We study the problem of estimating an unknown function from ergodic samples
corrupted by additive noise. It is shown that one can consistently recover
an unknown measurable function in this setting, if the one-dimensional (1-
D) distribution of the samples is comparable to a known reference distribut
ion, and the noise is independent of the samples and has known mixing rates
. The estimates are applied to deterministic sampling schemes, in which suc
cessive samples are obtained by repeatedly applying a fixed map to a given
initial vector, and it is then shown how the estimates can be used to recon
struct an ergodic transformation from one of its trajectories.