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12.2. Fourier Series

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12.2. Fourier Series 12.2. Fourier Series. While the need to solve physically interesting partial differential equations served as our (and Fourier’s) initial motivation, the remarkable range of applications qualifies Fourier’s discovery as one of the most important in all of mathematics. We therefore take some time to properly develop the basic theory of Fourier series and, in the following chapter, a number of important extensions. Then, properly equipped, we will be in a position to return to the source — solving partial differential equations. The starting point is the need to represent a given functionf(x), defined for π x π, as a convergent series in the elementary trigonometric functions: − ≤ ≤ � a ∞ f(x) = 0 + [a coskx+b sinkx]. (12.24) 2 k k k=1 Thefirst order of business is to determine the formulae for the Fourier coefficientsa k, bk. The key is orthogonality. We already observed, in Example 5.12, that the trigonometric functions are orthogonal with respect to the rescaled L2 inner product � 1 π f;g = f(x)g(x) dx (12.25) � � π π − on the interval† [ π,π ]. The explicit orthogonality relations are − coskx ; coslx = sinkx;sinlx =0, fork=l, � � � � � coskx;sinlx =0, for all k,l, (12.26) � � 1 = √2, coskx = sinkx =1, fork= 0, � � � � � � � wherek andl indicate non-negative integers. Remark: If we were to replace the constant function 1 by 1 , then the resulting √2 functions would form an orthonormal system. However, this extra √2 turns out to be utterly annoying, and is best omitted from the outset. Remark: Orthogonality of the trigonometric functions is not an accident, but follows from their status as eigenfunctions for the self-adjoint boundary value problem (12.13). The general result, to be presented in Section 14.7, is the function space analog of the orthogonality of eigenvectors of symmetric matrices, cf. Theorem 8.20. If we ignore convergence issues for the moment, then the orthogonality relations (12.26) serve to prescribe the Fourier coefficients: Taking the 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 2π periodic, any interval of length 2π will serve equally well. Adapting Fourier series to intervals of other lengths will be discussed in Section 12.4. 12/11/12 638 c 2012 Peter J. Olver � with coslx forl> 0, and invoking the underlying linearity ‡ of the inner product, yields � a ∞ f ; coslx = 0 1 ; coslx + [a coskx ; coslx +b sinkx ; coslx ] � � 2 � � k � � k � � k=1 =a coslx ; coslx =a , l � � l since, by the orthogonality relations (12.26), all terms but thel th vanish. This serves to pre- scribe the Fourier coefficienta l. A similar manipulation with sinlxfixesb l = f ;sinlx , while taking the inner product with the constant function 1 gives � � � a ∞ a f;1 = 0 1;1 + [a coskx;1 +b sinkx;1 ]= 0 1 2 =a , � � 2 � � k � � k � � 2 � � 0 k=1 which agrees with the preceding formula fora l whenl = 0, and explains why we include 1 the extra factor of 2 in the constant term. Thus, if the Fourier series converges to the functionf(x), then its coefficients are prescribed by taking inner products with the basic trigonometric functions. The alert reader may recognize the preceding argument — it is the function space version of our derivation or the fundamental orthonormal and orthogonal 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 offinite sums, and convergence issues must be properly addressed. However, we defer these more delicate 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. The Fourier series of a functionf(x) defined on π x π is the infinite trigonometric series − ≤ ≤ � a ∞ f(x) 0 + [a coskx+b sinkx], (12.27) ∼ 2 k k k=1 whose coefficients are given by the inner product formulae � 1 π ak = f(x) cos k x dx, k=0,1,2,3,..., π π �− (12.28) 1 π bk = f(x) sin k x dx, k=1,2,3,.... π π − Note that the functionf(x) cannot be completely arbitrary, since, at the very least, the integrals in the coefficient formulae must be well defined andfinite. Even if the coefficients (12.28) arefinite, there is no guarantee that the resulting Fourier series converges, and, even if it converges, no guarantee that it converges to the original functionf(x). For these reasons, we use the symbol instead of an equals sign when writing down a Fourier series. Before tackling these∼ key issues, let us look at an elementary example. ‡ More rigorously, linearity only applies tofinite linear combinations, not infinite series. Here, thought, we are just trying to establish and motivate the basic formulae, and can safely defer such technical complications until thefinal section. 