Fourier Series Expansions

Part 13, Chapter 13: Fourier Series and Transforms

Learning objectives

Write the Fourier series of a $2\pi$ -periodic function
Compute $a_n$ and $b_n$ via integral formulas
Use orthogonality of $\{1, \cos(nx), \sin(nx)\}$ to derive the coefficient formulas
Exploit even/odd symmetry to predict which coefficients vanish

If every reasonable $2\pi$ -periodic function is a sum of sines and cosines, how do you find the coefficients? The answer is one of the most elegant constructions in mathematics: the trigonometric system is orthogonal in the integral inner product, so each coefficient is recovered by a single integration. The Fourier-coefficient formula is just the inner product of $f$ with the basis function, divided by the basis function's norm. Once you internalize that one idea, every special-symmetry rule (odd functions have no cosines, even functions have no sines) becomes obvious.

The Fourier series

Let $f$ be a $2\pi$ -periodic function on $[-\pi, \pi]$ . Its Fourier series is the formal sum

$f(x) \sim \dfrac{a_0}{2} + \displaystyle\sum_{n=1}^{\infty} \bigl(a_n \cos(nx) + b_n \sin(nx)\bigr)$ bigl(ancos(nx)+bnsin(nx)bigr)

where the Fourier coefficients are computed by:

$a_n = \dfrac{1}{\pi} \displaystyle\int_{-\pi}^{\pi} f(x) \cos(nx) \, dx, \quad n = 0, 1, 2, \ldots$ cos(nx),dx,quadn=0,1,2,ldots

$b_n = \dfrac{1}{\pi} \displaystyle\int_{-\pi}^{\pi} f(x) \sin(nx) \, dx, \quad n = 1, 2, 3, \ldots$ sin(nx),dx,quadn=1,2,3,ldots

The factor of $1/\pi$ comes from the inner-product norm, and the constant term gets a factor of $a_0 / 2$ for tidiness. The $\sim$ symbol (instead of $=$ ) is a reminder that convergence of the right side to $f$ is a separate question, taken up in section 13.3.

Why the formula works: orthogonality

Equip the space of square-integrable functions on $[-\pi, \pi]$ with the inner product $\langle f, g \rangle = \int_{-\pi}^{\pi} f(x) g(x) \, dx$ −pipif(x)g(x),dx. The system $\{1, \cos(x), \sin(x), \cos(2x), \sin(2x), \ldots\}$ is orthogonal: any two distinct basis functions integrate to zero against each other. The product-to-sum identities give

$\displaystyle\int_{-\pi}^{\pi} \cos(mx) \cos(nx) \, dx = \begin{cases} 0, & m \neq n \\ \pi, & m = n \geq 1 \\ 2\pi, & m = n = 0 \end{cases}$

and analogous formulas for sine-sine and sine-cosine products. To find $a_m$ m, multiply the Fourier series by $\cos(mx)$ and integrate term by term: all terms vanish except the $m$ -th cosine term, which contributes $a_m \cdot \pi$ mcdotpi. Solving gives the formula $a_m = (1/\pi) \int f \cos(mx) \, dx$ m=(1/pi)intfcos(mx),dx.

The series summer lets you stack partial sums of a Fourier series. Try the square wave $f(x) = \text{sign}(x)$ on $[-\pi, \pi]$ , the first few odd harmonics with coefficients $4/(\pi n)$ already give a recognisable square shape. The amplitude of each new term decays as $1/n$ , so adding terms rapidly diminishes the visible error away from the jumps.

Even/odd shortcuts

The integrand $f(x) \cos(nx)$ is even if $f$ is even (even times even). Its integral over $[-\pi, \pi]$ equals twice the integral over $[0, \pi]$ , you save half the work. The integrand $f(x) \sin(nx)$ is odd if $f$ is even (even times odd), so it integrates to zero: every $b_n = 0$ n=0 for an even $f$ . Symmetrically, odd $f$ has every $a_n = 0$ n=0 and only sine terms appear. Always check the symmetry of $f$ before computing any integral.

