Hilbert-Valued Spherical Fields

• Hilbert-valued spherical fields are generalized random fields on spheres that take values in a separable Hilbert space, enabling the modeling of infinite-dimensional spatial data.
• Operator-valued Schoenberg theorems and spectral decompositions provide explicit covariance structures and facilitate the analysis of isotropic fields through trace-class operators and Gegenbauer polynomials.
• High-frequency asymptotics and criteria for Gaussian measure equivalence underpin rigorous statistical inference and identifiability in applications like stochastic geometry and functional data analysis.

Hilbert-valued spherical fields are generalized random fields defined on spheres (such as the unit sphere $\mathbb{S}^d$ in %%%%1%%%%) that take values in a real, separable Hilbert space $H$. These fields provide a rigorous framework for modeling infinite-dimensional spatial or functional data indexed by spherical coordinates, arising in domains such as spatial statistics, functional data analysis, and stochastic geometry on manifolds. Central topics include the operator-valued extension of Schoenberg’s theorem, spectral decompositions, Gaussian measure equivalence on function spaces, and high-frequency asymptotics. This field unites harmonic analysis, operator theory, and probability theory, allowing precise characterization, inference, and identification of Hilbert-valued phenomena on spherical domains.

1. Definition and Covariance Structure

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $H$ a real, separable Hilbert space with inner product $\langle \cdot, \cdot \rangle_H$ and norm $\| \cdot \|_H$. A Hilbert-valued spherical random field is a jointly measurable map

$T: \Omega \times \mathbb{S}^d \to H$

such that $\mathbb{E}\|T(x)\|_H^2 < \infty$ for all $x \in \mathbb{S}^d$. The auto-covariance kernel is the family of trace-class operators

$$R_{x, y} := \mathbb{E}[T(x) \otimes T(y)] \in \mathcal{L}^1(H), \quad x, y \in \mathbb{S}^d$$

acting as $$(T(x) \otimes T(y))(h) = T(x)\langle T(y), h \rangle_H$$ for $h \in H$. The field is called isotropic if $R_{x, y} = R_{\rho x, \rho y}$ for all rotations $\rho \in SO(d+1)$; equivalently, $R_{x, y}$ depends only on the geodesic distance or the inner product $t = x \cdot y$. This restriction ensures the existence of operator-valued analogues of positive-definite kernels and spectral decompositions on spherical domains (Caponera, 2022, Caponera et al., 28 Nov 2025).

2. Operator-Valued Schoenberg Theorem

Isotropic, positive-definite operator kernels on $\mathbb{S}^d$ admit a unique expansion in terms of trace-class, self-adjoint operators and orthogonal polynomials:

$$C(x, y) = \sum_{\ell=0}^\infty C_\ell \, G_\ell^{((d-1)/2)}(x \cdot y)$$

where $\{C_\ell\}$ are the $d$-Schoenberg operators (trace-class, positive semidefinite) and $G_\ell^{(\alpha)}$ are Gegenbauer polynomials. For $\mathbb{S}^2$, this specializes to Legendre polynomials $P_\ell(x \cdot y)$ and operators $F_\ell$:

$$R_{x, y} = \sum_{\ell=0}^\infty \frac{2\ell+1}{4\pi} P_\ell(x \cdot y) F_\ell$$

The coefficients must satisfy $$\sum_{\ell=0}^\infty \operatorname{tr}_{H}(C_\ell) < \infty$$ for absolute convergence. This operator-valued generalization bridges classical Schoenberg theory for scalar fields on spheres to infinite-dimensional Hilbert space contexts, providing the necessary spectral and measure-theoretic machinery for rigorous analysis (Caponera, 2022, Caponera et al., 28 Nov 2025).

3. Spectral Representation and Power-Spectrum Operators

The Cramér–Karhunen–Loève expansion yields for mean-zero, isotropic fields

$$T(x) = \sum_{\ell=0}^\infty \sum_{m=1}^{h(d, \ell)} a_{\ell m} Y_{\ell m}(x)$$

where $\{Y_{\ell m}\}$ are hyperspherical harmonics and $a_{\ell m} \in H$ are Bochner coefficients. The covariance structure satisfies

$$\mathbb{E}[a_{\ell m} \otimes a_{\ell' m'}] = \delta_{\ell \ell'} \delta_{m m'} F_\ell$$

with $F_\ell$ the operator-valued power spectrum. For Gaussian fields, the $\{a_{\ell m}\}$ are independent, centered H-valued Gaussian random variables. The empirical (sample) power spectrum operator is

$$\hat F_\ell := \frac{1}{h(d, \ell)} \sum_{m=1}^{h(d, \ell)} (a_{\ell m} \otimes a_{\ell m})$$

and $\hat F_\ell$ is an unbiased estimator of $F_\ell$ (Caponera, 2022).

4. Covariance Operators and Equivalence of Gaussian Measures

A Hilbert-valued Gaussian field on $\mathbb{S}^d$ induces a covariance operator $B$ on $L^2(\mathbb{S}^d; H)$:

$$(B f)(x) = \int_{\mathbb{S}^d} C(x, y) f(y) \, \sigma(dy)$$

Using harmonic analysis, $B$ admits a spectral decomposition

$B = \sum_{\ell=0}^\infty C_\ell P_\ell$

where $P_\ell$ are projections onto the span of $\{ Y_{\ell, m} \otimes H \}$.

