Functional Gaussian fields on hyperspheres with their equivalent Gaussian measures

We develop a general framework for isotropic functional Gaussian fields on the d-dimensional sphere Sᵈ , where the field takes values in a separable Hilbert space H . We establish an operator-valued extension of Schoenberg’s theorem and show that the covariance structure of such fields admits a representation in terms of a sequence of trace-class d-Schoenberg operators. This yields an explicit spectral decomposition of the covariance operator on L²(Sᵈ;H) . We then derive a functional version of the Feldman-Hájek criterion and prove that equivalence of the Gaussian measures induced by two Hilbert-valued spherical fields is determined by a Hilbert summability criterion that involves Schoenberg functional sequences, thereby extending classical results for scalar and vector fields on spheres 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 classes of models: (i) a multiquadratic bivariate family on Sᵈ , for which the equivalence region can be expressed in closed form in terms of cross-correlation and geodesic decay parameters, and (ii) an infinite-dimensional Legendre-Matérn construction, where operator-valued spectra lead to explicit identifiability conditions on smoothness and scale parameters. These examples demonstrate how the operator-valued Schoenberg coefficients govern both the geometry and the measure-theoretic behavior of functional spherical fields. Overall, the results provide a unified spectral framework for Gaussian measures on L²(Sᵈ;H) , bridging harmonic analysis, operator theory, and stochastic geometry on manifolds, and offering foundational tools for functional data analysis, spatial statistics, and kernel methods on spherical domains.