12/11/12 639 c 2012 Peter J. Olver � Example 12.2. Consider the functionf(x) =x. We may compute its Fourier coefficients directly, employing integration by parts to evaluate the integrals: � � � � � π π �π 1 1 1 x sinkx coskx � a0 = x dx=0, a k = x cos k x dx= + 2 � = 0, π π π π π k k x= π �− � − � � − π �π 1 1 x coskx sinkx � 2 k+1 bk = x sin k x dx= + 2 � = ( 1) . (12.29) π π π − k k x= π k − − − Therefore, the Fourier cosine coefficients of the functionx all vanish,a k = 0, and its Fourier series is � � sin 2x sin 3x sin 4x x 2 sinx + + . (12.30) ∼ − 2 3 − 4 ··· Convergence of this series is not an elementary matter. Standard tests, including the ratio and root tests, fail to apply. Even if we know that the series converges (which it does — for allx), it is certainly not obvious what function it converges to. Indeed, it cannot converge to the functionf(x) =x for all values ofx. If we substitutex=π, then every term in the series is zero, and so the Fourier series converges to 0 — which is not the same asf(π)=π. th Then partial sum of a Fourier series is the trigonometric polynomial† �n a s (x) = 0 + [a coskx+b sinkx]. (12.31) n 2 k k k=1 By definition, the Fourier series converges at a pointx if and only if the partial sums have a limit: � lim s (x) = f(x), (12.32) n n →∞ which may or may not equal the value of the original functionf(x). Thus, a key require- ment is to formulate conditions on the functionf(x) that guarantee that the Fourier series converges, and, even more importantly, the limiting sum reproduces the original function: � f(x) =f(x). This will all be done in detail below. Remark: The passage from trigonometric polynomials to Fourier series is similar to the passage from polynomials to power series. A power series � ∞ f(x) c +c x+ +c xn + = c xk ∼ 0 1 ··· n ··· k k=0 can be viewed as an infinite linear combination of the basic monomials 1, x, x2, x3,.... f (k)(0) According to Taylor’s formula, (C.3), the coefficientsc = are given in terms of k k! † The reason for the term “trigonometric polynomial” was discussed at length in Exam- ple 2.17(c). 12/11/12 640 c 2012 Peter J. Olver � the derivatives of the function at the origin. The partial sums �n s (x) =c +c x+ +c xn = c xk n 0 1 ··· n k k=0 of a power series are ordinary polynomials, and the same convergence issues arise. Although superficially similar, in actuality the two theories are profoundly different. Indeed, while the theory of power series was well established in the early days of the cal- culus, there remain, to this day, unresolved foundational issues in Fourier theory. A power series either converges everywhere, or on an interval centered 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 convergence properties of Fourier series led the nineteenth century German mathematician Georg Cantor to formulate mod- ern set theory, and, thus, played a seminal role in the establishment of the foundations of modern mathematics. Secondly, when a power series converges, it converges to an analytic function, which is infinitely differentiable, and whose derivatives 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 like the delta function! Therefore, the termwise differentiation of a Fourier series is a nontrivial issue. Once one comprehends how different the two subjects are, one begins to understand why Fourier’s astonishing claims were initially widely disbelieved. Before the advent of Fourier, mathematicians only accepted analytic functions as the genuine article. The fact that Fourier series can converge to nonanalytic, even discontinuous functions was extremely disconcerting, and resulted in a complete re-evaluation of function theory, culminating in the modern definition of function that you now learn infirst year calculus. Only through the combined efforts of many of the leading mathematicians of the nineteenth century was a rigorous theory of Fourier seriesfirmly established; see Section 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 2π.
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