Abstract
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.
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Scarpellini, B. Predicting the future of functions on flows. Math. Systems Theory 12, 281–296 (1978). https://doi.org/10.1007/BF01776579
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DOI: https://doi.org/10.1007/BF01776579