HarmonikAnalysen

HarmonikAnalysen is a field of mathematics and signal processing focused on representing functions or signals as sums or integrals of basic oscillatory components, such as sinusoids or complex exponentials. The central aim is to understand and manipulate the frequency content of data, by moving between representations in the time or spatial domain and the frequency domain.

Its historical roots lie in Fourier analysis, developed in the 18th and 19th centuries, with later formalization in the framework of function spaces, distribution theory, and representation theory. Modern harmonic analysis encompasses both continuous settings, using Fourier transforms, and discrete settings, using discrete Fourier transforms and related spectral methods.

Key techniques include Fourier series for periodic signals, the Fourier transform for nonperiodic signals, and related tools such as the short-time Fourier transform and wavelet transform for time–frequency analysis. In noncommutative contexts, harmonic analysis studies representations of groups and the resulting decompositions of functions into irreducible components.

Applications span signal processing, communications, audio and image processing, data compression, and scientific fields such as quantum mechanics and materials science. In mathematics, harmonic analysis informs areas such as number theory, spectral geometry, and analysis on manifolds.

Core theoretical results include Parseval's and Plancherel's identities, which relate energy in time and frequency domains, and the development of frameworks like Lp spaces and Besov spaces that describe function regularity. The field continues to evolve with research in time–frequency analysis, noncommutative harmonic analysis, and computational algorithms.