dualdensity

Duiddensity is a term used in several areas of mathematics and applied science to describe a density defined on a dual object associated with a given density on a primal space. Because duality appears in many forms—measure-theoretic duality, convex conjugacy, Fourier duality, and categorical duality—the precise meaning of dualdensity depends on the context.

In measure theory and probability, a dual density often refers to a density with respect to a reference measure on a dual space (such as a dual vector space or a dual parameter space) that corresponds to the original density under a chosen duality map. For example, if a density p on a space X induces a measure on a dual space X* through a transformation, and the induced measure is absolutely continuous with respect to a baseline on X*, the corresponding density on X* may be described as a dual density.

In convex analysis, dual densities can arise via the Legendre–Fenchel transform. A density defined on a primal space may be related to a density on the dual space through the conjugate function, with applications in variational problems and information-theoretic formulations.

In signal processing and time-series analysis, dualdensity can refer to densities in a dual representation such as the frequency domain. For stationary processes, the spectral density (the Fourier transform of the autocovariance function) serves as a dual object to time-domain correlations and is sometimes described in terms of duality between domains.

Properties of dualdensity depend on the specific duality chosen and the reference measure. Common considerations include existence (absolute continuity), normalization, and transformation rules under the duality map. See also density, dual space, Fourier transform, Legendre transform, and measure theory.