Frequenzmasse

Frequenzmasse is a term used in German-language mathematics and signal processing to denote a measure that describes how a quantity is distributed over frequencies. It is closely related to the concept of a spectral measure in Fourier analysis and to the notion of a power spectral density.

Definition and interpretation: For a function f in L2(R) with Fourier transform F, the energy allocated to a frequency set Ω is ∫Ω |F(ω)|^2 dω. This integral can be interpreted as the Frequenzmasse of Ω. If the total energy is ∥f∥2, then μ(Ω) = ∫Ω |F(ω)|^2 dω defines a finite positive measure μ on the real line. If f is normalized, μ becomes a probability measure, providing a distribution of energy across frequencies.

Connections to stochastic processes: In the context of stationary processes, the spectral measure Φ describes how the process's variance is distributed over frequencies. The autocovariance function is the Fourier transform of the spectral measure, linking time-domain behavior to a Frequenzmasse in the frequency domain.

Properties: A Frequenzmasse is countably additive and nonnegative. Its total mass equals the total energy (or one in the probabilistic normalization). When a density exists, the Frequenmasse has a density function with respect to Lebesgue measure, corresponding to the power spectral density.

Applications: Frequenzmasse concepts underlie spectral analysis, filter design, and estimation of power spectra in signal processing and time-series analysis. They provide a rigorous framework to quantify and manipulate how energy or probability is allocated across frequencies.

See also: spectral measure, power spectral density, Spektralmaß, Leistungsdichte, Fourier transform.