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Abstract

We introduce a nonparametric spectral density estimator for continuous-time and continuous-space processes measured at fully irregular locations. Our estimator is constructed using a weighted nonuniform Fourier sum whose weights yield a high-accuracy quadrature rule with respect to a user-specified window function. The resulting estimator significantly reduces the aliasing seen in periodogram approaches and least squares spectral analysis, sidesteps the dangers of ill-conditioning of the nonuniform Fourier inverse problem, and can be adapted to a wide variety of irregular sampling settings. We describe methods for rapidly computing the necessary weights in various settings, making the estimator scalable to large datasets. We then provide a theoretical analysis of sources of bias, and close with demonstrations of the method's efficacy, including for processes that exhibit very slow spectral decay and are observed at up to a million locations in multiple dimensions.