Where this shows up

MP3, AAC, Opus audio: Modern audio codecs apply a windowed Fourier-like transform (modified DCT) to short frames of a recording, quantise the coefficients aggressively, and store only the perceptually significant ones. The "frequency-domain compression" the standards talk about is exactly Fourier coefficient quantisation.
JPEG image compression: JPEG splits an image into 8x8 pixel blocks and computes a 2D discrete cosine transform of each block. Cosine basis functions are chosen because images tend to be smooth (the highest-frequency cosine coefficient is usually tiny), and the few large coefficients suffice to reconstruct the block.
Heat equation solutions: When you separate variables on the heat equation $u_t = u_{xx}$ on $[0, L]$ with zero boundary, the solution is a Fourier sine series in $x$ with time-dependent coefficients. Section 14.4 makes this explicit.
Music synthesis: Additive synthesisers build a tone by summing sinusoids at integer multiples of the fundamental, each with its own amplitude envelope. This is literally constructing a sound from its Fourier coefficients in real time.

Pause and think: Suppose you compute $a_n = 0$ n=0 for every $n$ . Does that force $f \equiv 0$ ? (Hint: think about what happens if $f$ is odd. Cosine coefficients vanish, but the function is non-zero.)

Try it

Compute $a_0$ for $f(x) = x^2$ on $[-\pi, \pi]$ . Use the formula $a_0 = (1/\pi) \int_{-\pi}^{\pi} x^2 \, dx$ ,dx and the symmetry of $x^2$ .

Before integrating: which Fourier coefficients of $f(x) = x^3$ on $[-\pi, \pi]$ vanish, and why?

Verify directly: $\int_{-\pi}^{\pi} \sin(2x) \cos(3x) \, dx = 0$ −pipisin(2x)cos(3x),dx=0. (Use the product-to-sum identity $\sin a \cos b = \tfrac{1}{2}[\sin(a+b) + \sin(a-b)]$ .)

On the series-summer widget, build $\sum_{n=1,3,5,\ldots}^{N} (4/(\pi n)) \sin(nx)$ n=1,3,5,ldotsN(4/(pin))sin(nx) for $N = 1, 3, 5, 9$ . Each partial sum is the Fourier series of the square wave truncated at $N$ terms. Sketch how the graph evolves.

Find the Fourier coefficients of the constant function $f(x) = 7$ on $[-\pi, \pi]$ .

A trap to watch for

The constant term $a_0/2$ (not $a_0$ ) is the average value of $f$ over $[-\pi, \pi]$ . Students routinely write the Fourier series as $a_0 + \sum (a_n \cos nx + b_n \sin nx)$ cosnx+bnsinnx), dropping the factor of $1/2$ . That extra factor exists because the cosine of $0x$ is the constant $1$ , and the inner-product norm of $1$ is $2\pi$ rather than $\pi$ , without the $1/2$ , the formula for $a_0$ would double-count the constant. The convention is annoying but universal; always include the $1/2$ .

What you now know

You can compute Fourier coefficients of a $2\pi$ -periodic function, use even/odd symmetry to predict which integrals vanish, and explain why the formulas come from inner-product projection onto an orthogonal basis. The next section asks the convergence question: does the Fourier series actually reconstruct $f$ , and in what sense?

Mark section complete →

References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, ch. 13.

Stein, E. M., Shakarchi, R. (2003). Fourier Analysis: An Introduction. Princeton University Press, ch. 2.

Folland, G. B. (1992). Fourier Analysis and Its Applications. Wadsworth & Brooks/Cole, ch. 2.

Bracewell, R. N. (1999). The Fourier Transform and Its Applications (3rd ed.). McGraw-Hill, ch. 2.

Korner, T. W. (1989). Fourier Analysis. Cambridge University Press, ch. 2 (orthogonality and Fourier coefficients).

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