(−π) =/tildewidef(π) =f(π). One could, alternatively, require/tildewidef(π) =/tildewidef(−π) =f(−π), which, if f(−π)/negationslash=f(π), leads to a slightly diﬀerent 2πperiodic extension of the function. There is no a priori reason to prefer one over the other. In fact, for Fourier theory, as we shall discover, one should use neither, but rather an “average” of the two. Thus, the preferred Fourier periodi c extension/tildewidef(x) will satisfy /tildewidef(π) =/tildewidef(−π) =1 2/bracketleftbig f(π)+f(−π)/bracketrightbig , (12.34) which then ﬁxes its values at the odd multiples of π. Example 12.4. The 2πperiodic extension /tildewidef(x) off(x) =xis the “sawtooth” function graphed in Figure 12.1. It agrees with xbetween −πandπ. Since f(π) = π, f(−π) =−π, the Fourier extension (12.34) sets /tildewidef(kπ) = 0 for any odd integer k. Explicitly, /tildewidef(x) =/braceleftbiggx−2mπ,(2m−1)π < x < (2m+1)π, 0, x = (2m−1)π,wheremis any integer. With this convention, it can be proved that the Fourier serie s (12.30) converges everywhere to the 2πperiodic extension /tildewidef(x). In particular, 2∞/summationdisplay k=1(−1)k+1sinkx k=/braceleftbiggx,−π < x < π, 0, x=±π.(12.35) 12/11/12 642 c/circleco√yrt2012 Peter J. Olver Even this very simple example has remarkable and nontrivial consequences. For in- stance, if we substitute x=1 2πin (12.30) and divide by 2, we obtain Gregory’s series π 4= 1−1 3+1 5−1 7+1 9− ···. (12.36) While this striking formula predates Fourier theory — it was , in fact, ﬁrst discovered by Leibniz — a direct proof is not easy. Remark: While numerologically fascinating, Gregory’s series is o f scant practical use for actually computing πsince its rate of convergence is painfully slow. The reader m ay wish to try adding up terms to see how far out one needs to go to a ccurately compute even the ﬁrst two decimal digits of π. Round-oﬀ errors will eventually interfere with any attempt to compute the complete summation to any reasonable degree of accuracy. Piecewise Continuous Functions As we shall see, all continuously diﬀerentiable, 2 πperiodic functions can be repre- sented as convergent Fourier series. More generally, we can allow the function to have some simple discontinuities. Although not the most general class of functions that pos- sess convergent Fourier series, such “piecewise continuou s” functions will suﬃce for all the applications we consider in this text. Deﬁnition 12.5. A function f(x) is said to be piecewise continuous on an interval [a,b] if it is deﬁned and continuous except possibly at a ﬁnite num ber of points a≤x1< x2< ... < xn≤b. At each point of discontinuity, the left and right hand limi ts† f(x− k) = lim x→x− kf(x), f (x+ k) = lim x→x+ kf(x), exist. Note that we do not require that f(x) be deﬁned at xk. Even if f(xk) is deﬁned, it does not necessarily equal either the left or the right hand l imit. A function f(x) deﬁned for all x∈Ris piecewise continuous provided it is piece- wise continuous on every bounded interval. In particular, a 2πperiodic function /tildewidef(x) is piecewise continuous if and only if it is piecewise continuo us on the interval [ −π,π]. A representative graph of a piecewise continuous function a ppears in Figure 12.2. The pointsxkare known as jump discontinuities off(x) and the diﬀerence βk=f(x+ k)−f(x− k) = lim x→x+ kf(x)−lim x→x− kf(x) (12 .37) between the left and right hand limits is the magnitude of the jump, cf. (11.49). If βk= 0, and so the right and left hand limits agree, then the disconti nuity is removable since redeﬁning f(xk) =f(x+ k) =f(x− k) makes fcontinuous at xk. We will assume, without signiﬁcant loss of generality, that our functions have no re movable discontinuities. †At the endpoints a,bwe only require one of the limits, namely f(a+) andf(b−), to exist. 12/11/12 643 c/circleco√yrt2012 Peter J. Olver -1 1 2 3 4 -1-0.50.51 Figure 12.2. Piecewise Continuous Function. The simplest example of a piecewise continuous function is t he step function σ(x) =/braceleftbigg1, x > 0, 0, x < 0.(12.38) It has a single jump discontinuity at x= 0 of magnitude 1, and is continuous — indeed, constant — everywhere else. If we translate and scale the ste p function, we obtain a function h(x) =βσ(x−y) =/braceleftbiggβ, x > y, 0, x < y,(12.39) with a single jump discontinuity of magnitude βat the point x=y. Iff(x) is any piecewise continuous function, then its Fourier coe ﬃcients are well- deﬁned — the integrals (12.28) exist and are ﬁnite. Continui ty, however, is not enough to ensure convergence of the resulting Fourier series. Deﬁnition 12.6. A function f(x) is called piecewise C1on an interval [ a,b] if it is deﬁned, continuous and continuously diﬀerentiable except possibly at a ﬁnite number of pointsa≤x1< x2< ... < xn≤b. At each exceptional point, the left and right hand limits†exist: f(x− k) = lim x→x− kf(x), f (x+ k) = lim x→x+ kf(x), f′(x− k) = lim x→x− kf′(x), f′(x+ k) = lim x→x+ kf′(x). See Figure 12.3 for a representative graph. For a piecewise c ontinuous C1function, an exceptional point xkis either •ajump discontinuity off, but where the left and right hand derivatives exist, or •acorner, meaning a point where fis continuous, so f(x− k) =f(x+ k), but has diﬀerent left and right hand derivatives: f′(x− k)/negationslash=f′(x+ k). †As before, at the endpoints we only require the appropriate one-sided limits, namely f(a+), f′(a+) andf(b−),f′(b−), to exist. 12/11/12 644 c/circleco√yrt2012 Peter J. Olver -1 1 2 3 4 -1-0.50.51 Figure 12.3. Piecewise C1Function. Thus, at each point, including jump discontinuities, the gr aph off(x) has well-deﬁned right and left tangent lines. For example, the function f(x) =|x|is piecewise C1since it is continuous everywhere and has a corner at x= 0, with f′(0+) = +1,f′(0−) =−1. There is an analogous deﬁnition of a piecewise Cnfunction. One requires that the function has ncontinuous derivatives, except at a ﬁnite number of points. Moreover, at every point, the function has well-deﬁned right and left han d limits of all its derivatives up to order n. The Convergence Theorem We are now able to state the fundamental convergence theorem for Fourier series. Theorem 12.7. If/tildewidef(x)is any2πperiodic, piecewise C1function, then, for any x∈R, its Fourier series converges to /tildewidef(x), if/tildewidefis continuous at x, 1 2/bracketleftbig/tildewidef(x+)+/tildewidef(x−)/bracketrightbig , ifxis a jump discontinuity. Thus, the Fourier series converges, as expected, to /tildewidef(x) at all points of continuity; at discontinuities, the Fourier series can’t decide whethe r to converge to the right or left hand limit, and so ends up “splitting the diﬀerence” by conve rging to their average; see Figure 12.4. If we redeﬁne /tildewidef(x) at its jump discontinuities to have the average limiting value, so /tildewidef(x) =1 2/bracketleftbig/tildewidef(x+)+/tildewidef(x−)/bracketrightbig , (12.40) — an equation that automatically holds at all points of conti nuity — then Theorem 12.7 would say that the Fourier series converges to /tildewidef(x) everywhere. We will discuss the ideas underlying the proof of the Convergence Theorem 12.7 at the e nd of Section 12.5. Example 12.8. Letσ(x) denote the step function (12.38). Its Fourier coeﬃcients 12/11/12 645 c/circleco√yrt2012 Peter J. Olver Figure 12.4. Splitting the Diﬀerence. are easily computed: a0=1 π/integraldisplayπ −πσ(x)dx=1 π/integraldisplayπ 0dx= 1, ak=1 π/integraldisplayπ −πσ(x)coskxdx=1 π/integraldisplayπ 0coskxdx= 0, bk=1 π/integraldisplayπ −πσ(x)sinkxdx=1 π/integraldisplayπ 0sinkxdx=  2 kπ, k= 2l+1 odd, 0, k = 2leven. Therefore, the Fourier series for the step function is σ(x)∼1 2+2 π/parenleftbigg sinx+sin3x 3+sin5x 5+sin7x 7+···/parenrightbigg . (12.41) According to Theorem 12.7, the Fourier series will converge to the 2πperiodic extension of the step function: /tildewideσ(x) =  0,(2m−1)π < x < 2mπ, 1,2mπ < x < (2m+1)π, 1 2, x=mπ,wheremis any integer, which is plotted in Figure 12.5. Observe that, in accordance with Theorem 12.7, /tildewideσ(x) takes the midpoint value1 2at the jump discontinuities 0 ,±π,±2π,.... It is instructive to investigate the convergence of this par ticular Fourier series in some detail. Figure 12.6 displays a graph of the ﬁrst few part ial sums, taking, respectively, n= 3,5, and 10 terms. The reader will notice that away from the disc ontinuities, the series does appear to be converging, albeit slowly. However, near t he jumps there is a consistent overshoot of about 9%. The region where the overshoot occurs becomes narrower and narrower as the number of terms increases, but the magnitude of the overshoot persists no matter how many terms are summed up. This was ﬁrst noted by the American physicist Josiah Gibbs, and is now known as the Gibbs phenomenon in his honor. The Gibbs overshoot is a manifestation of the subtle non-uniform conv ergence of the Fourier series. 12/11/12 646 c/circleco√yrt2012 Peter J. Olver -5 5 10 15 -1-0.50.51 Figure 12.5. Periodic Step Function. -3 -2 -1 1 2 3 -1-0.50.51 -3 -2 -1 1 2 3 -1-0.50.51 -3 -2 -1 1 2 3 -1-0.50.51 Figure 12.6. Gibbs Phenomenon. Even and Odd Functions We already noted that the Fourier cosine coeﬃcients of the fu nctionf(x) =xare all 0. This is not an accident, but rather a direct consequence of the fact that xis an odd function. Recall ﬁrst the basic deﬁnition: Deﬁnition 12.9. A function is called eveniff(−x) =f(x). A function is oddif f(−x) =−f(x). For example, the functions 1, cos kx, andx2are all even, whereas x, sinkx, and sign x are odd. We require two elementary lemmas, whose proofs are l eft to the reader. Lemma 12.10. The sum, f(x)+g(x), of two even functions is even; the sum of two odd functions is odd. The product f(x)g(x)of two even functions, or of two odd functions, is an even function. The product of an even and an odd function is odd. Remark:Everyfunctioncanberepresentedasthesumofanevenandanoddfun ction; see Exercise . 