Fourier Analysis glossary

Clear, one-line definitions of the Fourier Analysis terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.

convolution

The operation (f * g)(x) = ∫f(t)g(x - t)dt; blends two functions to produce a third, key to filtering and probability of sums.

See: Convolution from scratch, Convolution, the Machine

convolution theorem

The Fourier transform converts convolution into pointwise multiplication: F(f * g) = F(f) · F(g).

dirichlet conditions

Sufficient conditions on a periodic function (bounded, finitely many extrema and discontinuities per period) for its Fourier series to converge pointwise.

fourier coefficient

The number c_n = (1/T)∫f(x)e^(-2πinx/T)dx (or a_n, b_n form) giving the amplitude of the n-th harmonic in a periodic function.

fourier series

An expansion of a periodic function as an infinite sum of sines and cosines (or complex exponentials) with frequencies that are integer multiples of a fundamental.

See: Fourier Series Expansions, Convergence Issues for Fourier Series

fourier transform

An integral transform f̂(ξ) = ∫f(x)e^(-2πiξx)dx that decomposes a function into its frequency components.

See: The Fourier transform, slowly, The Fourier Transform on the Real Line

frequency

The number of oscillations per unit time, f = 1/T where T is the period; measured in hertz (Hz) for time-based signals.

See: Multi-scale frequency continuation, Waves: amplitude, frequency, wavelength

fundamental period

The smallest positive T for which f(x + T) = f(x) for all x; every other period is an integer multiple of T.

gibbs phenomenon

The persistent overshoot of about 9% of jump height in Fourier partial sums near a discontinuity, which does not vanish as more terms are added.

l2 convergence

Convergence of f_n to f in mean square: ∫|f_n - f|² → 0; the natural notion of convergence for Fourier series in L²(T).

orthogonal system

A family of functions {φ_n} with ∫φ_m φ_n w(x) dx = 0 for m ≠ n; the basis property underlying Fourier series and eigenfunction expansions.

parseval theorem

For a function f with Fourier coefficients c_n, ∫|f|² = Σ|c_n|²; energy in the time domain equals energy in the frequency domain.

periodic function

A function f with f(x + T) = f(x) for some positive T and all x; the smallest such T is its fundamental period.

See: Periodic Functions and Wave Phenomena