Fourier analysis - Wikipedia

Abstraction: Decomposing functions into sums of trigonometric or complex exponential components

Key points:

• Five variants: CTFT, CTFS, DTFT, DTFS, DFT; only the DFT is numerically computable and can approximate the others
• Exponential functions are eigenfunctions of differentiation, converting linear ODEs with constant coefficients into algebraic equations
• Convolution theorem: Fourier transform converts convolution to pointwise multiplication, enabling efficient filtering and polynomial multiplication
• FFT algorithm (O(N log N)) first discovered by Gauss ~1805 for asteroid orbit interpolation; modernly attributed to Cooley-Tukey
• JPEG compression uses the discrete cosine transform (a Fourier variant) on 8×8 image blocks
• Generalizes to harmonic analysis on locally compact Abelian groups via Pontryagin duality; the transform maps functions on a group to functions on the dual group

Connections: Fourier Analysis · Signal Processing · Harmonic Analysis

Source: http://en.wikipedia.org/wiki/Fourier_analysis