CHAPTER12 FourierSeries In 1807, the French mathematician and physicist Joseph Fourier submitted a paper on heat conduction to the Academy of Sciences of Paris. In this paper Fourier made » the claim that any function f º x canbeexpandedintoaninfinitesumoftrigonometric functions, ½ a0 » D C Ì º » C º »Í; : f º x a cos kx b sin kx for x 2 k k k D1 The paper was rejected after it was read by some of the leading mathematicians of his day. They objected to the fact that Fourier had not presented much in the way of proof for this statement, and most of them did not believe it. In spite of its less than glorious start, Fourier’s paper was the impetus for major developments in mathematics and in the application of mathematics. His ideas forced mathematicians to come to grips with the definition of a function. This, together with other metamathematical questions, caused nineteenth-century mathematicians to rethink completely the foundationsof their subject, and toputitonamorerigorous foundation. Fourier’s ideas gave rise to a new part of mathematics, called harmonic analysis or Fourier analysis. This, in turn, fostered the introduction at the end of the nineteenth century of a completely new theory of integration, now called the Lebesgue integral. The applications of Fourier analysis outside of mathematics continue to multiply. » One important application pertains to signal analysis. Here, f º x could represent the amplitude of a sound wave, such as a musical note, or an electrical signal from a CD playerorsomeotherdevice(in this case x representstime and is usually replaced by t). The Fourier series representation of a signal represents a decomposition of this signal into its various frequency components. The terms sin kx and cos kx 712 12.1 Computation of Fourier Series 713 1 oscillate with numerical frequency of k =2 . Signals are often corrupted by noise, which usually involves the high-frequency components (when k is large). Noise can sometimes be filtered out by setting the high-frequency coefficients (the ak and bk when k is large) equal to zero. Data compression is another increasingly important problem. One way to ac- complish data compression uses Fourier series. Here the goal is to be able to store or transmit the essential parts of a signal using as few bits of information as possible. The Fourier series approach to the problem is to store (or transmit) only those ak and bk that are larger than some specified tolerance and discard the rest. Fortunately, an important theorem (the Riemann–Lebesgue Lemma, which is our Theorem 2.10) as- suresusthatonly a small numberofFouriercoefficientsare significant, and therefore the aforementioned approach can lead to significant data compression. 12.1 Computation of Fourier Series » The problem that we wish to address is the one faced by Fourier. Suppose that f º x ; Í is a given function on the interval Ì . Can we find coefficients, an and bn, so that ½ a0 » D C C ; f º x [a cos nx b sin nx] for x ? (1.1) 2 n n n D1 Notice that, except for the term a0 =2, the seriesis aninfinite linear combinationofthe basic terms sin nx and cos nx for n a positive integer. These functions are periodic Ì ; Í with period 2 = n, so their graphs trace through n periods over the interval . Figure 1 shows the graphs of cos x and cos5x, and Figure 2 shows the graphs of sin x and sin5x. Notice how the functions become more oscillatory as n increases. 1 1 x x −π π −π π −1 −1 Figure 1 The graphs of cos x Figure 2 The graphs of sin x and cos 5x. and sin 5x. The orthogonality relations Our task of finding the coefficients an and bn for which (1.1) is true is facilitated by the following lemma. These orthogonality relations are one of the keys to the whole theory of Fourier series. 1 Be sure you know the difference between angular frequency, k in this case, and numerical frequency. It is explained in Section 4.1. 