12/11/12 647 c/circleco√yrt2012 Peter J. Olver Lemma 12.11. Iff(x)is odd and integrable on the symmetric interval [−a,a], then/integraldisplaya −af(x)dx= 0. Iff(x)is even and integrable, then/integraldisplaya −af(x)dx= 2/integraldisplaya 0f(x)dx. The next result is an immediate consequence of applying Lemm as 12.10 and 12.11 to the Fourier integrals (12.28). Proposition 12.12. Iff(x)is even, then its Fourier sine coeﬃcients all vanish, bk= 0, and so fcan be represented by a Fourier cosine series f(x)∼a0 2+∞/summationdisplay k=1akcoskx, (12.42) where ak=2 π/integraldisplayπ 0f(x)coskxdx, k = 0,1,2,3,... . (12.43) Iff(x)isodd, thenitsFouriercosinecoeﬃcientsvanish, ak= 0,andsofcanberepresented by aFourier sine series f(x)∼∞/summationdisplay k=1bksinkx, (12.44) where bk=2 π/integraldisplayπ 0f(x)sinkxdx, k = 1,2,3,... . (12.45) Conversely, a convergent Fourier cosine (respectively, sine )series alwaysrepresents an even (respectively, odd )function. Example 12.13. The absolute value f(x) =|x|is an even function, and hence has a Fourier cosine series. The coeﬃcients are a0=2 π/integraldisplayπ 0xdx=π, (12.46) ak=2 π/integraldisplayπ 0xcoskxdx=2 π/bracketleftbiggxsinkx k+coskx k2/bracketrightbiggπ x=0=  0, 0/negationslash=keven, −4 k2π, kodd. Therefore |x| ∼π 2−4 π/parenleftbigg cosx+cos3x 9+cos5x 25+cos7x 49+···/parenrightbigg . (12.47) AccordingtoTheorem12.7,thisFouriercosineseriesconve rgestothe2 πperiodicextension of|x|, the “sawtooth function” graphed in Figure 12.7. In particular, if we substitute x= 0, we obtain another interesting series π2 8= 1 +1 9+1 25+1 49+···=∞/summationdisplay n=01 (2n+1)2. (12.48) 12/11/12 648 c/circleco√yrt2012 Peter J. Olver -5 5 10 15 -3-2-1123 Figure 12.7. Periodic extension of |x|. ItconvergesfasterthanGregory’sseries(12.36),and, whi lefarfromoptimalinthisregards, can be used to compute reasonable approximations to π. One can further manipulate this result to compute the sum of the series S=∞/summationdisplay n=11 n2= 1 +1 4+1 9+1 16+1 25+1 36+1 49+···. We note that S 4=∞/summationdisplay n=11 4n2=∞/summationdisplay n=11 (2n)2=1 4+1 16+1 36+1 64+···. Therefore, by (12.48), 3 4S=S−S 4= 1 +1 9+1 25+1 49+···=π2 8, from which we conclude that S=∞/summationdisplay n=11 n2= 1 +1 4+1 9+1 16+1 25+···=π2 6. (12.49) Remark: The most famous function in number theory — and the source of the most outstanding problem in mathematics, the Riemann hypothesi s — is the Riemann zeta function ζ(s) =∞/summationdisplay n=11 ns. (12.50) Formula (12.49) shows that ζ(2) =1 6π2. In fact, the value of the zeta function at any even positive integer s= 2ncan be written as a rational polynomial in π. Iff(x) is any function deﬁned on [0 ,π], then its Fourier cosine series is deﬁned by the formulas (12.42–43); the resulting series represents i ts even, 2 πperiodic extension. For example, the cosine series of f(x) =xis given in (12.47); indeed the even, 2 πperiodic 12/11/12 649 c/circleco√yrt2012 Peter J. Olver extension of xcoincides with the 2 πperiodic extension of |x|. Similarly, the formulas (12.44,45) deﬁne its Fourier sine series , representing its odd, 2 πperiodic extension. In particular, since f(x) =xis already odd, its Fourier sine series concides with its ord inary Fourier series. Complex Fourier Series Analternative, andoftenmoreconvenient, approachtoFour ierseriesistousecomplex exponentials instead of sines and cosines. Indeed, Euler’s formula eikx= coskx+ i sinkx, e−ikx= coskx−i sinkx, (12.51) shows how to write the trigonometric functions coskx=eikx+e−ikx 2,sinkx=eikx−e−ikx 2i, (12.52) in terms of complex exponentials. Orthonormality with resp ect to the rescaled L2Hermi- tian inner product /angbracketleftf;g/angbracketright=1 2π/integraldisplayπ −πf(x)g(x)dx, (12.53) was proved by direct computation in Example 3.45: /angbracketlefteikx;eilx/angbracketright=1 2π/integraldisplayπ −πei(k−l)xdx=/braceleftBigg 1, k=l, 0, k/negationslash=l, /bardbleikx/bardbl2=1 2π/integraldisplayπ −π|eikx|2dx= 1.(12.54) Again, orthogonality follows from their status as (complex ) eigenfunctions for the periodic boundary value problem (12.13). Thecomplex Fourier series for a (piecewise continuous) real or complex function fis f(x)∼∞/summationdisplay k=−∞ckeikx=···+c−2e−2ix+c−1e−ix+c0+c1eix+c2e2ix+···.(12.55) The orthonormality formulae (12.53) imply that the complex Fourier coeﬃcients are ob- tained by taking the inner products ck=/angbracketleftf;eikx/angbracketright=1 2π/integraldisplayπ −πf(x)e−ikxdx (12.56) with the associated complex exponential. Pay attention to t he minus sign in the integrated exponential — the result of taking the complex conjugate of t he second argument in the inner product (12.53). It should be emphasized that the real (12.27) and complex (12.55) Fourier formulae are just two diﬀerent ways of writing the sameseries! Indeed, if we apply 12/11/12 650 c/circleco√yrt2012 Peter J. Olver Euler’s formula (12.51) to (12.56) and compare with the real Fourier formulae (12.28), we ﬁnd that the real and complex Fourier coeﬃcients are related by ak=ck+c−k, bk= i(ck−c−k),ck=1 2(ak−ibk), c−k=1 2(ak+ ibk),k= 0,1,2,... . (12.57) Remark: We already see one advantage of the complex version. The con stant function 1 =e0ixno longer plays an anomalous role — the annoying factor of1 2in the real Fourier series (12.27) has mysteriously disappeared! Example 12.14. Forthestepfunction σ(x)consideredinExample12.8,thecomplex Fourier coeﬃcients are ck=1 2π/integraldisplayπ −πσ(x)e−ikxdx=1 2π/integraldisplayπ 0e−ikxdx=  1 2, k = 0, 0, 0/negationslash=keven, 1 ikπ, kodd. Therefore, the step function has the complex Fourier series σ(x)∼1 2−i π∞/summationdisplay l=−∞e(2l+1)ix 2l+1. You should convince yourself that this is exactly the same series as the real Fourier series (12.41). We are merely rewriting it using complex exponenti als instead of real sines and cosines. Example 12.15. Let us ﬁnd the Fourier series for the exponential function eax. It is much easier to evaluate the integrals for the complex Four ier coeﬃcients, and so ck=/angbracketlefteax;eikx/angbracketright=1 2π/integraldisplayπ −πe(a−ik)xdx=e(a−ik)x 2π(a−ik)/vextendsingle/vextendsingle/vextendsingle/vextendsingleπ x=−π =e(a−ik)π−e−(a−ik)π 2π(a−ik)= (−1)keaπ−e−aπ 2π(a−ik)=(−1)k(a+ ik) sinhaπ π(a2+k2). Therefore, the desired Fourier series is eax∼sinhaπ π∞/summationdisplay k=−∞(−1)k(a+ ik) a2+k2eikx. (12.58) As an exercise, the reader should try writing this as a real Fo urier series, either by breaking up the complex series into its real and imaginary parts, or by direct evaluation of the real coeﬃcientsviatheirintegralformulae(12.28). According toTheorem12.7(whichisequally valid for complex Fourier series) the Fourier series conver ges to the 2 πperiodic extension of the exponential function, graphed in Figure 12.8. 12/11/12 651 c/circleco√yrt2012 Peter J. Olver -5 5 10 155101520 Figure 12.8. Periodic Extension of ex. The Delta Function Fourier series can even be used to represent more general obj ects than mere functions. The most important example is the delta function δ(x). Using its characterizing properties (11.37), the real Fourier coeﬃcients are computed as ak=1 π/integraldisplayπ −πδ(x)coskxdx=1 πcosk0 =1 π, bk=1 π/integraldisplayπ −πδ(x)sinkxdx=1 πsink0 = 0.(12.59) Therefore, δ(x)∼1 2π+1 π/parenleftbig cosx+cos2x+cos3x+···/parenrightbig . (12.60) Sinceδ(x) is an even function, it should come as no surprise that it has a cosine series. To understand in what sense this series converges to the delt a function, it will help to rewrite it in complex form δ(x)∼1 2π∞/summationdisplay k=−∞eikx=1 2π/parenleftbig ···+e−2ix+e−ix+1+eix+e2ix+···/parenrightbig .(12.61) where the complex Fourier coeﬃcients are computed†as ck=1 2π/integraldisplayπ −πδ(x)e−ikxdx=1 2π. †Or, we could use (12.57). 12/11/12 652 c/circleco√yrt2012 Peter J. Olver Figure 12.9. Partial Fourier Sums Approximating the Delta Function. Thenthpartial sum sn(x) =1 2πn/summationdisplay k=−neikx=1 2π/parenleftbig e−inx+···+e−ix+1+eix+···+einx/parenrightbig can, in fact, be explicitly evaluated. Recall the formula fo r the sum of a geometric series m/summationdisplay k=0ark=a+ar+ar2+···+arm=a/parenleftbiggrm+1−1 r−1/parenrightbigg . (12.62) Thepartialsum sn(x)hasthisform, with m+1 = 2n+1summands, initialterm a=e−inx, and ratio r=eix. Therefore, sn(x) =1 2πn/summationdisplay k=−neikx=1 2πe−inx/parenleftbiggei(2n+1)x−1 eix−1/parenrightbigg =1 2πei(n+1)x−e−inx eix−1 =1 2πei/parenleftbig n+1 2/parenrightbig x−e−i/parenleftbig n+1 2/parenrightbig x eix/2−e−ix/2=1 2πsin/parenleftbig n+1 2/parenrightbig x sin1 2x.(12.63) In this computation, to pass from the ﬁrst to the second line, we multiplied numerator and denominator by e−ix/2, after which we used the formula (3.86) for the sine function in terms of complex exponentials. Incidentally, (12.63) is equivalent to the intriguing trigonometric summation formula sn(x) =1 2π+1 π/parenleftbig cosx+cos2x+cos3x+···+cosnx/parenrightbig =1 2πsin/parenleftbig n+1 2/parenrightbig x sin1 2x.