For two such Gaussian measures $\mu_1, \mu_2$ with corresponding operator sequences $\{C_\ell^{(1)}\}$, $\{C_\ell^{(2)}\}$, Feldman–Hájek theory yields that $\mu_1 \equiv \mu_2$ iff

$$\sum_{\ell=0}^\infty h(d, \ell) \| (C_\ell^{(2)})^{-1/2} C_\ell^{(1)} (C_\ell^{(2)})^{-1/2} - I \|^2_{HS(H)} < \infty$$

where $\| \cdot \|_{HS(H)}$ is the Hilbert–Schmidt norm. This functional criterion encompasses and strictly dominates all scalar projection criteria: equivalence in $H$-valued law implies equivalence for every scalar projection (Caponera et al., 28 Nov 2025).

5. High-Frequency Regime and Quantitative Asymptotics

Under isotropic Gaussianity, sample power spectrum operators show ergodicity in the high-frequency regime:

$\|\hat F_\ell - F_\ell\|_p \to 0$

in probability and almost surely as $\ell \to \infty$, for any Schatten–$p$ norm, provided $\sum_\ell \| F_\ell \|_1 h(d, \ell)^{-1} < \infty$. The rate for the Hilbert–Schmidt norm is

$$\mathbb{E} \|\hat F_\ell - F_\ell\|_2^2 = \frac{\| F_\ell \|_2^2 + (\operatorname{Tr} F_\ell)^2}{h(d, \ell)}$$

Central limit theorems describe the convergence of normalized errors $F_\ell^{(n)}$ to Gaussian laws in operator norm, with explicit $d_2$-distance rates $O(1/\ell)$. Reduced (scalar) power spectrum estimators $$\hat C_\ell = h(d, \ell)^{-1} \sum_{m=1}^{h(d, \ell)} \| a_{\ell m} \|^2$$ converge in distribution with explicit bounds in total variation to standard normal (Caponera, 2022).

6. Illustrative Models and Identifiability

Two key models elucidate practical structure and identifiability:

Multiquadratic Bivariate Family ($H = \mathbb{R}^2$):

Define block Schoenberg coefficients as:

$$b_\ell(i, j) = \rho_{i,j} \sigma_i \sigma_j \binom{d+\ell-2}{\ell} \alpha_{i,j}^\ell (1-\alpha_{i,j})^{d-1}$$

with constraints $\alpha_{12}^2 \leq \alpha_{11}\alpha_{22}$ and $$\rho_{12} < ((1-\alpha_{11})(1-\alpha_{22})/(1-\alpha_{12})^2)^{(d-1)/2}$$. Equivalence of two fields with such parameters requires coincidence in variances and autocorrelation parameters; cross-correlation equivalence has closed-form characterization depending on additional constraints (Caponera et al., 28 Nov 2025).

Infinite-Dimensional Legendre–Matérn Construction ($H = L^2([0, 1])$):

Expand $C_\ell$ in Fourier basis:

$$C_\ell = \sum_{k \in \mathbb{Z}} \gamma_{\ell, k} e^{2\pi i k \cdot} \otimes e^{2\pi i k \cdot}$$

with spectral weights:

$$h(d, \ell)\, \gamma_{\ell, k} = \frac{\sigma^2}{(\alpha + k^2 + \ell^2)^{\nu + (d-1)/2}}$$

Functional measure equivalence reduces to summability of

$$\sum_{\ell, k} h(d, \ell) \left[ \gamma_{\ell, k}^{(1)} / \gamma_{\ell, k}^{(2)} - 1 \right]^2 < \infty$$

A plausible implication is that only variance and smoothness parameters are identifiable under infill asymptotics, while scale remains unidentifiable (Caponera et al., 28 Nov 2025).

7. Connections and Theoretical Significance

Hilbert-valued spherical fields unify stochastic geometry, harmonic analysis, and infinite-dimensional Gaussian measure theory. Operator-valued Schoenberg theorems generalize positive-definite kernel characterizations and support explicit spectral analysis for functional data on spheres. Feldman–Hájek equivalence conditions provide rigorous identification tools extending classical scalar cases. High-frequency asymptotics underpin consistency and statistical inference for functional observational processes, with theoretical results applicable to kernel methods and spatial statistics for spherical manifolds. Current research by Caponera, Ferreira, Porcu and others highlights ongoing integration of spectral methods and operator theory with functional statistics in spherical domains (Caponera, 2022, Caponera et al., 28 Nov 2025).

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Key References:

• "Asymptotics for isotropic Hilbert-valued spherical random fields" (Caponera, 2022)
• "Functional Gaussian Fields on Hyperspheres with their Equivalent Gaussian Measures" (Caponera et al., 28 Nov 2025)
• Caponera (2023); Marinucci–Peccati (2011); Hsing–Eubank (2015); Nourdin–Peccati (2012) (cited within articles).