Functional estimation of anisotropic covariance and autocovariance operators on the sphere

We propose nonparametric estimators for the second-order central moments of possibly anisotropic spherical random fields, within a functional data analysis context. consider measurement framework where each field among an identically distributed collection fields is sampled at few directions, subject to error. The could be i.i.d. or serially dependent. Though similar setups have already been explored functions defined on unit interval, proposed in literature often rely local polynomials, which do not readily extend (product) setting. therefore formulate our estimation procedure as variational problem involving generalized Tikhonov regularization term. latter favours smooth covariance/autocovariance functions, smoothness specified by means suitable Sobolev-like pseudo-differential operators. Using machinery reproducing kernel Hilbert spaces, we establish representer theorems that fully characterize form estimators. determine their uniform rates convergence number diverges, both dense (increasing spatial samples) and sparse (bounded regimes. moreover demonstrate computational feasibility practical merits simulation setting, assuming fixed samples per field. Our numerical leverages sparsity Kronecker structure setup reduce memory requirements approximately three orders magnitude compared naive implementation would require.