(12.64) Graphs of the partial sums sn(x) for several values of nare displayed in Figure 12.9. Note that the spike, at x= 0, progressively becomes taller and thinner, converging t o an 12/11/12 653 c/circleco√yrt2012 Peter J. Olver inﬁnitely tall, inﬁnitely thin delta spike. Indeed, by l’Hˆ opital’s Rule, lim x→01 2πsin/parenleftbig n+1 2/parenrightbig x sin1 2x= lim x→01 2π/parenleftbig n+1 2/parenrightbig cos/parenleftbig n+1 2/parenrightbig x 1 2cos1 2x=n+1 2 π−→ ∞ asn→ ∞. (An elementary proof of this formula is to note that, at x= 0, every term in the original sum (12.64) is equal to 1.) Furthermore, the integrals remai n ﬁxed, 1 2π/integraldisplayπ −πsn(x)dx=1 2π/integraldisplayπ −πsin/parenleftbig n+1 2/parenrightbig x sin1 2xdx=1 2π/integraldisplayπ −πn/summationdisplay k=−neikxdx= 1,(12.65) asrequiredfor convergence to thedeltafunction. However, awayfrom thespike, thepartial sums do notgo to zero! Rather, they oscillate more and more rapidly, mai ntaining an overall amplitude of1 2πcsc1 2x= 1//parenleftbig 2πsin1 2x/parenrightbig . Asngets large, the amplitude function appears as an envelope of the increasingly rapid oscillatio ns. Roughly speaking, the fact thatsn(x)→δ(x) asn→ ∞means that the “inﬁnitely fast” oscillations somehow cance l each other out, and the net eﬀect is zero away from the spike at x= 0. Thus, the convergence of the Fourier sums to δ(x) is much more subtle than in the original limiting deﬁnition (11.31). The technical term is weak convergence, which plays an very important role in advanced mathematical analysis, [ 159]; see Exercise below for additional details. Remark: Although we stated that the Fourier series (12.60,61) repr esent the delta function, this is not entirely correct. Remember that a Four ier series converges to the 2πperiodic extension of the original function. Therefore, (1 2.61) actually represents the periodic extension of the delta function: /tildewideδ(x) =···+δ(x+4π)+δ(x+2π)+δ(x)+δ(x−2π)+δ(x−4π)+δ(x−6π)+···,(12.66) consisting of a periodic array of delta spikes concentrated at all integer multiples of 2 π. 12.3. Diﬀerentiation and Integration. If a series of functions converges “nicely” then one expects to be able to integrate and diﬀerentiate it term by term; the resulting series shoul d converge to the integral and derivative of the original sum. Integration and diﬀerentia tion of power series is always valid within the range of convergence, and is used extensive ly in the construction of series solutions of diﬀerential equations, series for integrals o f non-elementary functions, and so on. The interested reader can consult Appendix C for further details. As we now appreciate, the convergence of Fourier series is a m uch more delicate mat- ter, andsoonemust beconsiderablymorecareful withtheir d iﬀerentiationandintegration. Nevertheless, in favorable situations, both operations le ad to valid results, and are quite useful for constructing Fourier series of more complicated functions. It is a remarkable, profound fact that Fourier analysis is completely compatib le with the calculus of general- ized functions that we developed in Chapter 11. For instance , diﬀerentiating the Fourier series for a piecewise C1function leads to the Fourier series for the diﬀerentiated f unction that has delta functions of the appropriate magnitude appea ring at each jump discontinu- ity. This fact reassures us that the rather mysterious const ruction of delta functions and 12/11/12 654 c/circleco√yrt2012 Peter J. Olver their generalizations is indeed the right way to extend calc ulus to functions which do not possess derivatives in the ordinary sense. Integration of Fourier Series Integration is a smoothing operation — the integrated funct ion is always nicer than the original. Therefore, we should anticipate being able to integrate Fourier series without diﬃculty. There is, however, one complication: the integra l of a periodic function is not necessarily periodic. The simplest example is the constant function 1, which is certainly periodic, but its integral, namely x, is not. On the other hand, integrals of all the other periodic sine and cosine functions appearing in the Fourier series are periodic. Thus, only theconstanttermmightcauseusdiﬃcultywhenwetrytointeg rateaFourierseries(12.27). According to (2.4), the constant term a0 2=1 2π/integraldisplayπ −πf(x)dx (12.67) is themeanoraverageof the function f(x) on the interval [ −π,π]. A function has no constant term in its Fourier series if and only if it has mean z ero. It is easily shown, cf. Exercise , that the mean zero functions are precisely the ones that rem ain periodic upon integration. Lemma 12.16. Iff(x)is2πperiodic, then its integral g(x) =/integraldisplayx 0f(y)dyis2π periodic if and only if/integraldisplayπ −πf(x)dx= 0, so that fhas mean zero on the interval [−π,π]. In particular, Lemma 12.11 implies that all odd functions au tomatically have mean zero, and hence periodic integrals. Since /integraldisplay coskxdx=sinkx k,/integraldisplay sinkxdx=−coskx k, (12.68) termwise integration of a Fourier series without constant t erm is straightforward. The resulting Fourier series is given precisely as follows. Theorem 12.17. Iffis piecewise continuous, 2πperiodic, and has mean zero, then its Fourier series f(x)∼∞/summationdisplay k=1[akcoskx+bksinkx], can be integrated term by term, to produce the Fourier series g(x) =/integraldisplayx 0f(y)dy∼m+∞/summationdisplay k=1/bracketleftbigg −bk kcoskx+ak ksinkx/bracketrightbigg , (12.69) for its periodic integral. The constant term m=1 2π/integraldisplayπ −πg(x)dx (12.70) is the mean of the integrated function. 12/11/12 655 c/circleco√yrt2012 Peter J. Olver In many situations, the integration formula (12.69) provid es a very convenient alter- native to the direct derivation of the Fourier coeﬃcients. Example 12.18. The function f(x) =xis odd, and so has mean zero:/integraldisplayπ −πxdx= 0. Let us integrate its Fourier series x∼2∞/summationdisplay k=1(−1)k−1 ksinkx (12.71) that we found in Example 12.2. The result is the Fourier serie s 1 2x2∼π2 6−2∞/summationdisplay k=1(−1)k−1 k2coskx =π2 6−2/parenleftbigg cosx−cos2x 4+cos3x 9−cos4x 16+···/parenrightbigg ,(12.72) whose the constant term is the mean of the left hand side: 1 2π/integraldisplayπ −πx2 2dx=π2 6. Let us revisit the derivation of the integrated Fourier seri es from a slightly diﬀerent standpoint. If we were to integrate each trigonometric summ and in a Fourier series (12.27) from 0 to x, we would obtain /integraldisplayx 0coskydy=sinkx k,whereas/integraldisplayx 0sinkydy=1 k−coskx k. The extra 1 /kterms arising from the deﬁnite sine integrals do not appear e xplicitly in our previous form for the integrated Fourier series, (12.69 ), and so must be hidden in the constant term m. We deduce that the mean value of the integrated function can be computed using the Fourier sine coeﬃcients of fvia the formula 1 2π/integraldisplayπ −πg(x)dx=m=∞/summationdisplay k=1bk k. (12.73) For example, the result of integrating both sides of the Four ier series (12.71) for f(x) =x from 0 to xis x2 2∼2∞/summationdisplay k=1(−1)k−1 k2(1−coskx). The constant terms sum up to yield the mean value of the integr ated function: 2/parenleftbigg 1−1 4+1 9−1 16+.../parenrightbigg = 2∞/summationdisplay k=1(−1)k−1 k2=1 2π/integraldisplayπ −πx2 2dx=π2 6,(12.74) which reproduces a formula established in Exercise . 12/11/12 656 c/circleco√yrt2012 Peter J. Olver More generally, if f(x) does not have mean zero, its Fourier series has a nonzero constant term, f(x)∼a0 2+∞/summationdisplay k=1[akcoskx+bksinkx]. In this case, the result of integration will be g(x) =/integraldisplayx 0f(y)dy∼a0 2x+m+∞/summationdisplay k=1/bracketleftbigg −bk kcoskx+ak ksinkx/bracketrightbigg ,(12.75) wheremis given in (12.73). The right hand side is not, strictly spea king, a Fourier series. There are two ways to interpret this formula within the Fouri er framework. Either we can write (12.75) as the Fourier series for the diﬀerence g(x)−a0 2x∼m+∞/summationdisplay k=1/bracketleftbigg −bk kcoskx+ak ksinkx/bracketrightbigg , (12.76) which is a 2 πperiodic function, cf. Exercise . Alternatively, one can replace xby its Fourier series (12.30), and the result will be the Fourier se ries for the 2 πperiodic extension of the integral g(x) =/integraldisplayx 0f(y)dy. Diﬀerentiation of Fourier Series Diﬀerentiation has the opposite eﬀect to integration. Diﬀe rentiation makes a function worse. Therefore, to justify taking the derivative of a Four ier series, we need to know that the diﬀerentiated function remains reasonably nice. Since we need the derivative f′(x) to be piecewise C1for the convergence Theorem 12.7 to be applicable, we must re quire that f(x) itself be continuous and piecewise C2. Theorem 12.19. Iffis2πperiodic, continuous, and piecewise C2, then its Fourier series can be diﬀerentiated term by term, to produce the Four ier series for its derivative f′(x)∼∞/summationdisplay k=1/bracketleftbig kbkcoskx−kaksinkx/bracketrightbig . (12.77) Example 12.20. The derivative (11.53) of the absolute value function f(x) =|x| is the sign function d dx|x|= signx=/braceleftbigg+1, x > 0 −1, x < 0. Therefore, if we diﬀerentiate its Fourier series (12.47), w e obtain the Fourier series signx∼4 π/parenleftbigg sinx+sin3x 3+sin5x 5+sin7x 7+···/parenrightbigg . (12.78) Notethat sign x=σ(x)−σ(−x) isthe diﬀerence of two step functions. Indeed, subtractin g the step function Fourier series (12.41) at xfrom the same series at −xreproduces (12.78). 12/11/12 657 c/circleco√yrt2012 Peter J. Olver Example 12.21. If we diﬀerentiate the Fourier series x∼2∞/summationdisplay k=1(−1)k−1 ksinkx= 2/parenleftbigg sinx−sin2x 2+sin3x 3−sin4x 4+···/parenrightbigg , we obtain an apparent contradiction: 1∼2∞/summationdisplay k=1(−1)k+1coskx= 2cosx−2cos2x+2cos3 x−2cos4x+···.