Fourier Series

📐 Definition

Let $f$ be a $2\pi$ -periodic function integrable on $[-\pi,\pi]$ . The Fourier series of $f$ is the trigonometric series

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

where the Fourier coefficients are defined by

a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\cos(nx)\,dx, \qquad b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\sin(nx)\,dx

The symbol “ $\sim$ ” denotes convergence in the appropriate sense.

Domain and Codomain

The Fourier series construction applies to real- or complex-valued functions defined on a finite interval with periodic extension. The series represents a function on $\mathbb{R}$ with period $2\pi$ .

⚙️ Key Properties

Orthogonality

The trigonometric basis functions satisfy the orthogonality relations

\int_{-\pi}^{\pi} \cos(nx)\cos(mx)\,dx = \int_{-\pi}^{\pi} \sin(nx)\sin(mx)\,dx = \pi\,\delta_{nm}

and

\int_{-\pi}^{\pi} \sin(nx)\cos(mx)\,dx = 0

These identities underlie the coefficient formulas. Here $n,m\ge 1$ ; the constant mode satisfies $\int_{-\pi}^{\pi} 1\cdot 1\,dx = 2\pi$ .

Complex Form

Equivalently, the Fourier series may be written in complex-exponential form:

f(x) \sim \sum_{n=-\infty}^{\infty} c_n e^{inx}

with coefficients

c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x)e^{-inx}\,dx

Linearity

The Fourier series map is linear:

\mathcal{F}(af + bg) = a\,\mathcal{F}(f) + b\,\mathcal{F}(g)

for scalars $a,b$ and admissible functions $f,g$ .

Differentiation and Integration

If $f$ is sufficiently smooth, termwise differentiation yields

\frac{d}{dx} \sum_{n=-\infty}^{\infty} c_n e^{inx} = \sum_{n=-\infty}^{\infty} (in)c_n e^{inx}

Integration acts by division by $in$ for $n \neq 0$ .

Convergence Behavior

If $f$ is piecewise smooth, the Fourier series converges pointwise to

\frac{1}{2}\bigl(f(x^+) + f(x^-)\bigr)

at each point $x$ .

Discontinuities produce the Gibbs phenomenon.

🎯 Special Cases

Even and Odd Functions

If $f$ is even, all sine coefficients vanish:

b_n = 0

If $f$ is odd, all cosine coefficients vanish:

a_n = 0

Pure Harmonics

For integer $k \ge 1$ ,

\cos(kx), \sin(kx)

have Fourier series consisting of a single nonzero mode.

The Fourier series is the discrete-frequency analogue of the Fourier transform. Complex exponentials form an orthogonal basis for periodic function spaces. Connections extend to eigenfunction expansions of linear differential operators.

Usage in Oakfield

Oakfield uses “Fourier series” most directly in its stimulus operators (explicit sums of harmonics / Fourier features):

Spectral line synthesis: stimulus_spectral_lines generates sums of spatial harmonics with optional temporal oscillation (for real fields the implied spectrum is Hermitian-symmetric; complex fields can represent single-sided lines).
Random Fourier features: stimulus_random_fourier synthesizes structured fields as sums of random cosine features with configurable band limits and spectral slope.
Single-mode waveforms: the sinusoidal stimulus family is effectively a one-term Fourier series (single harmonic with phase/velocity controls).

For full spectral transforms and mode-by-mode operators, Oakfield relies on discrete FFTs (see the Fourier transform docs).

Historical Foundations

📜 Origin (19th Century)

Joseph Fourier introduced trigonometric series to study heat conduction, asserting that arbitrary functions could be expanded in sines and cosines.

🔬 Rigorous Development

Dirichlet and others established precise convergence conditions, clarifying the mathematical foundations of Fourier analysis.

🌍 Modern Perspective

Fourier series are now understood as expansions in orthogonal bases of Hilbert spaces, forming a cornerstone of harmonic analysis, PDE theory, and numerical computation.

📚 References

J. Fourier, Théorie analytique de la chaleur (1822)
Dirichlet, Sur la convergence des séries trigonométriques
NIST Digital Library of Mathematical Functions (DLMF), §1.4