A canonical entropy-based multiscale decomposition of L1 functions

Abstract

We introduce a canonical entropy-based multiscale decomposition of nonnegative L1 functions that yields both physically meaningful approximations and an injective representation. The construction produces a recursive partition of the domain-organized as a Hahn tree-in which each region carries exact mass information together with an entropy-equivalent uniform surrogate. Unlike classical step-function approximations that rely on externally imposed partitions or fixed resolution scales, the present scheme is intrinsic, parameter-free, and adaptive: refinement occurs only where the distribution exhibits genuine informational non-uniformity. At finite depth, the resulting step functions provide entropy-aligned coarse-grained summaries that preserve the most relevant structural features of the original function while discarding information in a controlled and quantifiable way. Such approximations are well suited for applications in physics where probability densities arise as empirical, effective, or coarse-grained descriptions, including statistical mechanics, nonequilibrium systems, and quantum-mechanical measurement models. At infinite depth, the Hahn tree retains complete information and uniquely determines the original function up to sets of measure zero, allowing exact reconstruction. To our knowledge, this is the first entropy-driven multiscale decomposition that simultaneously provides adaptive step-function approximation, convergence, and injective representation of general nonnegative L1 functions. We prove almost-everywhere convergence and L1 convergence of the entropy-equivalent step-function approximations, together with injectivity of the infinite decomposition. A series of explicit examples illustrates how the construction concentrates resolution near regions of high probability and informational structure, while uniform or piecewise-uniform regions stabilize automatically. Together, these results establish the Hahn D– c decomposition as a canonical information-theoretic framework that unifies multiscale approximation and representation for L1 functions.