(12.79) But the Fourier series for 1 just consists of a single constan t term! (Why?) The resolution of this paradox is not diﬃcult. The Fourier se ries (12.30) does not converge to x, but rather to its periodic extension /tildewidef(x), which has a jump discontinuity of magnitude 2 πat odd multiples of π; see Figure 12.1. Thus, Theorem 12.19 is notdirectly applicable. Nevertheless, we can assign a consistent inter pretation to the diﬀerentiated series. The derivative /tildewidef′(x) of the periodic extension is notequal to the constant function 1, but, rather, has an additional delta function concentrat ed at each jump discontinuity: /tildewidef′(x) = 1−2π∞/summationdisplay j=−∞δ/parenleftbig x−(2j+1)π/parenrightbig = 1−2π/tildewideδ(x−π), where/tildewideδdenotes the 2 πperiodic extension of the delta function, cf. (12.66). The d if- ferentiated Fourier series (12.79) does, in fact, converge to this modiﬁed distributional derivative! Indeed, diﬀerentiation and integration of Fou rier series is entirely compatible with the calculus of generalized functions, as will be borne out in yet another example. Example 12.22. Let us diﬀerentiate the Fourier series (12.41) for the step f unction and see if we end up with the Fourier series (12.60) for the del ta function. We ﬁnd d dxσ(x)∼2 π/parenleftbig cosx+cos3x+cos5x+cos7x+···/parenrightbig , (12.80) which does notagree with (12.60) — half the terms are missing! The explanat ion is similar to the preceding example: the 2 πperiodic extension of the step function has two jump discontinuities, of magnitudes +1 at even multiples of πand−1 at odd multiples. Therefore, its derivative is the diﬀerence of the 2 πperiodic extension of the delta function at 0, with Fourier series (12.60) minus the 2 πperiodic extension of the delta function at π, with Fourier series δ(x−π)∼1 2π+1 π/parenleftbig −cosx+cos2x−cos3x+···/parenrightbig derived in Exercise . The diﬀerence of these two delta function series produces ( 12.80). 12.4. Change of Scale. So far, we have only dealt with Fourier series on the standard interval of length 2 π. (We chose [ −π,π] for convenience, but all of the results and formulas are eas ily adapted 12/11/12 658 c/circleco√yrt2012 Peter J. Olver to any other interval of the same length, e.g., [0 ,2π].) Since physical objects like bars and strings do not all come in this particular length, we need to u nderstand how to adapt the formulas to more general intervals. The basic idea is to resc ale the variable so as to stretch or contract the standard interval†. Any symmetric interval [ −ℓ,ℓ] of length 2 ℓcan be rescaled to the standard interval [−π,π] by using the linear change of variables x=ℓ πy,so that −π≤y≤πwhenever −ℓ≤x≤ℓ. (12.81) Given a function f(x) deﬁned on [ −ℓ,ℓ], therescaled function F(y) =f/parenleftbiggℓ πy/parenrightbigg lives on [−π,π]. Let F(y)∼a0 2+∞/summationdisplay k=1/bracketleftbig akcosky+bksinky/bracketrightbig , be the standard Fourier series for F(y), so that ak=1 π/integraldisplayπ −πF(y)coskydy, bk=1 π/integraldisplayπ −πF(y)sinkydy. (12.82) Then, reverting to the unscaled variable x, we deduce that f(x)∼a0 2+∞/summationdisplay k=1/bracketleftbigg akcoskπx ℓ+bksinkπx ℓ/bracketrightbigg . (12.83) The Fourier coeﬃcients ak,bkcan be computed directly from f(x). Indeed, replacing the integration variable in (12.82) by y=πx/ℓ, and noting that dy= (π/ℓ)dx, we deduce the adapted formulae ak=1 ℓ/integraldisplayℓ −ℓf(x) coskπx ℓdx, bk=1 ℓ/integraldisplayℓ −ℓf(x) sinkπx ℓdx, (12.84) for the Fourier coeﬃcients of f(x) on the interval [ −ℓ,ℓ]. All of the convergence results, integration and diﬀerentia tion formulae, etc., that are valid for the interval [ −π,π] carry over, essentially unchanged, to Fourier series on nonstandard intervals. In particular, adapting our basic c onvergence Theorem 12.7, we conclude that if f(x) is piecewise C1, then its rescaled Fourier series (12.83) converges to its 2ℓperiodic extension /tildewidef(x), subject to the proviso that /tildewidef(x) takes on the midpoint values at all jump discontinuities. Example 12.23. Let us compute the Fourier series for the function f(x) =xon the interval−1≤x≤1. Since fis odd, only the sine coeﬃcients will be nonzero. We have bk=/integraldisplay1 −1xsinkπxdx=/bracketleftbigg −xcoskπx kπ+sinkπx (kπ)2/bracketrightbigg1 x=−1=2(−1)k+1 kπ. †The same device was already used, in Section 5.4, to adapt the orthogonal Legen dre polyno- mials to other intervals. 12/11/12 659 c/circleco√yrt2012 Peter J. Olver -4 -2 2 4 6 -1-0.50.51 Figure 12.10. 2 Periodic Extension of x. The resulting Fourier series is x∼2 π/parenleftbigg sinπx−sin2πx 2+sin3πx 3− ···/parenrightbigg . The series converges to the 2 periodic extension of the funct ionx, namely /tildewidef(x) =/braceleftbiggx−2m,2m−1< x <2m+1, 0, x =m,wheremis an arbitrary integer, plotted in Figure 12.10. We can similarly reformulate complex Fourier series on the n onstandard interval [−ℓ,ℓ]. Using (12.81) to rescale the variables in (12.55), we ﬁnd f(x)∼∞/summationdisplay k=−∞ckeikπx/ℓ,where ck=1 2ℓ/integraldisplayℓ −ℓf(x)e−ikπx/ℓdx.(12.85) Again, this is merely an alternative way of writing the real F ourier series (12.83). When dealing with a more general interval [ a,b], there are two options. The ﬁrst is to takeafunction f(x)deﬁned for a≤x≤bandperiodicallyextendit toa function /tildewidef(x)that agrees with f(x) on [a,b] and has period b−a. One can then compute the Fourier series (12.83)for itsperiodicextension /tildewidef(x)onthe symmetricinterval [1 2(a−b),1 2(b−a)] ofwidth 2ℓ=b−a; the resulting Fourier series will (under the appropriate h ypotheses) converge to/tildewidef(x) and hence agree with f(x) on the original interval. An alternative approach is to translatetheintervalbyanamount1 2(a+b)soastomakeitsymmetric; thisisaccomplished by the change of variables /hatwidex=x−1 2(a+b). an additional rescaling will convert the interval into [−π,π]. The two methods are essentially equivalent, and full deta ils are left to the reader. 12.5. Convergence of the Fourier Series. The purpose of this ﬁnal section is to establish some basic co nvergence results for Fourier series. This is not a purely theoretical exercise, s ince convergence considerations impingedirectlyuponavarietyofapplicationsofFouriers eries. Oneparticularlyimportant consequence is the connection between smoothness of a funct ion and the decay rate of its high order Fourier coeﬃcients — a result that is exploited in signal and image denoising and in the analytical properties of solutions to partial diﬀ erential equations. 12/11/12 660 c/circleco√yrt2012 Peter J. Olver Be forewarned: the material in this section is more mathemat ical than we are used to, and the more applied reader may consider omitting it on a ﬁrst reading. However, a full understanding of the scope of Fourier analysis as well as its limitations does requires some familiarity with the underlying theory. Moreover, the requ ired techniques and proofs serve as an excellent introduction to some of the most important to ols of modern mathematical analysis. Anyeﬀort expended toassimilatethismaterialwi llbemorethanamplyrewarded in your later career. Unlike power series, which converge to analytic functions o n the interval of conver- gence, and diverge elsewhere (the only tricky point being wh ether or not the series con- verges at the endpoints), the convergence of a Fourier serie s is a much more subtle matter, and still not understood in complete generality. A large par t of the diﬃculty stems from the intricacies of convergence in inﬁnite-dimensional fun ction spaces. Let us therefore begin with a brief discussion of the fundamental issues. Convergence in Vector Spaces We assume that you are familiar with the usual calculus deﬁni tion of the limit of a sequence of real numbers: lim n→∞an=a⋆. In any ﬁnite-dimensional vector space, e.g., Rm, there is essentially only one way for a sequence of vectors v(0),v(1),v(2),...∈Rmto converge, which is guaranteed by any one of the following equ ivalent criteria: (a) The vectors converge: v(n)−→v⋆∈Rmasn→ ∞. (b) The individual components of v(n)= (v(n) 1,...,v(n) m) converge, so lim n→∞v(n) i=v⋆ ifor alli= 1,...,m. (c) The diﬀerence in norms goes to zero: /bardblv(n)−v⋆/bardbl −→0 asn→ ∞. The last requirement, known as convergence in norm , does not, in fact, depend on which norm is chosen. Indeed, Theorem 3.17 implies that, on a ﬁnite -dimensional vector space, all norms are essentially equivalent, and if one norm goes to zero, so does any other norm. The analogous convergence criteria are certainly not the same in inﬁnite-dimensional vector spaces. There are, in fact, a bewildering variety of c onvergence mechanisms in func- tion space, that include pointwise convergence, uniform co nvergence, convergence in norm, weak convergence, and many others. All play a signiﬁcant rol e in advanced mathematical analysis, and hence all are deserving of study. Here, though , we shall be content to learn just the most basic aspects of convergence of the Fourier ser ies, leaving further details to more advanced texts, e.g., [ 63,159,199]. The most basic convergence mechanism for a sequence of funct ionsvn(x) is called pointwise convergence , which requires that lim n→∞vn(x) =v⋆(x) for all x. (12.86) In other words, the functions’ values at each individual poi nt converge in the usual sense. Pointwiseconvergence isthefunctionspace versionofthec