dorsal/arxiv
View SchemaA Unified Spectral Framework for Aging, Heterogeneous, and Distributed Order Systems via Weighted Weyl-Sonine Operators
| Authors | Gustavo Dorrego |
|---|---|
| Categories | |
| ArXiv ID | 2601.05423vv1 |
| URL | https://arxiv.org/abs/2601.05423 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
While General Fractional Calculus has successfully expanded the scope of memory operators beyond power-laws, standard formulations remain predominantly restricted to the half-line via Riemann-Liouville or Caputo definitions. This constraint artificially truncates the system's history, limiting the thermodynamic consistency required for modeling processes on unbounded domains. To overcome these barriers, we construct the \textbf{Weighted Weyl-Sonine Framework}, a generalized formalism that extends non-local theory to the entire real line without history truncation. Unlike recent algebraic approaches based on conjugation for finite intervals, we develop a rigorous harmonic analysis framework. Our central contribution is the \textbf{Generalized Spectral Mapping Theorem}, which establishes the Weighted Fourier Transform as a unitary diagonalization map for these operators. This result allows us to rigorously classify and solve distinct physical regimes under a single algebraic structure. We explicitly derive exact solutions for \textit{diffusive relaxation} (governed by Complete Bernstein Functions), \textit{inertial wave propagation} (exhibiting oscillatory dynamics), and \textit{retarded aging} (via distributed order), proving that our framework unifies the description of anomalous transport and wave mechanics in complex, time-deformed media.
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"abstract": "While General Fractional Calculus has successfully expanded the scope of memory operators beyond power-laws, standard formulations remain predominantly restricted to the half-line via Riemann-Liouville or Caputo definitions. This constraint artificially truncates the system\u0027s history, limiting the thermodynamic consistency required for modeling processes on unbounded domains. To overcome these barriers, we construct the \\textbf{Weighted Weyl-Sonine Framework}, a generalized formalism that extends non-local theory to the entire real line without history truncation.\n Unlike recent algebraic approaches based on conjugation for finite intervals, we develop a rigorous harmonic analysis framework. Our central contribution is the \\textbf{Generalized Spectral Mapping Theorem}, which establishes the Weighted Fourier Transform as a unitary diagonalization map for these operators. This result allows us to rigorously classify and solve distinct physical regimes under a single algebraic structure. We explicitly derive exact solutions for \\textit{diffusive relaxation} (governed by Complete Bernstein Functions), \\textit{inertial wave propagation} (exhibiting oscillatory dynamics), and \\textit{retarded aging} (via distributed order), proving that our framework unifies the description of anomalous transport and wave mechanics in complex, time-deformed media.",
"arxiv_id": "2601.05423",
"authors": [
"Gustavo Dorrego"
],
"categories": [
"math.FA",
"math-ph",
"math.AP",
"math.MP"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "A Unified Spectral Framework for Aging, Heterogeneous, and Distributed Order Systems via Weighted Weyl-Sonine Operators",
"url": "https://arxiv.org/abs/2601.05423",
"version": "v1"
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