Closed-form Covariance Functions in Space and Time from B-spline Spectral Densities

The autocovariance function (ACF) is central to the analysis of time series, spatial, and spatio-temporal data. Classical parametric models are often too restrictive for multivariate and space-time data, which motivates non-parametric descriptions of the ACF. Bochner's Theorem guarantees that any positive spectral measure yields a valid ACF via the inverse Fourier transform; however, existing non-parametric approaches in the spectral domain rarely return closed-form expressions for the ACF itself. We develop a closed-form class of non-parametric ACFs by deriving the inverse Fourier transform of B-spline spectral bases with arbitrary degree and knot placement. This yields a family of closed-form stationary covariance models with three properties: (i) it is uniformly dense for the ACF, among the ACFs of weakly stationary processes that possess a spectral density; (ii) it accommodates univariate, multivariate, and multidimensional processes; and (iii) it is non-separable in general, with separability a special case. Jackson-type approximation bounds establish convergence rates, and simulation studies show accurate recovery of known univariate and bivariate ACFs. Applied to output from a high-resolution ocean simulation, the fit space-time ACF is non-separable and demonstrates accurate process recovery.