onvergence ofthecomponents of a vector. Indeed, pointwise convergence immediately impli es component-wise convergence of the sample vectors v(n)= (vn(x1),...,vn(xm))T∈Rmfor any choice of sample points x1,...,xm. 12/11/12 661 c/circleco√yrt2012 Peter J. Olver Figure 12.11. Uniform and Non-Uniform Convergence of Functions. On the other hand, convergence in norm of the function sequence requires lim n→∞/bardblvn−v⋆/bardbl= 0, where/bardbl·/bardblis a prescribed norm on the function space. As we have learned , not all norms on an inﬁnite-dimensional function space are equivalent: a function might be small in one norm, but large in another. As a result, convergence in norm willdepend upon the choice of norm. Moreover, convergence in norm does not necessarily imply pointwise convergence or vice versa. A variety of examples can be found in the exerci ses. Uniform Convergence Proving uniform convergence of a Fourier series is reasonab ly straightforward, and so we will begin there. You no doubt ﬁrst saw the concept of a un iformly convergent sequence of functions in your calculus course, although cha nces are it didn’t leave much of an impression. In Fourier analysis, uniform convergence begins to play an increasingly important role, and is worth studying in earnest. For the rec ord, let us restate the basic deﬁnition. Deﬁnition 12.24. A sequence of functions vn(x) is said to converge uniformly to a function v⋆(x) on a subset I⊂Rif, for every ε >0, there exists an integer N=N(ε) such that |vn(x)−v⋆(x)|< εfor allx∈Iand alln≥N. (12 .87) The key point — and the reason for the term “uniform convergen ce” — is that the integerNdepends only upon εand not on the point x∈I. Roughly speaking, the se- quence converges uniformly if and only if for any small ε, the graphs of the functions eventually lie inside a band of width 2 εcentered around the graph of the limiting func- tion; see Figure 12.11. Functions may converge pointwise, b ut non-uniformly: the Gibbs phenomenon is the prototypical example of a nonuniformly co nvergent sequence: For a givenε >0, the closer xis to the discontinuity, the larger nmust be chosen so that the inequality in (12.87) holds, and hence there is noconsistent choice of Nthat makes (12.87) valid for all xand alln≥N. A detailed discussion of these issues, including the proof s of the basic theorems, can be found in any basic real analysis te xt, e.g., [ 9,158,159]. A key consequence of uniform convergence is that it preserve s continuity. 12/11/12 662 c/circleco√yrt2012 Peter J. Olver Theorem 12.25. Ifvn(x)→v⋆(x)converges uniformly, and each vn(x)is continu- ous, then v⋆(x)is also a continuous function. The proof is by contradiction. Intuitively, if v⋆(x) were to have a discontinuity, then, as sketched in Figure 12.11, a suﬃciently small band around i ts graph would not connect together, and this prevents the graph of any continuous func tion, such as vn(x), from remaining entirely within the band. Rigorous details can be found in [ 9]. Warning : A sequence ofcontinuous functions canconverge non-uniformly to acontin- uous function. An example is the sequence vn(x) =2nx 1+n2x2, which converges pointwise tov⋆(x)≡0 (why?) but not uniformly since max |vn(x)|=vn/parenleftbig1 n/parenrightbig = 1, which implies that (12.87) cannot hold when ε <1. The convergence (pointwise, uniform, in norm, etc.) of a ser ies∞/summationdisplay k=1uk(x) is, by deﬁnition, governed by the convergence of its sequence of partial sums vn(x) =n/summationdisplay k=1uk(x). (12.88) The most useful test for uniform convergence of series of fun ctions is known as the Weier- strassM–test,inhonorofthenineteenthcenturyGermanmathematicianKa rlWeierstrass, known as the “father of modern analysis”. Theorem 12.26. LetI⊂R. Suppose the functions uk(x)are bounded by |uk(x)| ≤mkfor all x∈I, (12.89) where the mk≥0are ﬁxed positive constants. If the series ∞/summationdisplay k=1mk<∞ (12.90) converges, then the series ∞/summationdisplay k=1uk(x) =f(x) (12 .91) converges uniformly and absolutely†to a function f(x)for allx∈I. In particular, if the summands uk(x)in Theorem 12.26 are continuous, so is the sum f(x). †Recall that a series∞/summationdisplay n=1an=a⋆is said to converge absolutely if and only if∞/summationdisplay n=1|an| converges, [ 9]. 12/11/12 663 c/circleco√yrt2012 Peter J. Olver With some care, we are allowed to manipulate uniformly conve rgent series just like ﬁnite sums. Thus, if (12.91) is a uniformly convergent serie s, so is the term-wise product ∞/summationdisplay k=1g(x)uk(x) =g(x)f(x) (12 .92) withanybounded function: |g(x)| ≤Cforx∈I. Wecanintegrateauniformlyconvergent series term by term‡, and the resulting integrated series /integraldisplayx a/parenleftBigg∞/summationdisplay k=1uk(y)/parenrightBigg dy=∞/summationdisplay k=1/integraldisplayx auk(y)dy=/integraldisplayx af(y)dy (12.93) is uniformly convergent. Diﬀerentiation is also allowed — b ut only when the diﬀerentiated series converges uniformly. Proposition 12.27. If∞/summationdisplay k=1u′ k(x) =g(x)is a uniformly convergent series, then ∞/summationdisplay k=1uk(x) =f(x)is also uniformly convergent, and, moreover, f′(x) =g(x). We are particularly interested in applying these results to Fourier series, which, for convenience, we take in complex form f(x)∼∞/summationdisplay k=−∞ckeikx. (12.94) Sincexis real,/vextendsingle/vextendsingleeikx/vextendsingle/vextendsingle≤1, and hence the individual summands are bounded by /vextendsingle/vextendsingleckeikx/vextendsingle/vextendsingle≤ |ck|for allx. Applying the Weierstrass M–test, we immediately deduce the basic result on uniform convergence of Fourier series. Theorem 12.28. If the Fourier coeﬃcients cksatisfy ∞/summationdisplay k=−∞|ck|<∞, (12.95) then the Fourier series (12.94)converges uniformly to a continuous function /tildewidef(x)having the same Fourier coeﬃcients: ck=/angbracketleftf;eikx/angbracketright=/angbracketleft/tildewidef;eikx/angbracketright. Proof: Uniform convergence and continuity of the limiting functi on follow from Theo- rem 12.26. To show that the ckactually are the Fourier coeﬃcients of the sum, we multiply the Fourier series by e−ikxand integrate term by term from −πtoπ. As in (12.92,93), both operations are valid thanks to the uniform convergence of the series. Q.E.D. ‡Assuming that the individual functions are all integrable. 12/11/12 664 c/circleco√yrt2012 Peter J. Olver The one thing that the theorem does not guarantee is that the o riginal function f(x) used to compute the Fourier coeﬃcients ckis thesameas the function /tildewidef(x) obtained by summing the resulting Fourier series! Indeed, this may very well not be the case. As we know, the function that the series converges to is necessari ly 2πperiodic. Thus, at the very least,/tildewidef(x) will be the 2 πperiodic extension of f(x). But even this may not suﬃce. Two functions f(x) and/hatwidef(x) that have the same values except for a ﬁnite set of points x1,...,xmhave the sameFourier coeﬃcients. (Why?) Moregenerally, tw o functions which agree everywhere outside a set of “measure zero” will have th e same Fourier coeﬃcients. In this way, a convergent Fourier series singles out a distin guished representative from a collection of essentially equivalent 2 πperiodic functions. Remark: The term “measure” refers to a rigorous generalization of t he notion of the length of an interval to more general subsets S⊂R. In particular, Shasmeasure zero if it can be covered by a collection of intervals of arbitrari ly small total length. For example, any collection of ﬁnitely many points, or even coun tably many points, e.g., the rational numbers, has measure zero. The proper development of the notion of measure, and the consequential Lebesgue theory of integration, is pr operly studied in a course in real analysis, [ 158,159]. As a consequence of Theorem 12.28, Fourier series cannot con verge uniformly when discontinuities are present. Non-uniform convergence is t ypically manifested by some form of Gibbs pehnomenon at the discontinuities. However, it can be proved, [ 33,63,199], that even when the function fails to be everywhere continuous, it s Fourier series is uniformly convergent on any closed subset of continuity. Theorem 12.29. Letf(x)be2πperiodic and piecewise C1. Iffis continuous for a < x < b , then its Fourier series converges uniformly to f(x)on any closed subinterval a+δ≤x≤b−δ, withδ >0. For example, the Fourier series (12.41) for the step functio n does converge uniformly if we stay away from the discontinuities; for instance, by re striction to a subinterval of the form [ δ,π−δ] or [−π+δ,−δ] for any 0 < δ <1 2π. This reconﬁrms our observation that the nonuniform Gibbs behavior becomes progressively m ore and more localized at the discontinuities. Smoothness and Decay The uniform convergence criterion (12.95) requires, at the very least, that the Fourier coeﬃcients decay to zero: ck→0 ask→ ±∞. In fact, the Fourier coeﬃcients cannot tend to zero too slowly. For example, the individual summand s of the inﬁnite series ∞/summationdisplay k=−∞1 |k|α(12.96) go to 0 as k→ ∞whenever α >0, but the series only converges when α >1. (This follows from the standard integral convergence test for series, [ 9,159].) Thus, if we can bound 12/11/12 665 c/circleco√yrt2012 Peter J. Olver the Fourier coeﬃcients by |ck| ≤M |k|αfor all |k| ≫0, (12.97) for some power α >1 and some positive constant M >0, then the Weierstrass Mtest will guarantee that the Fourier series converges uniformly to a c ontinuous function. An important consequence