  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.
