By Claude Dellacherie

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X-; r 1? . , ri)(pi(t,rl9. . , xt) dxi . . dxt. This outline of the theory of finding the best approximation of the functions r\(t) and £(£) covers extrapolation, smoothing and so on.. Thus, if we consider an advance tn in time, we have Accordingly, :(t)=i(t+t0). , the function £(£). is the sum of the useful signal and noise £(t) = 8(t) + n(t), which are given by the combined moments m^ (r)n . The function which we require to approximate is: t(t) = Ls(t). £,(j)£ a ^ so * This theory is equally applicable to non-stationary random t i m e series.

Dx{. (6a) Thus, the form of the non-linear transformation is fully defined by the given functions a0(t), ax(t, r x ), a2(t, xv r 2 ), . . which should, generally speaking, be independent of | . Using this circumstance it is convenient to make g(t) quite small in magnitude and go in quest of the solution rj(t) in the form of a series expansion in powers of this quantity. We thereby find functions a0, alf a2, . . , which hold for any function £(t). The method is illustrated by an example in the next section.

We can again apply the same method as in find ing the functions of a formal series (see above). By way of illustration we will take as our example _ / ( * ) = a*2, u0=0, on 0