of the diﬀerentiation formulae (1 2.77) for Fourier series is the fact that the faster the Fourier coeﬃcients of a function tend to zero as k→ ∞, the smoother the function is. Thus, one can detect the degree of s moothness of a function by seeing how rapidly its Fourier coeﬃcients decay to zero. Mor e rigorously: Theorem 12.30. If the Fourier coeﬃcients satisfy ∞/summationdisplay k=−∞kn|ck|<∞, (12.98) then the Fourier series (12.55)converges to an ntimes continuously diﬀerentiable 2π periodic function f(x)∈Cn. Moreover, for any m≤n, themtimes diﬀerentiated Fourier series converges uniformly to the corresponding derivativ ef(m)(x). Proof: This is an immediate consequence of Proposition 12.27 comb ined with Theo- rem 12.28. Application of the Weierstrass Mtest to the diﬀerentiated Fourier series based on our hypothesis (12.98) serves to complete the proof. Q.E.D. Corollary 12.31. If the Fourier coeﬃcients satisfy (12.97)for some α > n+1, then the function f(x)isntimes continuously diﬀerentiable. Thus, stated roughly, the smaller its high frequency Fourie r coeﬃcients, the smoother the function. If the Fourier coeﬃcients go to zero faster tha n any power of k, e.g., ex- ponentially fast, then the function is inﬁnitely diﬀerenti able. Analyticity is a little more delicate, and we refer the reader to [ 63,199] for details. Example 12.32. The 2πperiodic extension of the function |x|is continuous with piecewise continuous ﬁrst derivative. Its Fourier coeﬃcie nts (12.46) satisfy the estimate (12.97) for α= 2, which is not quite fast enough to ensure a continuous seco nd derivative. On the other hand, the Fourier coeﬃcients (12.29) of the step function σ(x) only tend to zero as 1 /k, soα= 1, reﬂecting the fact that its periodic extension is only pi ecewise continuous. Finally, the Fourier coeﬃcients (12.59) for th e delta function do not tend to zero at all, indicative of the fact that it is not an ordinary f unction, and its Fourier series does not converge in the standard sense. Hilbert Space In order to make further progress, we must take a little detou r. The proper setting for the rigorous theory of Fourier series turns out to be the m ost important function space in modern physics and modern analysis, known as Hilbert space in honor of the great German mathematician David Hilbert. The precise deﬁnition of this inﬁnite-dimensional inner product space is rather technical, but a rough version goes as follows: 12/11/12 666 c/circleco√yrt2012 Peter J. Olver Deﬁnition 12.33. A complex-valued function f(x) is called square-integrable on the interval [ −π,π] if it satisﬁes /bardblf/bardbl2=1 2π/integraldisplayπ −π|f(x)|2dx <∞. (12.99) TheHilbert space L2= L2[−π,π] is the vector space consisting of all complex-valued square-integrable functions. Note that (12.99) is the L2norm based on the standard Hermitian inner product /angbracketleftf;g/angbracketright=1 2π/integraldisplayπ −πf(x)g(x)dx. (12.100) The triangle inequality /bardblcf+dg/bardbl ≤ |c|/bardblf/bardbl+|d|/bardblg/bardbl, implies that the Hilbert space is, as claimed, a complex vect or space, i.e., if f,g∈L2, so /bardblf/bardbl,/bardblg/bardbl<∞, then any linear combination cf+dg∈L2since/bardblcf+dg/bardbl<∞. The Cauchy–Schwarz inequality |/angbracketleftf;g/angbracketright| ≤ /bardblf/bardbl/bardblg/bardbl, implies that the inner product of two square-integrable fun ctions is well-deﬁned and ﬁnite. In particular, the Fourier coeﬃcients of a function f(x) are deﬁned as inner products ck=/angbracketleftf;eikx/angbracketright=1 2π/integraldisplayπ −πf(x)e−ikxdx offwith the complex exponentials (which are continuous and so i n L2), and hence are well-deﬁned for any f∈L2. There are some interesting analytical subtleties that aris e when one tries to prescribe precisely which functions are to be admitted to Hilbert spac e. Every piecewise continuous function belongs to L2. But some functions with singularities are also members. Fo r example, the power function |x|−αbelongs to L2for anyα <1 2, but not if α≥1 2. Analysis requires limiting procedures, and Hilbert space m ust be “complete” in the sense thatappropriately convergent†sequences offunctions have a limit. Thecompleteness requirement is not elementary, and relies on the developmen t of the more sophisticated Lebesgue theory of integration, which was formalized in the early part of the twentieth century by the French mathematician Henri Lebesgue. Any fun ction which is square- integrable in the Lebesgue sense is admitted into L2. This includes such non-piecewise continuous functions as sin1 xandx−1/3, as well as the strange function r(x) =/braceleftbigg1 ifxis a rational number , 0 ifxis irrational .(12.101) †The precise technical requirement is that every Cauchy sequence of functions vk(x)∈L2 converges to a function v⋆(x)∈L2; see Exercise for details. 12/11/12 667 c/circleco√yrt2012 Peter J. Olver One soon discovers that square-integrable functions can be quite bizarre. A second complication is that (12.99) does not, strictly spe aking, deﬁne a norm once we allow discontinuous functions into the fold. For example , the piecewise continuous function f0(x) =/braceleftbigg1, x= 0, 0, x/negationslash= 0,(12.102) has norm zero, /bardblf0/bardbl= 0, even though it is not zero everywhere. Indeed, any functi on which is zero except on a set of measure zero also has norm zero , including the function (12.101). Therefore, in order to make (12.99) into a legitim ate norm on Hilbert space, we must agree to identify any two functions which have the same v alues except on a set of measure zero. For instance, the zero function 0 and the prece ding examples f0(x) andr(x) are all viewed as deﬁning the sameelement of Hilbert space. Thus, although we treat them as if they were ordinary functions, each element of Hilbert s pace is not, in fact, a function, but, rather, an equivalence class of functions all diﬀering on a set of measure zero. All this might striketheapplied reader asbecoming much tooabstrac t and arcane. Inpractice, you will not lose much by assuming that the “functions” in L2are always piecewise continuous and square-integrable. Nevertheless, the full analytical power of Hilbert space theory is only unleashed by including completely general functions i n L2. After its invention by pure mathematicians around the turn o f the twentieth century, physicists in the 1920’s suddenly realized that Hilbert spa ce was the correct setting to establish the modern theory of quantum mechanics. A quantum mechanical wave function is a element†ϕ∈L2that has unit norm: /bardblϕ/bardbl= 1. Thus, the set of wave functions is merely the unit sphere in Hilbert space. Quantum mechanic s endows each physical wave function with a probabilistic interpretation. Suppos e the wave function represents a single subatomic particle — photon, electron, etc. The mod ulus|ϕ(x)|of the wave function quantiﬁes the probability of ﬁnding the particle a t the position x. More cor- rectly, the probability that the particle resides in a presc ribed interval [ a,b] is equal to/radicalBigg 1 2π/integraldisplayb a|ϕ(x)|2dx. In particular, the wave function has unit norm /bardblϕ/bardbl=/radicalBigg 1 2π/integraldisplayπ −π|ϕ(x)|2dx= 1 because the particle must certainly, i.e., with probabilit y 1, besomewhere ! Convergence in Norm We are now in a position to discuss convergence in norm of the F ourier series. We begin with the basic deﬁnition, which makes sense on any norm ed vector space. Deﬁnition 12.34. LetVbe a normed vector space. A sequence v(n)is said to converge in norm tov⋆∈Vif/bardblv(n)−v⋆/bardbl →0 asn→ ∞. †Here we are acting as if the physical space were represented by the on e-dimensional interval [−π,π]. Themoreaptcaseofthree-dimensionalphysical spaceisdevelopedan alogously, replacing the single integral by a triple integral over all of R3. 12/11/12 668 c/circleco√yrt2012 Peter J. Olver As we noted earlier, on ﬁnite-dimensional vector spaces, co nvergence in norm is equiv- alent to ordinary convergence. On the other hand, on inﬁnite -dimensional function spaces, convergence in norm is very diﬀerent from pointwise converg ence. For instance, it is pos- sible, cf. Exercise , to construct a sequence of functions that converges in norm to 0, but does not converge pointwise anywhere ! We are particularly interested in the convergence in norm of the Fourier series of a square integrable function f(x)∈L2. Let sn(x) =n/summationdisplay k=−nckeikx(12.103) be thenthpartial sum of its Fourier series (12.55). The partial sum (1 2.103) belongs to the subspace T(n)⊂L2of all trigonometric polynomials of degree at most n, spanned by e−inx,...,einx. It is, in fact, distinguished as the function in T(n)that lies the closest tof, where the distance between functions is measured by the L2norm of their diﬀer- ence:/bardblf−g/bardbl. This important characterization of the Fourier partial su ms is, in fact, an immediate consequence of the orthonormality of the trigono metric basis. Theorem 12.35. Thenthorder Fourier partial sum sn∈ T(n)is the best least squares approximation to f∈L2, meaning that it minimizes the distance, as measured by theL2norm of the diﬀerence /bardblf−pn/bardbl2=1 2π/integraldisplayπ −π|f(x)−pn(x)|2dx, (12.104) among all possible degree ntrigonometric polynomials pn(x) =n/summationdisplay k=−ndkeikx∈ T(n). (12.105) Proof: Theproofis, infact, anexactreplicaofthatoftheﬁnite-d imensionalTheorems 5.37 and 5.39. Note ﬁrst that, owing to the orthonormality of the basis exponentials, (12.54), we can compute the norm