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The least squares linear filter, also called the Wiener filter, is a populartool to predict the next element(s) of time series by linear combination oftime-delayed observations. We consider observation sequences of deterministicdynamics, and ask: Which pairs of observation function and dynamics arepredictable? If one allows for nonlinear mappings of time-delayed observations,then Takens' well-known theorem implies that a set of pairs, large in aspecific topological sense, exists for which an exact prediction is possible.We show that a similar statement applies for the linear least squares filter inthe infinite-delay limit, by considering the forecast problem for invertiblemeasure-preserving maps and the Koopman operator on square-integrablefunctions.

A Koopman-Takens theorem: Linear least squares prediction of nonlinear time series