# Fourier Transform

**Type:** Concept  
**Domain:** Applied Mathematics  
**Codex URL:** /mathematics/applied/fourier-transform/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Decomposes any signal into a sum of simple sine/cosine waves of different frequencies. Developed by Fourier (1807) for heat flow; essential to signal processing, imaging, quantum mechanics.

## Fourier series
f(x) = a₀/2 + Σ[aₙcos(nx)+bₙsin(nx)]

## Fourier Transform
F(ω) = ∫f(x)e⁻ⁱωˣdx — continuous version for non-periodic functions

## Fast Fourier Transform (FFT)
Cooley-Tukey (1965): O(n log n) vs naive O(n²). Traces to unpublished Gauss work (~1805).

## Uncertainty principle
Time-frequency trade-off mathematically identical in structure to Heisenberg Uncertainty Principle.

## Quantum mechanics
Position and momentum wavefunctions are Fourier Transform pairs — deep origin of the Uncertainty Principle.

## Sources
### Tier 1
- Körner, T.W. (1988). *Fourier Analysis*. Cambridge University Press.
- Stein, E.M. and Shakarchi, R. (2003). *Fourier Analysis: An Introduction*.
### 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/*