arXiv subjects

Gustavo Dorrego

Publications and source records attributed to Gustavo Dorrego.

8 recordsLinked to original sources

arXiv2026

A Semigroup Approach to Weighted Weyl-Sonine Operators: Bypassing Bernstein Functions for Well-Posedness

This paper establishes a rigorous functional analytic framework for weighted Weyl-Sonine fractional operators on semi-infinite intervals. While the classical Phillips functional calculus relies strictly on completely monotonic Bernstein functions to subordinate linear semigroups, we demonstrate that this structural restriction can be entirely bypassed. By exploiting the intrinsic topological decay induced by an admissible amnesia weight ($ω$), we prove that the abstract integral representation converges absolutely for general classes of tempered Sonine kernels, thereby unifying the treatment of both diffusive relaxation and oscillatory non-local dynamics within a single framework. Employing the theory of strongly continuous semigroups, we characterize these operators as the infinitesimal generators of a deformed translation semigroup that simultaneously incorporates subjective aging scales ($ψ$) and history-attenuating amnesia effects. This similarity-based approach not only provides a sound abstract foundation for the hypersingular Marchaud representation of these operators but also guarantees their closedness and the analytical well-posedness of the associated abstract fractional Cauchy problems in general Banach spaces. Furthermore, we introduce the associated generalized fractional integral and prove the fundamental inversion theorem, demonstrating the complete analytical symmetry of the theory. Finally, we establish pointwise decay estimates for the solutions via weighted Sobolev embeddings.

math.FA↗

arXiv2026

A Fractional Fox H-Function Kernel for Support Vector Machines: Robust Classification via Weighted Transmutation Operators

Support Vector Machines (SVMs) rely heavily on the choice of the kernel function to map data into high-dimensional feature spaces. While the Gaussian Radial Basis Function (RBF) is the industry standard, its exponential decay makes it highly susceptible to structural noise and outliers, often leading to severe overfitting in complex datasets. In this paper, we propose a novel class of non-stationary kernels derived from the fundamental solution of the generalized time-space fractional diffusion-wave equation. By leveraging a structure-preserving transmutation method over Weighted Sobolev Spaces, we introduce the Amnesia-Weighted Fox Kernel, an exact analytical Mercer kernel governed by the Fox H-function. Unlike standard kernels, our formulation incorporates an aging weight function (the "Amnesia Effect") to penalize distant outliers and a fractional asymptotic power-law decay to allow for robust, heavy-tailed feature mapping (analogous to Lévy flights). Numerical experiments on both synthetic datasets and real-world high-dimensional radar data (Ionosphere) demonstrate that the proposed Amnesia-Weighted Fox Kernel consistently outperforms the standard Gaussian RBF baseline, reducing the classification error rate by approximately 50\% while maintaining structural robustness against outliers.

cs.LG↗

arXiv2026

Weighted Sobolev Spaces and Distributional Spectral Theory for Generalized Aging Operators via Transmutation Methods

The spectral analysis of operators in heterogeneous and aging media typically requires a functional framework that extends beyond the standard Hilbertian setting. In this paper, we establish a rigorous distributional theory for a class of non-local operators, termed Weighted Weyl-Sonine operators, by employing a structure-preserving transmutation method. We construct the Weighted Schwartz Space $\mathcal{S}_{ψ,ω}$ and its topological dual, the space of Weighted Tempered Distributions $\mathcal{S}'_{ψ,ω}$, ensuring that the underlying Fréchet topology is consistent with the infinitesimal generator of the aging dynamics. This topological foundation allows us to: (i) extend the Weighted Fourier Transform to generalized functions as a unitary isomorphism; (ii) provide an explicit spectral characterization of the weighted Dirac delta $δ_{ψ,ω}$ and its scaling laws under geometric dilations; and (iii) introduce a scale of Weighted Sobolev Spaces $H^{s}_{ψ,ω}$ defined via spectral multipliers. A central result is the derivation of a sharp embedding theorem, $|u(t)| \le C ω(t)^{-1} \|u\|_{H^s_{ψ,ω}}$, which rigorously connects abstract spectral energy to the pointwise decay induced by the weight $ω$. This framework provides a unified geometric characterization of several fractional regimes, including the Hadamard and Riemann-Liouville cases, within a single operator-theoretic architecture.

