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On optimal recovery methods in Hardy-Sobolev spaces. (English. Russian original) Zbl 1017.41024
Sb. Math. 192, No. 2, 225-244 (2001); translation from Mat. Sb. 192, No. 2, 67-86 (2001).
Let \(W\) be a convex central symmetric set in a linear space \(X\), and \(I:= (\ell_1, \ell_2, \dots, \ell_n)\) be the information operator. Here \(\ell_j \in X^*\), \(1\leq i\leq d\). Let \(S_0\): image \(I\to\mathbb{R}\) (or \(\mathbb{C})\) be a solution of the problem \(\sup_{x\in W} |Lx-S(Ix) |\to \inf\) over the set \(S\): image \(I\to \mathbb{R}\). Then \(S_0\) is called an optimal recovery method for \(L\in X^*\) on the set \(W\). The authors proposed a general approach to find out \(S_0\) basing on a solution to the dual extremal problem. The approach applied to optimal recovery problems in Hardy-Sobolev classes with \(I\) defined by Fourier coefficients and evaluations.

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
30E10 Approximation in the complex plane
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