Optimal Recovery of Linear Functionals on Sets of Finite Dimension

Suppose that $X$ is a linear space and $L_1,\dots,L_n$ is a system of linearly independent functionals on $P$, where $P\subset X$ is a bounded set of dimension $n+1$. Suppose that the linear functional $L_0$ is defined in $X$. In this paper, we find an algorithm that recovers the functional $L_0$ on the set $P$ with the least error among all linear algorithms using the information $L_1f,\dots,L_nf$, $f\in P$.