Fourieranalyse

Fourieranalyse, or Fourier analysis, is a branch of mathematical analysis that studies how a function or signal can be expressed as a sum or integral of sinusoidal components. The central tool is the Fourier transform, F(ω) = ∫ f(t) e^{-i ω t} dt, with the inverse f(t) = (1/2π) ∫ F(ω) e^{i ω t} dω, under suitable conditions. For periodic functions, Fourier series represent f(t) as ∑_{n=-∞}^{∞} c_n e^{i n ω0 t} with ω0 = 2π/T; the real form uses sums of cosines and sines.

In discrete contexts, the discrete Fourier transform X_k = ∑_{n=0}^{N-1} x_n e^{-i 2π kn / N}, with inverse x_n = (1/N) ∑ X_k e^{i 2π kn / N}, provides a finite-dimensional analogue. The subject also includes harmonic analysis on spaces such as L^2, with convergence often understood in the sense of distributions and governed by results like Plancherel’s theorem.

A key computational development is the fast Fourier transform (FFT), an algorithmic family that reduces the complexity of computing the DFT dramatically, enabling practical use across many fields. Fourier analysis thus serves as a bridge between time (or spatial) domains and frequency domains.

Applications are widespread: signal processing, acoustics, optics, image and data compression, solving differential equations, and quantum mechanics. The theory also informs mathematical areas such as spectral theory and representation theory, highlighting deep connections between analysis and structure.

Originating with Joseph Fourier in the early 19th century to model heat flow, Fourieranalyse has grown into a foundational tool in science and engineering, enabling precise analysis of signals and systems through their frequency content.