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Optimal recovery of linear functionals on sets of finite dimension. (English. Russian original) Zbl 1159.65018
Math. Notes 84, No. 4, 561-567 (2008); translation from Mat. Zametki 84, No. 4, 602-608 (2008).
Let \(X\) be a linear space of dimension \(n+1\) generated by \(f_0, f_1,\dots,f_n\). Let \(L_1,\dots,L_n\) be linear functionals linearly independent defined on \(X\) and \(L_0\neq L_i\), \(i=1,\dots,n\), another linear functional on \(X\). Let \(P=\{\sum^n_{i=0}a_if_i:a_i\in\mathbb R\), \(|a_i|\leq |\beta_i|\), \(i=0,\dots,n\}\), where \(\beta n=1\) and \((\beta_0,\dots,\beta n_1)\) is the solution of the system \(\sum^n_{i=0} \beta j\) \(L_jf_i=0\), \(j=1,\dots,n\). The author presents an algorithm that recovers the functional \(L_0\) on \(P\) with the least error among all linear algorithms using the information \((L_1f,\dots,L_nf)\).
MSC:
65D15 Algorithms for approximation of functions
65D05 Numerical interpolation
65Y20 Complexity and performance of numerical algorithms
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