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The spectral density function describes the second-order properties of astationary stochastic process on $\mathbb{R}^d$. This paper considers thenonparametric estimation of the spectral density of a continuous-timestochastic process taking values in a separable Hilbert space. Our estimator isbased on kernel smoothing and can be applied to a wide variety of spatialsampling schemes including those in which data are observed at irregularspatial locations. Thus, it finds immediate applications in Spatial Statistics,where irregularly sampled data naturally arise. The rates for the bias andvariance of the estimator are obtained under general conditions in amixed-domain asymptotic setting. When the data are observed on a regular grid,the optimal rate of the estimator matches the minimax rate for the class ofcovariance functions that decay according to a power law. The asymptoticnormality of the spectral density estimator is also established under generalconditions for Gaussian Hilbert-space valued processes. Finally, with a viewtowards practical applications the asymptotic results are specialized to thecase of discretely-sampled functional data in a reproducing kernel Hilbertspace.

Spectral Density Estimation of Function-Valued Spatial Processes