Abstract:
Let $\xi=Af+\nu$ be the result of measuring a signal $f\in R$ that is not directly observable, where $A\in\mathbf B(R\to\widetilde R)$, $R$ and $\widetilde R$ are Hilbert spaces, and $\nu$ is a random element of $\widetilde R$ giving an error in the measurement of $Af$. Let $\mathbf U$ be the class of Hilbert–Schmidt operators acting from $R$ to $U$, and let $q(\,\cdot\,)$ be a vector-valued function on $\mathbf U$ giving the quality of the “instruments” in $\mathbf U$ in such a way that $q(U_1)<q(U_2)$ if the quality of $U_1$ is higher than that of $U_2$. If $R_{\varepsilon,\delta}$; $U_{\varepsilon,\delta}$ forms a solution of the minimum problem $\min\{\|RA-U\|\mid R\in\mathbf H_-$, $\mathbf E\|R\nu\|^2\leqslant\varepsilon$, $U\in\mathbf U$, $q(U)\leqslant\delta\}=\rho_{\varepsilon,\delta}$, then $R_{\varepsilon,\delta}\xi$ is interpreted as the output signal, distorted by the noise $R_{\varepsilon,\delta}\nu$, of an “instrument” $R_{\varepsilon,\delta}A$ which to within $\rho_{\varepsilon,\delta}$ coincides with an “instrument” $U_{\varepsilon,\delta}$ of guaranteed quality $q(U_{\varepsilon,\delta})\leqslant\delta$. The properties of the measurement reduction $\xi\to R_{\varepsilon,\delta}\xi$ are studied, and questions of optimal design of measurements are considered.
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