The least squares linear filter, also called the Wiener
filter, is a popular tool
to predict the next element(s) of time series by linear combination of
time-delayed observations.
We consider observation sequences of deterministic dynamics, and ask:
Which pairs of observation function and dynamics are predictable?
If one allows for nonlinear mappings of time-delayed observations,
then Takens' well-known theorem implies that a set of pairs, large in a
specific topological sense,
exists for which an exact prediction is possible. In joint work with
Péter Koltai
we show that a similar statement applies for the linear least squares
filter in the infinite-delay limit, by considering the forecast problem for
invertible measure-preserving maps
and the Koopman operator on square-integrable functions.