of a trigonometric polynomi al (12.105) by summing the squared moduli of its Fourier coeﬃcients: /bardblpn/bardbl2=/angbracketleftpn;pn/angbracketright=n/summationdisplay k,l=−ndkdl/angbracketlefteikx;eilx/angbracketright=n/summationdisplay k=−n|dk|2, reproducing our standard formula (5.5) for the norm with res pect to an orthonormal basis in this situation. Therefore, employing the identity in Exe rcise 3.6.43( a), /bardblf−pn/bardbl2=/bardblf/bardbl2−2 Re/angbracketleftf;pn/angbracketright+/bardblpn/bardbl2=/bardblf/bardbl2−2 Ren/summationdisplay k=−ndk/angbracketleftf;eikx/angbracketright+/bardblpn/bardbl2 =/bardblf/bardbl2−2n/summationdisplay k=−nRe/parenleftbig ckdk/parenrightbig +n/summationdisplay k=−n|dk|2=/bardblf/bardbl2−n/summationdisplay k=−n|ck|2+n/summationdisplay k=−n|dk−ck|2; 12/11/12 669 c/circleco√yrt2012 Peter J. Olver the last equality results from adding and subtracting the sq uared norm /bardblsn/bardbl2=n/summationdisplay k=−n|ck|2(12.106) of the Fourier partial sum. We conclude that /bardblf−pn/bardbl2=/bardblf/bardbl2−/bardblsn/bardbl2+n/summationdisplay k=−n|dk−ck|2. (12.107) The ﬁrst and second terms on the right hand side of (12.107) ar e uniquely determined by f(x) and hence cannot be altered by the choice of trigonometric p olynomial pn(x), which only aﬀects the ﬁnal summation. Since the latter is a sum of no nnegative quantities, it is minimized by setting all the summands to zero, i.e., setting dk=ck. We conclude that /bardblf−pn/bardblis minimized if and only if dk=ckare the Fourier coeﬃcients, and hence the least squares minimizer is the Fourier partial sum: pn(x) =sn(x). Q.E.D. Settingpn=sn, sodk=ck, in (12.107), we conclude that the least squares error for the Fourier partial sum is 0≤ /bardblf−sn/bardbl2=/bardblf/bardbl2−/bardblsn/bardbl2=/bardblf/bardbl2−n/summationdisplay k=−n|ck|2. Therefore, the Fourier coeﬃcients of the function fmust satisfy the basic inequality n/summationdisplay k=−n|ck|2≤ /bardblf/bardbl2. Consider what happens in the limit as n→ ∞. Since we are summing a sequence of non-negative numbers with uniformly bounded partial sums, the limiting summation must exist, and be subject to the same bound. We have thus proved Bessel’s inequality : ∞/summationdisplay k=−∞|ck|2≤ /bardblf/bardbl2, (12.108) which is an important waystation on the road to the general th eory. Now, as noted earlier, if a series is to converge, the individual summands must go to zero:|ck|2→0. Therefore, Bessel’s inequality immediately implies the following sim pliﬁed form of the Riemann– Lebesgue Lemma . Lemma 12.36. Iff∈L2is square integrable, then its Fourier coeﬃcients satisfy ck=1 2π/integraldisplayπ −πf(x)e−ikxdx−→0as|k| → ∞, (12.109) 12/11/12 670 c/circleco√yrt2012 Peter J. Olver which is equivalent to the decay of the real Fourier coeﬃcien ts ak=1 π/integraldisplayπ −πf(x)coskxdx bk=1 π/integraldisplayπ −πf(x)sinkxdx  −→0ask→ ∞. (12.110) Remark: As before, the convergence of the sum (12.108) requires tha t the coeﬃcients ckcannot tend to zero too slowly. For instance, assuming the po wer bound (12.97), namely |ck| ≤M|k|−α,then requiring α >1 2is enough to ensure that∞/summationdisplay k=−∞|ck|2<∞. Thus, as we should expect, convergence in norm imposes less r estrictive requirements on the decay of the Fourier coeﬃcients than uniform convergenc e — which needed α >1. Indeed, a Fourier series may very well converge in norm to a di scontinuous function, which is not possible under uniform convergence. In fact, there ev en exist bizarre continuous functions whose Fourier series do not converge uniformly, e ven failing to converge at all at some points. A deep result says that the Fourier series of a co ntinuous function converges except possiblyonaset ofmeasurezero, [ 199]. Again, thesubtledetailsoftheconvergence of Fourier series are rather delicate, and lack of space and a nalytical savvy prevents us from delving any further into these topics. Completeness As we know, speciﬁcation of a basis enables you to prescribe a ll elements of a ﬁnite- dimensional vector space as linear combinations of the basi s elements. The number of basis elements dictates the dimension. In an inﬁnite-dimen sional vector space, there are, by deﬁnition, inﬁnitely many linearly independent element s, and no ﬁnite collection can serve to describe the entire space. The question then arises to what extent an inﬁnite col- lection of linearly independent elements can be considered as a basis for the vector space. Mere counting will no longer suﬃce, since omitting one, or tw o, or any ﬁnite number — or even certain inﬁnite subcollections — from a purported basi s will still leave inﬁnitely many linearly independent elements; but, clearly, the reduced c ollection should, in some sense, no longer serve as a complete basis. The curse of inﬁnity stri kes again! For example, while the complete trigonometric collection 1 ,cosx,sinx,cos2x,sin2x,...will represent any 2 π periodic L2function as a Fourier series, the subcollection cos x,sinx,cos2x,sin2x,...can only represent functions with mean zero, while the subcolle ction sin x,sin2x,...only rep- resents odd functions. All three consist of inﬁnitely many l inearly independent functions, but only the ﬁrst could possibly be deemed a basis of L2. In general, just because we have found a inﬁnite collection of independent elements in an inﬁ nite-dimensional vector space, how do we know that we have enough, and are not missing one or tw o or 10,000 or even inﬁnitely many additional independent elements? The concept of “completeness” serves to properly formalize the notion of a “basis” of an inﬁnite-dimensional vector space. We shall discuss co mpleteness in a general, ab- stract setting, but the key example is, of course, the Hilber t space L2and the system of 12/11/12 671 c/circleco√yrt2012 Peter J. Olver trigonometric(orcomplexexponential)functionsforming aFourierseries. Otherimportant examples arising in later applications include wavelets, B essel functions, Legendre polyno- mials, spherical harmonics, and other systems of eigenfunc tions of self-adjoint boundary value problems. For simplicity, we only deﬁne completeness in the case of ort honormal systems. (Sim- ilar arguments will clearly apply to orthogonal systems, bu t normality helps to streamline thepresentation.) Let Vbeaninﬁnite-dimensional complex†innerproduct space. Suppose thatu1,u2,u3,...∈Vform an orthonormal collection of elements of V, so /angbracketleftui;uj/angbracketright=/braceleftbigg1i=j, 0, i/negationslash=j.(12.111) A straightforward argument, cf. Proposition 5.4, proves th at theuiare linearly indepen- dent. Given f∈V, we form its generalized Fourier series f∼∞/summationdisplay k=1ckuk,where ck=/angbracketleftf;uk/angbracketright, (12.112) which is our usual orthonormal basis coeﬃcient formula (5.4 ), and is obtained by formally taking the inner product of the series with ukand invoking the orthonormality conditions (12.111). Deﬁnition 12.37. An orthonormal system u1,u2,u3,...∈Vis called complete if the generalized Fourier series (12.112) of any f∈Vconverges in norm to f: /bardblf−sn/bardbl −→0,asn→ ∞, where sn=n/summationdisplay k=1ckuk (12.113) is thenthpartial sum of the generalized Fourier series (12.112). Thus, completeness requires that every element can be arbit rarily closely approxi- mated (in norm) by a suitable linear combination of the basis elements. A complete or- thonormal system should be viewed as the inﬁnite-dimension al version of an orthonormal basis of a ﬁnite-dimensional vector space. The key result for classical Fourier series is that the compl ex exponentials, or, equiv- alently, the trigonometric functions, form a complete syst em. An indication of its proof will appear below. Theorem 12.38. The complex exponentials eikx,k= 0,±1,±2,..., form a com- plete orthonormal system in L2= L2[−π,π]. In other words, if sn(x)denotes the nthpartial sum of the Fourier series of the square-integrable f unctionf(x)∈L2, then lim n→∞/bardblf−sn/bardbl= 0. †The results are equally valid in real inner product spaces, with sl ightly simpler proofs. 12/11/12 672 c/circleco√yrt2012 Peter J. Olver Inordertounderstandcompleteness, letusdescribesomeeq uivalentcharacterizations. ThePlancherel formula is the inﬁnite-dimensional counterpart of our formula (5.5 ) for the norm of a vector in terms of its coordinates with respect to an orthonormal basis. Theorem 12.39. The orthonormal system u1,u2,u3,...∈Vis complete if and only if the Plancherel formula /bardblf/bardbl2=∞/summationdisplay k=1|ck|2=∞/summationdisplay k=1/angbracketleftf;uk/angbracketright2, (12.114) holds for every f∈V. Proof: We begin by computing†the Hermitian norm /bardblf−sn/bardbl2=/bardblf/bardbl2−2 Re/angbracketleftf;sn/angbracketright+/bardblsn/bardbl2. Substituting the formula (12.113) for the partial sums, we ﬁ nd, by orthonormality, /bardblsn/bardbl2=n/summationdisplay k=1|ck|2,while /angbracketleftf;sn/angbracketright=n/summationdisplay k=1ck/angbracketleftf;uk/angbracketright=n/summationdisplay k=1|ck|2. Therefore, 0≤ /bardblf−sn/bardbl2=/bardblf/bardbl2−n/summationdisplay k=1|ck|2. (12.115) The fact that the left hand side of (12.115) is non-negative f or allnimplies the abstract form ofBessel inequality ∞/summationdisplay k=1|ck|2≤ /bardblf/bardbl2, (12.116) which is valid for anyorthonormal system of elements in an inner product space. Th e trigonometric Bessel inequality (12.108) is a particular c ase of this general result. As we noted above, Bessel’s inequality implies that the generali zed Fourier coeﬃcients ck→0 must tend to zero reasonably rapidly in order that the sum of t heir squares converges. Plancherel’s Theorem 12.39, thus, states that the system of functions is complete if and only if the Bessel inequality is, in fact, an equality! In deed, letting n→ ∞in (12.115), we have lim n→∞/bardblf−sn/bardbl2=/bardblf/bardbl2−∞/summationdisplay k=1|ck|2. Therefore, the completeness condition (12.113) holds if an d only if the right hand side vanishes, which is the Plancherel identity (12.114). Q.E.D. †We are, in essence, repeating the proofs of Theorem 12.35 and the subseq uent trigonometric Bessel inequality (12.108) in a more abstract setting. 