funktionsspektrum

Funktionsspektrum, often called frequency spectrum of a function, describes how the function f(t) distributes its energy over frequency. It is obtained by transforming the function from the time (or spatial) domain to the frequency domain through a Fourier transform. For a continuous-time function f(t) that is integrable, the Fourier transform F(ω) is defined by F(ω) = ∫_{-∞}^{∞} f(t) e^{-i ω t} dt. The inverse transform reconstructs f via f(t) = (1/2π) ∫_{-∞}^{∞} F(ω) e^{i ω t} dω.

The spectrum is generally complex; its magnitude |F(ω)| is called the amplitude spectrum and its phase ∠F(ω) the phase spectrum. In many analyses one focuses on the power spectrum P(ω) = |F(ω)|^2, which describes how signal energy is distributed across frequencies.

For discrete signals, the discrete Fourier transform (DFT) or the discrete-time Fourier transform (DTFT) is used, computed efficiently by the fast Fourier transform (FFT). The frequency axis depends on sampling rate and observation duration; frequency resolution Δf = 1/T for a window of length T.

Applications include signal processing, acoustics, image analysis, and system identification. The spectrum aids in filtering, compression, and spectral estimation. In statistics, the Wiener-Khinchin theorem connects the spectrum to the autocorrelation function of a stationary process.

In German-language literature the term Funktionsspektrum is used interchangeably with Spektrum einer Funktion or Frequenzspektrum; it is a general concept rather than a single mathematical object, and its precise definition may vary with context.