This paper considers the problem of reconstructing missing parts of functions based on their observed segments. It provides, for Gaussian processes and arbitrary bijective transformations thereof, theoretical expressions for the $L^2$-optimal reconstruction of the missing parts. These functions are obtained as solutions of explicit integral equations. In the discrete case, approximations of the solutions provide consistent expressions of all missing values of the processes. Rates of convergence of these approximations, under extra assumptions on the transformation function, are provided. In the case of Gaussian processes with a parametric covariance structure, the estimation can be conducted separately for each function, and yields nonlinear solutions in presence of memory. Simulated examples show that the proposed reconstruction indeed fares better than the conventional interpolation methods in various situations.

翻译：本文研究了基于观测片段重构函数缺失部分的问题。针对高斯过程及其任意双射变换，我们给出了缺失部分$L^2$最优重构的理论表达式。这些函数可通过求解显式积分方程获得。在离散情形下，解的近似提供了过程所有缺失值的一致表达式。在变换函数满足额外假设的条件下，我们给出了这些近似解的收敛速率。对于具有参数化协方差结构的高斯过程，估计可对每个函数单独进行，并在存在记忆时产生非线性解。仿真实验表明，在各种情境下，所提出的重构方法确实优于传统的插值方法。