Robust Linear Interpolation and Extrapolation of Stationary Time Series in L^p

To deal with uncertainty of the spectral distribution, we consider minimax interpolation and extrapolation problems in L^p for stationary processes. The interpolation and extrapolation problems can be regarded as a linear approximation problem on the unit disk in the complex plane. Although the robust one-step-ahead predictor and interpolator has already been considered separately in the previous literature, we give two conditions for the uncertainty class to find the minimax interpolator and extrapolator in the general framework from both the point of view of the observation set and the point of view of evaluation on the interpolation and extrapolation error under the L^p-norm. We show that there exists a minimax interpolator and extrapolator for the class of spectral densities ε-contaminated by unknown spectral densities under our conditions. When the uncertainty class contains spectral distribution functions which are not absolutely continuous to the Lebesgue measure, we show that there exists an approximate interpolator and extrapolator in L^p such that its maximal interpolation and extrapolation error is arbitrarily close to the minimax error when the spectral distributions have densities. Our results are applicable to the stationary harmonizable stable processes.