Predicting the future of functions on flows

Let Ω be a topological space,S _t ∈ R (R the reals) a homeomorphism group on Ω andμ a Borel measure invariant with respect toS _t , (μ(Ω)=1); forP ∈Ω putS _t (P)=P _t . Assumef ∈L ₂(Ω,μ); according to E. Hopf there is for almost everyP ∈ Ω a well-determined spectral function σ(P,λ),λ ∈ R with lim \(T^{ - 1} \int_0^T {f(P_{t + s} )\overline {f(P_t )} dt = \int_{ - \infty }^{ + \infty } {e^{i\lambda s} d\sigma (P\lambda )} }\) . The question to be considered is:^*) if for a fixedP ∈ Ω we know the “past”f(P _t ), t ≦ 0, is it then possible to compute (or “predict”) the future valuesf(P _t ), t > 0? By using ideas from linear prediction theory we show that if \(\int_{ - \infty }^{ + \infty } {(1 + \lambda ^2 )\log \frac{d}{{d\lambda }}\sigma (P,\lambda )} d\lambda = - \infty\) then the prediction required by^*) is possible. An algorithm is described which accomplishes the prediction.