Fourier coefficient

A scalar multiplying a basis function in a Fourier expansion, measuring the represented function’s contribution at a particular frequency. Its value depends on the chosen Fourier basis and normalization.

Fourier series: A representation of a periodic function as a sum of sinusoidal or complex exponential terms. Fourier coefficients are the weights that assemble its frequency components.
Periodic function: A function whose values repeat after a fixed nonzero interval. Classical Fourier-series coefficients describe periodic functions.
Harmonic analysis: The study of functions and signals through oscillatory components such as exponentials and waves. Fourier coefficients are basic coordinates in this broader study of frequency structure.
Fourier transform coefficient: A value of a Fourier transform describing a function’s contribution at a continuous frequency. Unlike Fourier-series coefficients, transform values typically vary over a continuous frequency axis.
Inner product: An operation that combines two vectors or functions to produce a scalar measuring their alignment. For orthogonal Fourier bases, an inner product extracts a component’s coefficient.
Frequency: The number of cycles of a periodic variation per unit of time or space. Each coefficient is associated with a specific frequency in the chosen expansion.
Signal processing: The analysis and transformation of signals to extract, modify, or transmit information. Coefficients expose a signal’s periodic components for filtering and spectral analysis.
Fourier cosine series: A Fourier expansion using only cosine functions on an interval. Its coefficients represent even-reflected components rather than the full sine-and-cosine basis.
Orthogonality: A relation in which vectors or functions have zero inner product. Orthogonality lets each Fourier coefficient be determined independently of the others.
Fourier basis: A collection of sinusoidal or complex exponential functions used to represent functions by frequency. The basis determines which contribution each coefficient measures.

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