Harmonic analysis

Harmonic analysis is an area of mathematical analysis that emerged from the study of harmonic functions, and especially their boundary behavior. The methods of harmonic analysis decompose functions and related objects, such as measures, into components based on symmetries, scales, spectra, or oscillation. It is also concerned with the analytic estimates for operators arising from such decompositions. Basic examples include Fourier series and the Fourier transform, while modern harmonic analysis also studies maximal functions, singular integrals, oscillatory integrals, Fourier multipliers, Littlewood–Paley theory, and spectral decompositions. A related tradition is abstract harmonic analysis where the emphasis is on functions and representations on topological groups, including Pontryagin duality, the Peter–Weyl theorem, and Plancherel-type theorems. Harmonic analysis overlaps substantially with Fourier analysis, real analysis, functional analysis, partial differential equations, potential theory, ergodic theory, representation theory, and number theory.