Math 489AB A Very Brief Intro to Fourier Series Fall 2008

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1 Math 489AB A Very Brief Intro to Fourier Series Fall 8 Contents Fourier Series. The coefficients Convergence Convergence theorems Bessel s inequality Proofs of convergence theorems and The complex form of the Fourier Series Exercises 3 Solutions to selected exercises Fourier Series Many phenomena in nature are periodic, and best described by periodic functions. For example, sound waves are made up of tones described by sine waves with various frequencies. Light waves have similar properties. Additionally, phenomena with boundary conditions, such as the concentration of a substance in an enclosed area in which the concentrations at the boundaries are fixed, are also periodic, and are naturally described by sums of cosines and sines. We thus attempt to write a given function f as a linear combination of cosines and sines: f(x) a + [a k cos(kx) + b k sin(kx)]. () k (Throughout this section we will always assume that f(x) is periodic. We can assume without loss of generality that the period is π. If the period of f is different we can always make a transformation (exercise).) The linear combination () is known as a Fourier Series, and plays an enormously important role in many areas of mathematics (PDE s, signal processing, etc.). Our aim here will be to determine the conditions under which the series above converges to f(x), and in what sense? That is, when can we find constants a k and b k so that the above equality holds?. The coefficients One important and amazing fact is that the trigonometric functions are orthogonal. Let f, g f(x)g(x)dx

2 FOURIER SERIES be the standard inner product on L [, π]. Then cos(nx), cos(mx) ( if n m) ( if n m) [ cos(nx) cos(mx)dx [cos(nx mx) + cos(nx + mx)] dx sin((n m)x) + n + m sin((n + m)x) ] xπ A similar result holds for sin(mx) and sin(nx). Thus { if n m cos(nx), cos(mx) π if n m { if n m sin(nx), sin(mx) π if n m () n m x [ + cos(nx)] dx π (3) cos(nx), sin(mx) for all integers m and n (6) (4) (5) Therefore, IF f(x) a + [a k cos(kx) + b k sin(kx)], (7) k THEN we can use the orthogonality of the cos(kx) and sin(kx) functions to determine the coefficients. Take the inner product of both sides of (7) with cos(mx): f(x), cos(mx) a + [a k cos(kx) + b k sin(kx)], cos(mx) k a, cos(mx) + a m π [a k cos(kx), cos(mx) + b k sin(kx), cos(mx) ] k (8) We get a similar result if we take the inner product with sin(mx). Thus we can calculate all of the coefficients: a π f, π a k π f,cos kx π b k π f,sinkx π f(x)dx (9) f(x) cos(kx) dx () f(x) sin(kx) dx () So, as long as the series converges we can compute the coefficients explicitly. Often we only compute the first several terms in the series. That is, we approximate f by a trigonometric polynomial, i.e. a truncated Fourier series. Take the first N terms: f(x) a + (a k cos(kx) + b k sin(kx)) : T N (x). () k

3 FOURIER SERIES 3 Then, as N increases we expect that the approximation should get better and better. In fact, we can glean a lot of information about f just from its first few terms. Figure shows various functions and the truncated Fourier series. We see how the approximation gets better and better as N gets larger and larger, even if the function is discontinuous. N 3 f(x) x.5.5 f(x) sign(x) Bessel function 4 6 f(x) piecewise N N N Figure : Trigonometric polynomial approximations. The figures show the partial sums T N (x) of the Fourier Series.

