MENTAL MODEL · M6084

Fourier Series

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Version 1.0.0 · Updated 2026-07-28

CORE DEFINITION

In mathematics, a Fourier series is a way of representing a wave-like function as a combination of simple sine waves. More formally, for a periodic function satisfying the Dirichlet conditions, its Fourier series is a weighted sum of sine and cosine functions. The Fourier series is closely related to the Fourier transform, which is used to find the frequency information of non-periodic functions. The Fourier series is a branch of Fourier analysis and is central to the original proof of the sampling theorem. Fourier series have wide applications in number theory, combinatorics, signal processing, probability theory, statistics, cryptography, acoustics, optics, and other fields.

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A periodic function satisfying certain conditions can be expanded into an infinite series of sines and cosines (fundamental and harmonics). A complex waveform is shown to be a linear superposition of a few simple periodic waves, allowing the original signal to be approximated and reconstructed using a finite number of frequency components.

MINIMUM ACTION

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Practice this model in one real situation:

Compute the coefficients (Fourier coefficients) for each order of a periodic signal
Observe the fundamental and main harmonics to identify hidden periodicities
Truncate to a few dominant components for approximation or compression
Use reconstruction error to judge whether the truncation order is sufficient.

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zh.wikipedia.orghttps://zh.wikipedia.org/wiki/%E5%82%85%E9%87%8C%E5%8F%B6%E7%BA%A7%E6%95%B0ZH · Explicit
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