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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2505.22325 |
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| _version_ | 1866917299421708288 |
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| author | Caputo, Antonio |
| author_facet | Caputo, Antonio |
| contents | Classical Graph Signal Processing (GSP) provides a robust framework for analyzing signals on irregular domains, utilizing the graph Fourier transform as a cornerstone for spectral analysis and filtering. However, as data structures grow in complexity, there is an increasing need to handle multi-dimensional information. In this paper, we propose a generalization of the GSP framework by introducing vector-valued graph signals which take values in arbitrary Banach spaces. We define and investigate the fundamental operators of vertex-frequency analysis within this broader setting, including the Fourier transform, convolution, and translation operators. A key contribution of this work is the derivation of operator norm estimates and the establishment of graph-theoretic versions of classical uncertainty principles. We demonstrate how these results depend on the choice of the orthonormal basis and on the underlying $L^p$ norms. By modeling multiple scalar signals as a single vector-valued entity, this framework facilitates the study of inter-signal correlations, providing a flexible and mathematically grounded environment for analyzing multivariate time-series and time-varying signals on complex networks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_22325 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fourier transform of vector-valued graph signals Caputo, Antonio Classical Analysis and ODEs Classical Graph Signal Processing (GSP) provides a robust framework for analyzing signals on irregular domains, utilizing the graph Fourier transform as a cornerstone for spectral analysis and filtering. However, as data structures grow in complexity, there is an increasing need to handle multi-dimensional information. In this paper, we propose a generalization of the GSP framework by introducing vector-valued graph signals which take values in arbitrary Banach spaces. We define and investigate the fundamental operators of vertex-frequency analysis within this broader setting, including the Fourier transform, convolution, and translation operators. A key contribution of this work is the derivation of operator norm estimates and the establishment of graph-theoretic versions of classical uncertainty principles. We demonstrate how these results depend on the choice of the orthonormal basis and on the underlying $L^p$ norms. By modeling multiple scalar signals as a single vector-valued entity, this framework facilitates the study of inter-signal correlations, providing a flexible and mathematically grounded environment for analyzing multivariate time-series and time-varying signals on complex networks. |
| title | Fourier transform of vector-valued graph signals |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2505.22325 |