4 FOURIER SERIES 4 We still have to verify that if condition from page. The basic notions are due to Joseph Fourier, who developed his ideas on the convergence of trigonometric series while studying heat flow. His initial 87 paper was rejected by the scientific community for being too imprecise, and was not published until 8.. Convergence When we say that T N converges to f, written T N f, we can mean that lim N T N (x) f(x) at every value of x, or we could mean that lim N T N f, for some norm. So this is just convergence of a sequence of real numbers, so all the standard tools to test for convergence can be used. However, the notion of convergence depends on the norm that is used. Consider, then Three notions of convergence:. Definition. We say that an infinite series f n (x) converges to f(x) pointwise in (a, b) if it converges to f(x) for each a < x < b. That is, for each a < x < b, we have. Case : N f(x) f n (x) as N. (3) Definition. We say that an infinite series f n (x) converges uniformly to f(x) in [a, b] if N max f(x) f n (x) as N. (4) 3. Case : a x b Definition. We say that an infinite series f n (x) converges in the mean-square (or L ) sense to f(x) in (a, b) if b a N f(x) f n (x) Example Let f n (x) ( x)x n on the interval < x <. Then, ( f n (x) x n x n) x N. dx as N. (5) So the series converges pointwise on (,) to f(x). Does it converge uniformly? In the meansquare sense?

5 FOURIER SERIES 5.3 Convergence theorems Now let f(x) be any function defined on a x b. WOLG we can take a, b π (exercise). Consider the Fourier Series (7), with coefficients given by (9 - ). We have the following theorems for each of the three notions of convergence. Theorem. Pointwise Convergence If f(x) is a function of period b a on the line for which f(x) and f (x) are piecewise continuous, then the Fourier Series converges to [f(x+) + f(x )] for < x <. In particular, if f (x) is continuous then the Fourier Series converges to f(x). Theorem. Uniform Convergence The Fourier Series converges to f(x) uniformly on [a, b] provided that f C [a, b]. In other words, if f(x), f (x), and f (x) exist and are continuous on [a, b], then the Fourier Series (7) converges uniformly to f(x). Theorem 3. L Convergence The Fourier Series converges to f(x) in the mean-square sense in (a, b) provided only that f(x) is integrable, that is, f is measurable and b f(x) dx is finite. a (Note: L convergence means that T N (x) g(x) where f g. That is, the Fourier series converges to a function that is equal to f almost everywhere.) We will prove Theorems and here. The proof of Theorem 3 is the subject of Section 3.5 of your text. Note that if f(x) is an odd function, then a k for all k, so the Fourier series is a sine series. Likewise, if f(x) is an even function, then b k for all k, so the Fourier series is a cosine series. Example. Consider the function f(x) on (, π) and f(x) on (, ). Then f(x) is odd, so all the a k s are zero, and the b k s are given by b k f(x)sin kx dx { π 4/kπ k odd sinkx dx π π k even Therefore, according to Theorem, the Fourier sine series of the function f(x) on (, π) is 4 sin nx. nπ n odd It converges at each point, but not uniformly on [, π]. For one thing, at x, π, the series! For any x in the open interval, the series converges, however. We may put x π/, for example, to get 4 π n odd ( )(n )/, so we get the convergent series π Since < < π, we can put x to get π 4 sin + 3 sin sin5 +.

6 FOURIER SERIES 6.3. Bessel s inequality Before we proceed to the proof of Theorem, we will need to develop some further theory, which turns out to be fundamental for all inner product spaces with an infinite orthogonal set. Consider an orthogonal set {X n }. These could be the sine and cosine functions in the Fourier series (or not). For a given f in the span of the X n s, i.e. the coordinates (Fourier coefficients) of f are f(x) A X (x) + A X (x) + A 3 X 3 (x) +, A n f, X n X n. For the sake of simplicity, we assume we are working in L [a, b], with the standard L inner product. Theorem Bessel s Inequality If {X n } is an orthogonal set and the coefficients A n are defined as above, then b b X n (x) dx f(x) dx. (6) A n Proof. We calculate the norm of the difference: f A n X n n N a f A n f, X n + n N a n,m N f A n X n + A n X n n N n N A n A m X n, X m f A n X n n N The inequality thus holds for every N. Taking N completes the proof. Note that this also implies the following Theorem The Fourier series of f(x) converges to f(x) in the mean-square sense if and only if b b A n X n (x) dx f(x) dx. (7) a a Proof. Mean-square convergence means that the error term goes to zero, which in turn means (7), known as Parseval s identity. Definition. The infinite set of functions {X (x), X (x),... } is called an orthogonal basis (also sometimes called complete) if Parseval s identity is true for all f with f <. Thus, to prove Theorem 3, it suffices to show that the set of trigonometric functions is a basis for L.

