Optimal recovery of linear operators from information of random functions
K. Yu. Osipenko
TL;DR
This work addresses the problem of optimally recovering a linear operator on a function class from information about random functions with noise. It presents a general Bayesian-like recovery framework on a weighted $L_2$ space $W$ with a diagonal operator $\\Lambda x=\\mu x$, and derives an explicit optimal recovery rule and error $E(\\Lambda,W,\\delta)$ under mild monotonicity conditions on $|\\mu|/\\sqrt{\\nu}$. The theory is then specialized to the Fourier-domain setting, yielding a closed-form optimal error and a band-limited estimator for recovering derivatives from noisy Fourier data, with the band limit $t_\\delta$ determined by $f(s)=\\delta^{-2}$ and $f$ expressed in terms of $\\mu$ and $\\nu$. A parallel development for recovering the solution of the heat equation from noisy initial Fourier data yields analogous optimal rules: the estimator uses a multiplier $\\alpha(t)$ supported on $|t|\le t_\\delta$ and achieves a computable error $E(T,W_2^r,\\delta)$. Across all results, the optimal methods selectively use a finite portion of the available noisy information, with the usable band expanding as measurement accuracy improves, illustrating a principled trade-off in stochastic recovery.
Abstract
The paper concerns problems of the recovery of linear operators defined on sets of functions from information of these functions given with stochastic errors. The constructed optimal recovery methods, in general, do not use all the available information. As a consequence, optimal methods are obtained for recovering derivatives of functions from Sobolev classes by the information of their Fourier transforms given with stochastic errors. A similar problem is considered for solutions of the heat equation.