Spectral Theory of the Weighted Fourier Transform with respect to a Function in $\mathbb{R}^n$: Uncertainty Principle and Diffusion-Wave Applications
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| Format: | Preprint |
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2025
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| _version_ | 1866912758357819392 |
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| author | Dorrego, Gustavo Luque, Luciano |
| author_facet | Dorrego, Gustavo Luque, Luciano |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_10880 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral Theory of the Weighted Fourier Transform with respect to a Function in $\mathbb{R}^n$: Uncertainty Principle and Diffusion-Wave Applications Dorrego, Gustavo Luque, Luciano Classical Analysis and ODEs Mathematical Physics Analysis of PDEs Functional Analysis 42B10, 26A33, 35S05 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. |
| title | Spectral Theory of the Weighted Fourier Transform with respect to a Function in $\mathbb{R}^n$: Uncertainty Principle and Diffusion-Wave Applications |
| topic | Classical Analysis and ODEs Mathematical Physics Analysis of PDEs Functional Analysis 42B10, 26A33, 35S05 |
| url | https://arxiv.org/abs/2512.10880 |