7 FOURIER SERIES 7.4 Proofs of convergence theorems and PROOF OF POINTWISE CONVERGENCE (Theorem ) For the moment let f(x) be a π periodic C function on the whole real line. Consider the series (7), with coefficients (9 - ). The Nth partial sum of the series is given by S N (x) a + (a k cos(kx) + b k sin(kx)). k We want to show that S N (x) f(x) as N. The first step is to do a bit of rearranging by plugging in the formulas for the coefficients, and using some trig identities: S N (x) where K N (θ) + [ + ] (cos(ky) cos(kx) + sin(ky) sin(kx)) f(y) dy π k K N (x y)f(y) dy π, cos kθ k K N (θ) is called the Dirichlet kernel. We will use some remarkable properties of this function to proceed. For one thing, K N(θ)dθ π, which follows from the evenness of all the cosine terms. Secondly, (( ) ) K N (θ) sin N + θ ( ) sin θ (8) The proof of this is left as an exercise. The figure below shows sketches of K N for N 5 and N. The idea is that as N gets larger, K N becomes more sharply peaked around, and the oscillations tend to cancel out in the integral away from. K 5 (t) K (t) To make these ideas more precise, make the change of variables θ y x in the integral above, and use the evenness of K N to deduce that S N (x) K N (θ)f(x + θ) dθ π.

8 FOURIER SERIES 8 Therefore, S N (x) f(x) K N (θ)[f(x + θ) f(x)] dθ π g(θ)sin[(n + /)θ] dθ π, (9) where g(θ) f(x + θ) f(x) sin θ. Now, we fix x, since we are proving pointwiese convergence. So we just have to show that the integral goes to zero as N. This is the next step. Notice that the functions φ N (θ) sin[(n + /)θ], N,, 3,..., form an orthogonal set on the interval (, π) (exercise). Therefore, Bessel s inequality (6) holds: N g, φ N φ n g. A direct calculation shows that φ N π. If g <, the series above is convergent and its terms tend to zero. So g, φ N, which says exactly that the integral in (9) tends to zero. It remains to show that g <. We have g [f(x + θ) f(x)] sin θ dθ Since the numerator is continuous, the only possible difficulty could occur where the sine vanishes, namely at θ. At that point, f(x + θ) f(x) θ lim g(θ) lim θ θ θ sin θ f (x). Therefore g is everywhere continuous, so that the integral g is finite. This completes the proof of pointwise convergence for C functions. PROOF FOR DISCONTINUOUS FUNCTIONS In the case where f(x) and f (x) are continuous except at a finite number of points, we proceed as before, with minor modifications. In particular, instead of (9), we have S N (x) [f(x+) f(x )] where g ± (θ) + K N (θ)[f(x + θ) f(x+)] dθ π K N (θ)[f(x + θ) f(x )] dθ π g + (θ)sin[(n + )θ]dθ π + g (θ)sin[(n + )θ]dθ π,() f(x + θ) f(x±) sin θ.

