is a tool used for analyzing an arbitrary periodic function by decomposing it into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. The weights, or coefficients, of the modes, are a one-to-one mapping of the original function. These modal coefficients are sometimes themselves confusingly referred to as "modes" for brevity, especially in physics literature. Fourier series serve many useful purposes, as manipulation and conceptualization of the modal coefficients are often easier than with the original function.

Areas of application include electrical engineering, vibration analysis, acoustics, optics, signal and image processing, and data compression. Using the tools and techniques of spectroscopy, for example, astronomers can deduce the chemical composition of a star by analyzing the frequency components, or spectrum, of the star's emitted light.