# Fourier Transform
Confidence: high
Last verified: 2026-05-22
Generation: human_only


## TL;DR

The Fourier Transform (Joseph Fourier, 1807) decomposes a signal into its constituent frequencies. It is fundamental to signal processing, image compression (JPEG), audio processing (MP3), and MRI reconstruction. FFT (Fast Fourier Transform, Cooley-Tukey 1965) reduces complexity from O(n²) to O(n log n).

## Core Explanation

Continuous FT: F(ω) = ∫ f(t)e^(-iωt) dt. Discrete FT: X_k = Σ x_n e^(-i2πkn/N). FFT is one of the most important algorithms of the 20th century (Strang). Applications: audio equalization (frequency domain filtering), JPEG compression (DCT, a Fourier variant), MRI (k-space to image space), radio astronomy, quantum mechanics.

## Further Reading

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