9 FOURIER SERIES 9 As before, we use Bessel s inequality, which proves the result so long as both g ± <. The only possible reason for the divergence of these integrals could come from the vanishing of sin θ at zero. Now, the one-sided limit of g + is lim g f(x + θ) f(x) θ +(θ) lim θց θց θ sin θ f (x+) if x is a point where the one-sided derivative f (x+) exists. If f (x+) does not exist (because of, e.g., a jump at x), then f is still differentiable at nearby points (Why?). By the mean value theorem, [f(x + θ) f(x)]/θ f (θ ) for some point θ between x and x + θ. Since the derivative is bounded, it follows that [f(x+θ) f(x)]/θ is bounded as well for θ small and positive. So g + (θ) is bounded and the integral g + (θ) dθ is finite. The case for g is similar. PROOF OF UNIFORM CONVERGENCE (Theorem ) Now assume that f is C. We know from the preceeding that the Fourier series converges pointwise to f(x) for all x (a, b). To show uniform convergence we need to show that max f(x) S N (x) as N. () The idea is to show that the Fourier coefficients go to zero pretty fast. Let a k and b k be the Fourier coefficients of f(x), and let a k and b k be the Fourier coefficients of f (x). Integrate by parts to get a k f(x)cos nx dx π π f(x)sin nx nπ a n n b n for n >. f (x)sin nxdx nπ Similarly, b n n a n. Now, if we apply Bessel s inequality to f (x), we have that Therefore, ( a n cos nx + b n sin nx ) ( a n + b n ) <. ( ( a n + b n ) a n n + b n ) ( ) / [ / ( a n n + b n )] <, where we have used the l Schwartz inequality. This means that the Fourier series converges absolutely. Thus, max f(x) S N (x) max nn+ a n cos nx + b n sin nx nn+ ( a n + b n ). () The last term above is the tail of an absolutely convergent series, and hence it tends to zero as N. This completes the proof.

10 FOURIER SERIES.5 The complex form of the Fourier Series It is sometimes easier to work with exponentials than cosines and sines. We can use all of the results we have developed so far, along with the De Moivre formulas sinθ eiθ e iθ i, cos θ eiθ + e iθ. So, instead of cosnx and sinnx, we can use e ±inx, as long as we use complex coefficients. The trigonometric polynomial T N (x) a + a + a + (a k cos(kx) + b k sin(kx)) k ( k k a k e ikx + e ikx e ikx e ikx ) + b k i [ (a k ib k )e ikx + (a k + ib k )e ikx ] k N c k e ikx where c a, c k (a k ib k ), c k (a k + ib k ) Or, since {, e ix, e ix, e ix, e ix,... } are orthogonal on L [, π], (Note: f, g b a c k f, eikx e ikx π f(x)e ikx dx. (3) π f(x)g(x)dx.) Thus, the Fourier Series of f(x) can be written as where the coefficients c k are as in (3). f(x) k c k e ikx, (4)

11 EXERCISES Exercises. Derive the Fourier series for f(x) and the coefficients for the series on a general interval (a, b).. Consider the functions f n (x) n + n x n + (n ) x. Show that f n (x) converges pointwise to f(x) in the interval < x < l, but that it does not converge uniformly or in the mean-square sense. 3. Use the Fourier Series for sinx on (, π) to find the sums 4n and ( ) n 4n. 4. Let φ(x) x for x π. Calculate (a) its Fourier sine series, and (b) its Fourier cosine series. 5. Prove formula (8) for the Dirichlet kernel. Hint: Write the cosines as exponentials using De Moivre s formula and then sum the geometric series. 6. Show that the functions φ N (θ) sin[(n + /)θ], N,, 3,..., form an orthogonal set on the interval (, π). Hint: If they are orthogonal on (, π), they are orthogonal on (, π). Why? 7. Prove that the Fourier Series of x in the interval (, π) converges uniformly to x in [, π]. 8. Show that if f(x) is a C function in [, π] and if f(x)dx, then f(x) dx f (x) dx. (Hint: Use Parseval s Identity eqn (7).) 9. Show that a complex-valued function f(x) is real valued if and only if its complex Fourier coefficients satisfy c n c n.. Fill in the details of a slick proof of pointwise convergence of Fourier Series, due to P. Chernoff (American Mathematical Monthly, May 98): (a) Let f(x) be a C function of period π. First show that we might as well assume that f() and we need only show that the Fourier series converges to zero at x. (b) Let g(x) f(x)/(e ix ). Show that g(x) is a continuous function. (c) Let C n be the complex Fourier coefficients of f(x) and D n the coefficients of g(x). Show that D n. (d) Show that C n D n D n so that the series C n is telescoping. (e) Deduce that the Fourier series of f(x) at x converges to zero.

