# Fourier Series

**Type:** Concept  
**Domain:** Analysis  
**Codex URL:** /mathematics/analysis/fourier-series/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Expresses periodic functions as sums of sine and cosine waves. Underlies MP3/JPEG compression.

## Formula
f(x) = a₀/2 + Σ[aₙcos(nx)+bₙsin(nx)], coefficients via integration exploiting orthogonality.

## Fourier's 1807 skeptical reception
Lagrange objected that discontinuous functions couldn't be represented by smooth trig sums. Published 1822 (Théorie analytique de la chaleur).

## Convergence resolved over 150+ years
Dirichlet (1829): sufficient conditions for pointwise convergence. Carleson (1966): general square-integrable case — Abel Prize 2006.

## Gibbs phenomenon
Persistent ~9% overshoot near jump discontinuities, regardless of term count (Wilbraham 1848, explained rigorously by Gibbs 1899).

## Fast Fourier Transform (FFT)
Cooley-Tukey (1965) — reduces computation from O(n²) to O(n log n). Gauss had related unpublished ideas c.1805, predating Fourier himself.

## Sources
### Tier 1
- Fourier, J. (1822). Théorie analytique de la chaleur.
- Carleson, L. (1966). Acta Mathematica.
### Tier 2
- Cooley, J.W. and Tukey, J.W. (1965). Mathematics of Computation.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*