Fourier series

A representation of a periodic function as a sum of sines and cosines, or complex exponentials, whose frequencies are integer multiples of the fundamental frequency.

Fourier coefficient: A scalar measuring a function’s contribution at a particular frequency in its Fourier series. Each coefficient sets the amplitude and phase of one harmonic in the series.
Periodic function: A function whose values repeat after a fixed positive interval called its period. Fourier series represent periodic behavior by repeating frequency components.
Joseph Fourier: French mathematician and physicist who developed trigonometric series to study heat conduction. His heat theory made the expansion central to mathematical physics.
Signal processing: The analysis and modification of signals to extract, transmit, or encode information. Fourier series identify the frequency content of periodic signals.
Fourier transform: An integral transform representing a function through a continuous range of frequencies. It is the continuous-frequency counterpart used chiefly for nonperiodic functions.
Fourier basis: A set of orthogonal functions used to represent other functions through weighted sums. Sines and cosines form the orthogonal basis used for periodic-function expansions.
Fundamental frequency: The lowest positive frequency associated with a periodic signal's repetition rate. It determines the integer-spaced frequencies allowed in the series.
Théorie analytique de la chaleur: Joseph Fourier’s 1822 treatise presenting a mathematical theory of heat conduction. It contains Fourier’s influential use of trigonometric expansions for heat problems.
Spectral analysis: Methods for describing a signal or system through its frequency components. A Fourier series gives a discrete spectrum for a periodic function.
Fourier integral: An integral representation of a function as a continuous superposition of sinusoidal frequencies. Unlike a Fourier series, it uses a continuous spectrum rather than integer-spaced harmonics.

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