12 3 SOLUTIONS TO SELECTED EXERCISES 3 Solutions to selected exercises. Consider the functions f n (x) n + n x n + (n ) x. Show that f n (x) converges pointwise to f(x) in the interval < x < l, but that it does not converge uniformly or in the mean-square sense. Solution. Note that the sum is telescoping, so the partial sums are So, for any < x <, S N (x) f n (x) S N (x) [ ] n + n x n + (n ) x n N + n x Thus the series converges pointwise to in < x <. N n + n x n N + N x + x N + N x n + (n ) x n + n x N + N x Nx + Nx N x as N However, when we consider uniform convergence, we notice that max S N(x) max N x (,) x (,) + N x N which obviously does not converge to zero. Thus the series does not converge uniformly. Now we consider L convergence. We have S N (x) dx N + N x dx N ( + N x ) dx N N dy N ( + y ) N N ( + y ) dy where we have made the change of variables y Nx. Since ( + y ) dy π 4,

13 3 SOLUTIONS TO SELECTED EXERCISES 3 The integral of the partial sums squared approaches Nπ/4, which tends to as N. Thus the series does not converge in the L sense. 3. Use the Fourier Series for sinx on (, π) to find the sums 4n and ( ) n 4n. Solution. sinx is an even function, so its Fourier series is a cosine series. We calculate he coefficients a k : a k sinx cos(kx)dx sin(x) cos(kx)dx π π { 4 + coskπ k even π k π( k ) k odd and a /π. Therefore, for < x < π, sinx π + π k even 4 k cos(kx) 4 ( π We can plug x, so sinx, cos nx, and 4n 4n. If we use x π/, then cos nx cos nπ ( ) n, and sinx, so 4 π ( ( ) n ) 4n ) 4n cos(nx) ( ) n 4n π 4 π 4 4. Let φ(x) x for x π. Calculate (a) its Fourier sine series, and (b) its Fourier cosine series. (a) The Fourier series of x on (, π) is a cosine series since x is even. We can calculate the coefficients a k : a π a k π x dx π 3 x cos(kx)dx 4 cos(kπ) k 4( )k k (k > ) Therefore, for < x < π, x π 3 + 4( ) k cos(kx) k

14 3 SOLUTIONS TO SELECTED EXERCISES 4 (b) To find the sine series expansion we form the function f(x) x sign(x), where sign(x) if x > and if x <. Then f(x) x on x π, and f( x) f(x), so f is an odd function. Thus, its Fourier series on (, π) will be a sine series. We calculate the coefficients b k : b k π f(x) sin(kx)dx ( + ( k π ) cos(kπ) ) x sin(kx)dx π π ( + ( k π ) ( ) k) k 3 So, for < x < π, π x k 3 ( + ( k π ) ( ) k) π k 3 sin(kx) Notice that, for < x < π we have the interesting relationship ( + ( k π ) ( ) k) π k 3 sin(kx) π 3 + 4( ) k k cos(kx) Note, also, that the sine series is discontinuous since sin kπ. The sine expansion will thus converge more slowly than the cosine series, and will also exhibit the Gibbs effect, due to the discontinuity. The figures below show plots of x (green), and the cosine (blue) and sine (red) series, taking different numbers of terms. Notice that with terms, the sine series is very close to x for most of the interval, but near x π, we see the overshoot typical of truncated Fourier series of a discontinuous function. 5 terms in the Fourier Series terms in the Fourier Series terms in the Fourier Series terms in the Fourier Series