Functional Gaussian Fields on Hyperspheres with their Equivalent Gaussian Measures
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arXiv
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| Format: | Preprint |
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2025
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| author | Caponera, Alessia Ferreira, Vinicius Porcu, Emilio |
| author_facet | Caponera, Alessia Ferreira, Vinicius Porcu, Emilio |
| contents | We develop a general framework for isotropic functional Gaussian fields on the $d$-dimensional sphere $\mathbb{S}^{d}$, where the field takes values in a separable Hilbert space $\mathcal{H}$. We establish an operator-valued extension of Schoenberg's theorem and show that the covariance structure of such fields admits a representation through a sequence of trace-class $d$-Schoenberg operators, yielding an explicit spectral decomposition of the covariance operator on $L^{2}(\mathbb{S}^{d};\mathcal{H})$. We derive a functional version of the Feldman-H'ajek criterion and prove that equivalence of the Gaussian measures induced by two Hilbert-valued spherical fields is determined by a Hilbert summability condition involving Schoenberg functional sequences, extending classical results for scalar and vector fields to the infinite-dimensional setting. We further show how equivalence of all scalar projections is contained within, and dominated by, the functional criterion. The theory is illustrated through two models: (i) a multiquadratic bivariate family on $\mathbb{S}^{d}$, where the equivalence region has a closed-form description in terms of cross-correlation and geodesic decay parameters, and (ii) an infinite-dimensional Legendre-Mat'ern construction, where operator-valued spectra yield identifiability conditions on smoothness and scale. These examples show how operator-valued Schoenberg coefficients govern both geometry and measure-theoretic behavior of functional spherical fields. Overall, the results provide a unified spectral framework for Gaussian measures on $L^{2}(\mathbb{S}^{d};\mathcal{H})$, bridging harmonic analysis, operator theory, and stochastic geometry on manifolds, and offering tools for functional data analysis, spatial statistics, and kernel methods on spherical domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_23008 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Functional Gaussian Fields on Hyperspheres with their Equivalent Gaussian Measures Caponera, Alessia Ferreira, Vinicius Porcu, Emilio Statistics Theory We develop a general framework for isotropic functional Gaussian fields on the $d$-dimensional sphere $\mathbb{S}^{d}$, where the field takes values in a separable Hilbert space $\mathcal{H}$. We establish an operator-valued extension of Schoenberg's theorem and show that the covariance structure of such fields admits a representation through a sequence of trace-class $d$-Schoenberg operators, yielding an explicit spectral decomposition of the covariance operator on $L^{2}(\mathbb{S}^{d};\mathcal{H})$. We derive a functional version of the Feldman-H'ajek criterion and prove that equivalence of the Gaussian measures induced by two Hilbert-valued spherical fields is determined by a Hilbert summability condition involving Schoenberg functional sequences, extending classical results for scalar and vector fields to the infinite-dimensional setting. We further show how equivalence of all scalar projections is contained within, and dominated by, the functional criterion. The theory is illustrated through two models: (i) a multiquadratic bivariate family on $\mathbb{S}^{d}$, where the equivalence region has a closed-form description in terms of cross-correlation and geodesic decay parameters, and (ii) an infinite-dimensional Legendre-Mat'ern construction, where operator-valued spectra yield identifiability conditions on smoothness and scale. These examples show how operator-valued Schoenberg coefficients govern both geometry and measure-theoretic behavior of functional spherical fields. Overall, the results provide a unified spectral framework for Gaussian measures on $L^{2}(\mathbb{S}^{d};\mathcal{H})$, bridging harmonic analysis, operator theory, and stochastic geometry on manifolds, and offering tools for functional data analysis, spatial statistics, and kernel methods on spherical domains. |
| title | Functional Gaussian Fields on Hyperspheres with their Equivalent Gaussian Measures |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2511.23008 |