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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 102, 244-250 (1984) Estimates of Derivatives of Random Functions I

By A. G. Ramm

Abstract

If one looks for an optimal (by criterion of minimal variance) linear estimate of s’(t) from the observation of u(t) = s(t) + n(t), where n(t) is noise and s(t) is useful signal, then one can derive an integral equation for the weight function of optimal estimate. This integral equation is often difftcult to solve and, even if one can solve it, it is difficult to construct the corresponding filter. In this paper an optimal estimate of s ’ on a subset of all linear estimates in sought and it is shown that this quasioptimal estimate is easy to calculate, the corresponding filter is easy to construct, and the error of this estimate differs little from the error of optimal estimates. It is also shown that among all estimates (linear and nonlinear) of s ’ for InI <6 and Is”1 <M the best estimate is given by A,u = (2h)- ’ [u(t + h)u(t- h)] with h = (26/M)“2. 1. DETERMINISTIC CASE Let u = s + rz, 1 nl < 6, Is”1 GM. Let us assume that u(t) and s(t) are continuous functions defined on R ’ and II. 11 denotes the norm in C(R ‘). Let A,u = (2h)- ’ [u(t + h)- u(t- h)], (1) h(6) = (26/M)“*, E(B) = (2M6)“2. (2) Let A denote the set of all operators T: C(lR’) + C(IR ‘). THEOREM 1. Under the above assumptions the following estimates hold IlA h(S) u- s ’ II G &(a (3) inf sup II Tu- ~‘11 = c(8), TEA IS”1 (‘44 (4

Year: 2012
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