Harmonic analysis - Wikipedia

Abstraction: Mathematical analysis decomposing functions via symmetries scales and spectra

Key points:

• Harmonic analysis decomposes functions and measures into components based on symmetries, scales, spectra, or oscillations; includes Fourier series, maximal functions, singular integrals, Littlewood-Paley theory
• Key tool: Calderon-Zygmund decomposition splits functions into "good" (bounded) and "bad" (localized, zero-mean) parts, enabling Lp estimates for singular integrals
• Littlewood-Paley theory decomposes by frequency scale using almost-orthogonality, extending Fourier methods to situations where strict orthogonality fails
• Abstract harmonic analysis studies functions on topological groups; Pontryagin duality and Peter-Weyl theorem generalize Fourier analysis to locally compact groups
• Fourier restriction problems ask whether Fourier transforms can be meaningfully restricted to curved surfaces (spheres, cones); curvature enables cancellation
• Connected to Sobolev spaces, PDEs, ergodic theory (maximal ergodic theorem), and number theory via Tate's thesis and L-functions

Connections: Harmonic Analysis · Fourier Transform · Singular Integrals · Representation Theory

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