Restoration of the Wiener field in the \(3D\) space from observations on the surface of rectangular parallelepiped (Q2736991)

Let \(W(x,y,z),\;x\geq 0,\;y\geq 0,\;z\geq 0,\) be a Wiener field in the \(3D\) space and let \(S\) be the surface of a rectangular parallelepiped. Let \(\mathcal F\) be a \(\sigma\)-algebra generated by \(W(x,y,z),\;(x,y,z) \in S\). The best mean square estimate of \(W(x_1,y_1,z_1),\;(x_1, y_1, z_1)\not\in S\) is determined by the formula NEWLINE\[NEWLINE\hat{W}(x_1,y_1,z_1)=E\{W(x_1,y_1,z_1) |\mathcal F\}NEWLINE\]NEWLINE and the mean square error is determined by the formula NEWLINE\[NEWLINED(x_1,y_1,z_1)=E\{(\hat{W}(x_1,y_1,z_1)-W(x_1,y_1,z_1))^2 |\mathcal F\}.NEWLINE\]NEWLINE The author proposes formulas for the calculation of the best mean square estimate \(\hat{W}(x_1,y_1,z_1)\) and the mean square error \(D(x_1,y_1,z_1)\).