To deal with uncertainty of the spectral distribution, we consider minimax interpolation and extrapolation problems in Lp 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 Lp‐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 Lp 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.

中文翻译：

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L p 中平稳时间序列的鲁棒线性插值和外插

为了处理谱分布的不确定性，我们考虑了平稳过程的 Lp 中的极小极大插值和外插问题。插值和外插问题可以看作是复平面单位圆盘上的线性逼近问题。虽然在之前的文献中已经分别考虑了稳健的一步法预测器和内插器，但我们给出了不确定性类的两个条件，以便从观测集的角度寻找通用框架中的极小极大内插器和外插器以及对 Lp 范数下的内插和外推误差的评估观点。我们表明，对于在我们的条件下被未知光谱密度污染的光谱密度 ε 类，存在一个极大极小内插器和外推器。当不确定性类包含与 Lebesgue 测度不绝对连续的谱分布函数时，我们证明了 Lp 中存在一个近似内插器和外插器，使得当谱分布具有以下条件时，其最大内插和外推误差任意接近于极大极小误差密度。我们的结果适用于平稳的可协调稳定过程。