12/11/12 673 c/circleco√yrt2012 Peter J. Olver Corollary 12.40. Letf,g∈V. Then their Fourier coeﬃcients ck=/angbracketleftf;ϕk/angbracketright,dk= /angbracketleftg;ϕk/angbracketrightsatisfyParseval’s identity /angbracketleftf;g/angbracketright=∞/summationdisplay k=1ckdk. (12.117) Proof: Using the identity in Exercise 3.6.43( b), /angbracketleftf;g/angbracketright=1 4/parenleftbig /bardblf+g/bardbl2−/bardblf−g/bardbl2+ i/bardblf+ ig/bardbl2−i/bardblf−ig/bardbl2/parenrightbig . Parseval’s identity results from applying the Plancherel f ormula (12.114) to each term on the right hand side: /angbracketleftf;g/angbracketright=1 4∞/summationdisplay k=−∞/parenleftbig |ck+dk|2−|ck−dk|2+ i|ck+ idk|2−i|ck−idk|2/parenrightbig =∞/summationdisplay k=−∞ckdk, by a straightforward algebraic manipulation. Q.E.D. In particular, inthe case ofthe complex exponential basis o fL2[−π,π], thePlancherel and Parseval formulae tell us that 1 2π/integraldisplayπ −π|f(x)|2dx=∞/summationdisplay k=−∞|ck|2,1 2π/integraldisplayπ −πf(x)g(x)dx=∞/summationdisplay k=−∞ckdk,(12.118) in which ck=/angbracketleftf;eikk/angbracketright, dk=/angbracketleftg;eikk/angbracketrightare the ordinary Fourier coeﬃcients of the complex-valued functions f(x) andg(x). Note that the Plancherel formula is a special case of the Parseval identity, obtained by setting f=g. In Exercise , you are asked to rewrite these formulas in terms of the real Fourier coeﬃcien ts. Completeness also tells us that a function is uniquely deter mined by its Fourier coef- ﬁcients. Proposition 12.41. If the orthonormal system u1,u2,...∈Vis complete, then the only element f∈Vwith all zero Fourier coeﬃcients, 0 =c1=c2=···, is the zero element:f= 0. More generally, two elements f,g∈Vhave the same Fourier coeﬃcients if and only if they are the same :f=g. Proof: The proof is an immediate consequence of the Plancherel for mula. Indeed, if ck= 0, then (12.114) implies that /bardblf/bardbl= 0. The second statement follows by applying the ﬁrst to their diﬀerence f−g. Q.E.D. Another way of stating this result is that the only function w hich is orthogonal to every element of a complete orthonormal system is the zero fu nction†. Interpreted in yet another way, a complete orthonormal system is maximal in the sense that no further orthonormal elements can be appended to it. †Or, to be more technically accurate, any function which is zero outsi de a set of measure zero. 12/11/12 674 c/circleco√yrt2012 Peter J. Olver Let us now discuss the completeness of the Fourier trigonome tric/complex exponential functions. We shall prove the completeness criterion only f or continuous functions, leaving the harder general proof to the references, [ 63,199]. According to Theorem 12.28, if f(x) is continuous, 2 πperiodic, and piecewise C1, its Fourier series converges uniformly to f(x), so f(x) =∞/summationdisplay k=−∞ckeikxfor all −π≤x≤π. The same holds for its complex conjugate f(x). Therefore, |f(x)|2=f(x)f(x) =f(x)∞/summationdisplay k=−∞cke−ikx=∞/summationdisplay k=−∞ckf(x)e−ikx, which also converges uniformly by (12.92). Equation (12.93 ) permits us to integrate both sides from −πtoπ, yielding /bardblf/bardbl2=1 2π/integraldisplayπ −π|f(x)|2dx=∞/summationdisplay k=−∞1 2π/integraldisplayπ −πckf(x)e−ikxdx=∞/summationdisplay k=−∞ckck=∞/summationdisplay k=−∞|ck|2. Therefore, Plancherel’s identity (12.114) holds for any co ntinuous function. With some additionaltechnical work, thisresultisusedtoestablish thevalidityofPlancherel’sformula for allf∈L2, the key step being to suitably approximate fby continuous functions. With this in hand, completeness is an immediate consequence of Th eorem 12.39. Q.E.D. Pointwise Convergence Let us ﬁnally turn to the proof of the Pointwise Convergence T heorem 12.7. The goal is to prove that, under the appropriate hypotheses on f(x), namely 2 πperiodic and piecewise C1, the limit of the partial Fourier sums is lim n→∞sn(x) =1 2/bracketleftbig f(x+)+f(x−)/bracketrightbig . (12.119) We begin by substituting the formulae (12.56) for the comple x Fourier coeﬃcients into the formula (12.103) for the nthpartial sum: sn(x) =n/summationdisplay k=−nckeikx=n/summationdisplay k=−n/parenleftbigg1 2π/integraldisplayπ −πf(y)e−ikydy/parenrightbigg eikx =1 2π/integraldisplayπ −πf(y)n/summationdisplay k=−neik(x−y)dy. Thenthpartial sum sn(x) =1 2πn/summationdisplay k=−neikx=1 2π/parenleftbig e−inx+···+e−ix+1+eix+···+einx/parenrightbig 12/11/12 675 c/circleco√yrt2012 Peter J. Olver can, in fact, be explicitly evaluated. Recall the formula fo r the sum of a geometric series m/summationdisplay k=0ark=a+ar+ar2+···+arm=a/parenleftbiggrm+1−1 r−1/parenrightbigg . (12.120) Thepartialsum sn(x)hasthisform, with m+1 = 2n+1summands, initialterm a=e−inx, and ratio r=eix. Therefore, sn(x) =1 2πn/summationdisplay k=−neikx=1 2πe−inx/parenleftbiggei(2n+1)x−1 eix−1/parenrightbigg =1 2πei(n+1)x−e−inx eix−1 =1 2πei/parenleftbig n+1 2/parenrightbig x−e−i/parenleftbig n+1 2/parenrightbig x eix/2−e−ix/2=1 2πsin/parenleftbig n+1 2/parenrightbig x sin1 2x.(12.121) In this computation, to pass from the ﬁrst to the second line, we multiplied numerator and denominator by e−ix/2, after which we used the equation for the sine function in ter ms of complex exponentials. Incidentally, (12.121) is equivale nt to the intriguing trigonometric summation formula sn(x) =1 2π+1 π/parenleftbig cosx+cos2x+cos3x+···+cosnx/parenrightbig =1 2πsin/parenleftbig n+1 2/parenrightbig x sin1 2x.(12.122) We can then use the geometric summation formula (12.121) to e valuate the result: sn(x) =1 2π/integraldisplayπ −πf(y)sin/parenleftbig n+1 2/parenrightbig (x−y) sin1 2(x−y)dy =1 2π/integraldisplayx+π x−πf(x+y)sin/parenleftbig n+1 2/parenrightbig y sin1 2ydy=1 2π/integraldisplayπ −πf(x+y)sin/parenleftbig n+1 2/parenrightbig y sin1 2ydy. The second equality is the result of changing the integratio n variable from ytox+y; the ﬁnal equality follows since the integrand is 2 πperiodic, and so its integrals over any interval of length 2 πall have the same value; see Exercise . Thus, to prove (12.119), it suﬃces to show that lim n→∞1 π/integraldisplayπ 0f(x+y)sin/parenleftbig n+1 2/parenrightbig y sin1 2ydy=f(x+), lim n→∞1 π/integraldisplay0 −πf(x+y)sin/parenleftbig n+1 2/parenrightbig y sin1 2ydy=f(x−).(12.123) The proofs of the two formulae are identical, and so we concen trate on the ﬁrst. Since the integrand is even, 1 π/integraldisplayπ 0sin/parenleftbig n+1 2/parenrightbig y sin1 2ydy=1 2π/integraldisplayπ −πsin/parenleftbig n+1 2/parenrightbig y sin1 2ydy= 1, 12/11/12 676 c/circleco√yrt2012 Peter J. Olver by equation (12.65). Multiplying this formula by f(x+) and subtracting oﬀ the right hand side leads to lim n→∞1 π/integraldisplayπ 0f(x+y)−f(x+) sin1 2ysin/parenleftbig n+1 2/parenrightbig y dy= 0, (12.124) which we now proceed to prove. We claim that, for each ﬁxed value of x, the function g(y) =f(x+y)−f(x+) sin1 2y is piecewise continuous for all 0 ≤y≤π. Owing to our hypotheses on f(x), the only problematic point is when y= 0, but then, by l’Hˆ opital’s rule (for one-sided limits), lim y→0+g(y) = lim y→0+f(x+y)−f(x+) sin1 2y= lim y→0+f′(x+y) 1 2cos1 2y= 2f′(x+). Consequently, (12.124) will be established if we can show th at lim n→∞1 π/integraldisplayπ 0g(y)sin/parenleftbig n+1 2/parenrightbig y dy= 0 (12 .125) whenever gis piecewise continuous. Were it not for the extra1 2, this would immediately follow from the simpliﬁed Riemann–Lebesgue Lemma 12.36. Mo re honestly, we can invoke the addition formula for sin/parenleftbig n+1 2/parenrightbig yto write 1 π/integraldisplayπ 0g(y)sin/parenleftbig n+1 2/parenrightbig ydy=1 π/integraldisplayπ 0/parenleftbig g(y)sin1 2y/parenrightbig cosnydy+1 π/integraldisplayπ 0/parenleftbig g(y)cos1 2y/parenrightbig sinnydy The ﬁrst integral is the nthFourier cosine coeﬃcient for the piecewise continuous func tion g(y)sin1 2y, while the second integral is the nthFourier sine coeﬃcient for the piecewise continuous function g(y)cos1 2y. Lemma 12.36 implies that both of these converge to zero asn→ ∞, and hence (12.125) holds. This completes the proof, establ ishing pointwise convergence of the Fourier series. Q.E.D. Remark: An alternative approach to the last part of the proof is to us e the general Riemann–Lebesgue Lemma , whose proof can be found in [ 63,199]. Lemma 12.42. Suppose g(x)is piecewise continuous on [a,b]. Then 0 = lim ω→∞/integraldisplayb ag(x)eiωxdx=/integraldisplayb ag(x)cosωxdx+ i/integraldisplayb ag(x)sinωxdx. (12.126) Intuitively, as thefrequency ωgetslarger andlarger, the increasingly rapidoscillation s ineiωxtend to cancel each other out. In mathematical language, the Riemann–Lebesgue formula (12.126) says that, as ω→ ∞, the integrand g(x)eiωxconverges weakly to 0; we saw the same phenomenon in the weak convergence of the Four ier series of the delta function (12.61). 12/11/12 677 c/circleco√yrt2012 Peter J. Olver