Understanding Fourier Series: From Trigonometric Expansions to Hilbert‑Space Orthogonality

Fourier series let us write a periodic function as an infinite sum of sinusoids. The theory is elegant, the calculations are systematic, and the connection to linear algebra is striking. Below is a self‑contained guide that walks through the derivation of the coefficients, discusses when the series converges, shows how symmetry simplifies the expansion, and ties everything back to orthogonal bases in a Hilbert space.

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1. Deriving the Trigonometric Coefficients

Assume a well‑behaved L‑periodic function (f(x)). We look for constants (a_0, a_n, b_n) such that

[ f(x) ;=; \frac{a_0}{2} + \sum_{n=1}^{\infty}\bigl(a_n\cos\tfrac{2\pi n}{L}x + b_n\sin\tfrac{2\pi n}{L}x\bigr). ]

The constant term is written as (a_0/2) because the average value of (f) over one period is

[ a_0 = \frac{2}{L}\int_{-L/2}^{L/2} f(x),dx. ]

Step‑by‑step coefficient extraction

1. Integrate both sides over one period. All integrals of sines and cosines vanish (Appendix A), leaving the formula for (a_0).

2. Multiply by (\cos\tfrac{2\pi m}{L}x) and integrate. Orthogonality of sines and cosines (Appendix B) kills every term except the one with (a_m), giving

   [ a_m = \frac{2}{L}\int_{-L/2}^{L/2} f(x)\cos\tfrac{2\pi m}{L}x,dx. ]

3. Multiply by (\sin\tfrac{2\pi m}{L}x) and integrate. The same reasoning yields

   [ b_m = \frac{2}{L}\int_{-L/2}^{L/2} f(x)\sin\tfrac{2\pi m}{L}x,dx. ]

These three formulas give the full set of Fourier coefficients.

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2. When Does the Series Converge?

The derivation assumes that (f) is square‑integrable ((f\in L^2[-L/2,L/2])) and piecewise smooth – each piece has continuous derivatives, and only finitely many jump discontinuities exist. Under these conditions:

• At every point where (f) is continuous, the Fourier series converges to (f(x)).
• At a jump, the series converges to the midpoint of the jump (the classic Gibbs phenomenon).

These are the standard Dirichlet conditions; they cover most functions used in engineering and physics.

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3. Exploiting Symmetry: Cosine and Sine Series

If (f) is even ((f(-x)=f(x))), the sine coefficients vanish because the integrand becomes odd. The series reduces to a cosine series:

[ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n\cos\tfrac{2\pi n}{L}x. ]

If (f) is odd ((f(-x)=-f(x))), the cosine coefficients disappear, leaving a sine series:

[ f(x) = \sum_{n=1}^{\infty} b_n\sin\tfrac{2\pi n}{L}x. ]

These reductions are useful because they halve the number of integrals you need to evaluate.

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4. Extending Non‑Periodic Functions on a Finite Interval

Suppose we have a function defined only on ([0,,L]). To apply Fourier analysis we must create a periodic extension. Common choices are:

Extension type	Definition on ([-L,0))	Resulting series
Direct periodic repetition	(f(x+L)=f(x))	General Fourier series
Even extension	(f(-x)=f(x))	Cosine series
Odd extension	(f(-x)=-f(x))	Sine series

The series always converges to the periodic extension on the interior of ([0,,L]). The choice of extension determines which coefficients are zero.

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5. Worked Example: Triangular Wave (Period 4)

Consider the odd triangular wave defined on ([-2,2]) by

[ f(x) = \begin{cases} x+2, & -2\le x < 0,\ 2-x, & 0\le x \le 2. \end{cases} ]

Because the function is odd, only sine terms survive. With (L=4) (so the fundamental frequency is (\pi/2)), the coefficient is

[ b_n = \frac{2}{4}\int_{-2}^{2} f(x)\sin\frac{\pi n}{2}x,dx. ]

The integrand is even, so we double the integral over ([0,2]). Splitting at the kink (x=1) and using integration by parts (Appendix C) yields

[ b_n = \frac{8}{\pi^2 n^2}\bigl[1-(-1)^n\bigr]. ]

Thus (b_n=0) for even (n) and for odd (n)

[ b_{2k+1}=\frac{16}{\pi^2 (2k+1)^2}. ]

The Fourier series becomes

[ f(x) = \sum_{k=0}^{\infty}\frac{16}{\pi^2 (2k+1)^2}\sin\bigl((2k+1)\tfrac{\pi}{2}x\bigr). ]

An interactive plot (see the original notes) shows rapid convergence as the number of terms grows.