714 Chapter 12 Fourier Series LEMMA 1.2 Let p and q be positive integers. Then we have the following orthogonality relations. D sin pxdx D cos pxdx 0 (1.3) sin px cos qxdx D 0 (1.4) ´ D ; if p q cos px cos qxdx D (1.5) ; 6 D 0 if p q ´ D ; if p q sin px sin qxdx D (1.6) ; 6 D 0 if p q We will leave the proof of these identities for the exercises. Computation of the coefficients The orthogonality relations enable us to find the coefficients an and bn in (1.1). Suppose we are given a function f that can be expressed as ½ a0 » D C Ì C Í f º x a cos kx b sin kx (1.7) 2 k k k D1 ; Í on the interval Ì . To find a0, we simply integrate the series (1.7) term by term. Using the orthogonality relation (1.3), we see that » D : f º x dx a0 (1.8) To find an for n 1, we multiply both sides of (1.7) by cos nx and integrate term by term, getting ½ a0 » D C Ì C Í f º x cos nxdx ak cos kx bk sin kx cos nxdx 2 k D1 a0 D cos nxdx 2 ½ (1.9) C ak cos kx cos nxdx k D1 ½ : C bk sin kx cos nxdx k D1 Using the orthogonality relations in Lemma 1.2, we see that all the terms on the right-hand side of (1.9) are equal to zero, except for ¡ : an cos nx cos nxdx D an 12.1 Computation of Fourier Series 715 Hence, equation (1.9) becomes » D ¡ ; ; f º x cos nxdx an for n 1 so, including equation (1.8), 2 1 º » ; : an D f x cos nxdx for n 0 (1.10) To find bn, we multiply equation(1.7)bysin nx andthenintegrate. By reasoning similar to the computation of an, we obtain 1 º » ; bn D f x sin nxdx for n 1. (1.11) Definition of Fourier series ; Í If f is a piecewise continuous function on the interval Ì , we can compute the coefficients an and bn using (1.10) and (1.11). Thus we can define the Fourier series for any such function. DEFINITION 1.12 Suppose that f is a piecewise continuous function on the ; Í interval Ì . With the coefficients computed using (1.10) and (1.11) , we define the Fourier series associated to f by ½ a0 » C C : f º x [a cos nx b sin nx] (1.13) 2 n n n D1 The finite sum N a0 » D C C S º x [a cos nx b sin nx] (1.14) N 2 n n n D1 is called the partial sum of order N for the Fourier series in (1.13). We say that the Fourier series converges at x if the sequence of partial sums converges at x as ½: N ! We use the symbol in (1.13) because we cannot be sure that the series converges. We will explore the question of convergence in the next section, and we will see in Theorem 2.3 that for functions that are minimally well behaved, the can be replaced by an equals sign for most values of x. EXAMPLE 1.15 ◆ Find the Fourier series associated with the function ´ < 0 ; for x 0, » D f º x ; : x for 0 x 2 We used the expression a0 =2 instead of a0 for the constant term in the Fourier series (1.7) so formulas like equation (1.10) would be true for n D 0 as well as for larger n. 716 Chapter 12 Fourier Series We compute the coefficient a0 using (1.8) or (1.10). We have 1 1 º » D º » D : a0 D f x dx x dx 0 2 For n 1, we use (1.10), and integrate by parts to get 1 1 º » D º » an D f x cos nxdx x cos nxdx 0 1 º » D x d sin nx n 0 ¬ 1 ¬ 1 ¬ º » C D x sin nx sin nxdx ¬ n 0 n 0 1 D »: º1 cos n 2 n n D º » D Thus, since cos n 1 , the even numbered coefficients are a2n 0, and the 2 D =Ì º C » Í : odd numbered coefficients are a2n C1 2 2n 1 for n 0 We compute bn using (1.11). Again we integrate by parts to get 1 1 º » D º » bn D f x sin nxdx x sin nxdx 0 1 º » D x d cos nx n 0 ¬ 1 ¬ 1 ¬ º » D x cos nx cos nxdx ¬ n 0 n 0 |an| and |bn| 1 : D n j The magnitude of the coefficients is plotted in Figure 3, with jan in black and 1 j jbn in blue. Notice how the coefficients decay to 0. The Fourier series for f is ½ ½ º C » 2 cos 2n 1 x sin nx » C C : f º x (1.16) n 2 º C » 0 4 2n 1 n D 0 10 20 30 n D0 n 1 ◆ Figure 3 The Fourier coefficients for the function in Example 1.15. Let’s examine the experimental evidence for convergence of the Fourier series in Example 1.15. The partial sums of orders 3, 30, and 300 for the Fourier series in Example 1.15 are shown in Figures 4, 5, and 6, respectively.