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, where the field takes values in a separable Hilbert space. 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. 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, 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, 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.