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6. Compact Phase‑Shifted Form

A pair of sine and cosine terms at the same frequency can be merged into a single sinusoid with amplitude (A_n) and phase (\phi_n):

[ a_n\cos\omega_n x + b_n\sin\omega_n x = A_n\sin(\omega_n x + \phi_n), ]

where

[ A_n = \sqrt{a_n^2 + b_n^2}, \qquad \phi_n = \operatorname{atan2}(a_n, b_n). ]

This representation is handy in signal‑processing contexts because it directly reveals the magnitude and phase of each harmonic.

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7. Complex Exponential Form

Euler's formula (e^{i\theta}=\cos\theta+i\sin\theta) lets us rewrite the series as

[ f(x) = \sum_{n=-\infty}^{\infty} c_n e^{i\frac{2\pi n}{L}x}, ]

with coefficients

[ c_n = \frac{1}{L}\int_{-L/2}^{L/2} f(x) e^{-i\frac{2\pi n}{L}x},dx. ]

For real‑valued (f), the coefficients satisfy (c_{-n}=\overline{c_n}), guaranteeing that the sum is real. The complex form is often more convenient for algebraic manipulation and for connecting Fourier series to the continuous Fourier transform.

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8. Fourier Series as an Orthogonal Basis in a Hilbert Space

The set

[ \bigl{1,;\cos\tfrac{2\pi n}{L}x,;\sin\tfrac{2\pi n}{L}x\bigr}_{n\ge 1} ]

forms an orthogonal basis of the Hilbert space (L^2[-L/2, L/2]) equipped with the inner product

[ \langle g, h \rangle = \int_{-L/2}^{L/2} g(x)h(x),dx. ]

Orthogonality follows from the integrals in Appendix B:

[ \langle \cos\tfrac{2\pi n}{L}x,; \cos\tfrac{2\pi m}{L}x \rangle = \begin{cases} L/2, & n=m\neq0,\ 0, & n\neq m,\end{cases} ]

and similarly for sines, while sine–cosine inner products are always zero. The Fourier coefficients are precisely the projections of (f) onto these basis vectors:

[ a_n = \frac{2}{L}\langle f,\cos\tfrac{2\pi n}{L}x\rangle, \qquad b_n = \frac{2}{L}\langle f,\sin\tfrac{2\pi n}{L}x\rangle. ]

Thus Fourier analysis is nothing more than expanding a vector in an infinite‑dimensional Euclidean space.

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9. Appendices (Reference Formulas)

Appendix A – Integrals of single sinusoids

[ \int_{-L/2}^{L/2}!\cos\tfrac{2\pi n}{L}x,dx = 0, \quad \int_{-L/2}^{L/2}!\sin\tfrac{2\pi n}{L}x,dx = 0 \quad (n\neq0). ]

Appendix B – Orthogonality of sine and cosine products

[ \int_{-L/2}^{L/2}!\cos\tfrac{2\pi n}{L}x,\cos\tfrac{2\pi m}{L}x,dx = \begin{cases}\frac{L}{2}, & n=m\neq0,\ 0, & n\neq m,\end{cases} ]

(and analogous formulas for sine–sine and sine–cosine).

Appendix C – Useful integral

[ \int x\cos(ax),dx = \frac{x\sin(ax)}{a}+\frac{\cos(ax)}{a^2}+C. ]

Appendix D – Converting a phase‑shifted sinusoid

[ A\sin(\omega x+\phi) = A\cos\phi,\sin\omega x + A\sin\phi,\cos\omega x. ]

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Further Reading

• Eli Bender’s original notes: https://eli.thegreenplace.net/2020/fourier-series
• Classic textbook: Fourier Analysis by T. W. Körner
• Hilbert‑space primer: https://en.wikipedia.org/wiki/Hilbert_space

Feel free to reach out with questions or comments via the author's email address listed on the original site.