math.FA↗

arXiv2026

Weighted fractional ultrahyperbolic diffusion on geometrically deformed domains

Standard fractional models on manifolds often conflate geometric anisotropy with medium heterogeneity. In this Letter, we overcome this rigidity by deriving the fundamental solution for a weighted space-time fractional ultrahyperbolic operator, denoted by $(-\Box_{ϕ,ω})^β$. Using a novel spectral approach based on the Weighted Fourier Transform, we explicitly \textbf{decouple the medium density from the geometric deformation}. A crucial finding is the emergence of a \textbf{geometry-independent drift mechanism} driven purely by the inhomogeneity of the medium. The Green's function is obtained in closed form via the Fox H-function, providing a unified and computable framework for anomalous transport in complex, structurally deformed media.

math.AP↗

arXiv2026

A Unified Spectral Framework for Aging, Heterogeneous, and Distributed Order Systems via Weighted Weyl-Sonine Operators

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.

math.FA↗

arXiv2026

Spectral Analysis of Weighted Weyl Fractional Operators: Aging, Infinite Memory, and the Amnesia Effect

This paper establishes a rigorous spectral framework for the Weighted Weyl Fractional Calculus, designed to model non-local systems exhibiting aging and subjective time scales. By constructing a conjugation map involving a time-dependent weight $ω(t)$ and a scale function $ψ(t)$, we define a new class of fractional operators that preserve the spectral tractability of time-invariant systems. We derive the Spectral Mapping Theorem for these operators and prove that Weighted Mittag-Leffler functions act as their fundamental eigenfunctions, demonstrating that the Weighted Fourier Transform naturally diagonalizes the associated evolution equations. As a physical application, we formulate a constitutive law for aging viscoelastic materials with infinite memory. Crucially, we analytically demonstrate the "Amnesia Phenomenon": we prove that rapid aging modulates the system's history, effectively transforming the hereditary power-law decay into a short-range exponential relaxation. This result provides a closed-form explanation for the loss of memory in fast-aging media, overcoming the computational bottlenecks of standard discretization methods.

math.SP↗

arXiv2025

Spectral Theory of the Weighted Fourier Transform with respect to a Function in $\mathbb{R}^n$: Uncertainty Principle and Diffusion-Wave Applications

In this paper, we generalize the weighted Fourier transform with respect to a function, originally proposed for the one-dimensional case in \cite{Dorrego}, to the $n$-dimensional Euclidean space $\mathbb{R}^{n}$. We develop a comprehensive spectral theory on a weighted Hilbert space, establishing the Plancherel identity, the inversion formula, the convolution theorem, and a Heisenberg-type uncertainty principle depending on the geometric deformation. Furthermore, we utilize this framework to rigorously define the weighted fractional Laplacian with respect to a function, denoted by $(-Δ_{ϕ,ω})^{s}$. Finally, we apply these tools to solve the generalized time-space fractional diffusion-wave equation, demonstrating that the fundamental solution can be expressed in terms of the Fox H-function, intrinsically related to the generalized $ω$-Mellin transform introduced in \cite{Dorrego}. In this paper, we generalize the weighted Fourier transform with respect to a function, originally proposed for the one-dimensional case, to the n-dimensional Euclidean space $\mathbb{R}^n$. We develop a comprehensive spectral theory on a weighted Hilbert space, establishing the Plancherel identity, the inversion formula, the convolution theorem, and a Heisenberg-type uncertainty principle depending on the geometric deformation. Furthermore, we utilize this framework to rigorously define the weighted fractional Laplacian with respect to a function, denoted by $(-Δ_{ϕ,ω})^s$. Finally, we apply these tools to solve the generalized time-space fractional diffusion-wave equation involving the weighted Hilfer derivative. We demonstrate that the fundamental solution can be explicitly expressed in terms of the Fox H-function, revealing an intrinsic connection with the generalized Mellin transform.

math.CA↗

arXiv2025

The $ψ$-$ω$-Mellin transform

This manuscript introduces a generalization of the Mellin integral transform within the framework of weighted fractional calculus with respect to an increasing function. The proposed transform is much more suitable for working with fractional operators that involve a weight and are defined with respect to a function. This work also explores the connections between the Laplace and Fourier integral transforms in the same context. To achieve this, a new formulation of the weighted Fourier integral transform with respect to a function is presented, along with a new version of the bilateral Laplace transform. We study some of the properties of these new operators, obtain an inversion formula and a convolution theorem, and also present a practical application as an illustrative example.

math.FA↗