Constrained best approximation in Hilbert space. II
[For part I see Constructive Approximation 6, No. 1, 35-64 (1990; Zbl 0682.41034).] Let \(X\) and \(Y\) be Hilbert spaces with \(Y\) being finite-dimensional. Suppose \(A\) a bounded linear operator from \(X\) into \(Y\), \(C\) a closed convex cone in \(X\), \(d\in AC\) and \(K=C\cap A^{-1}(d)\). In this paper the authors firstly prove that for each \(x\in X\), \(P_ K(x)= P_{C_ F} (x+A^*y)\) for any \(y\in Y\) which satisfies \(A(P_{C_ F} (x+A^* y))=d\), where \(P_ Z(z)\) is the unique best approximation of \(z\) in \(Z\), \(F\) is the minimal convex extremal subset of \(AC\) which contains \(d\) and \(C_ F= C\cap A^{-1} (F)\). The main results of the paper are three applications of the above results. The first application concerns the case when \(X= L_ 1(\mu)\), \(C\) is the cone of positive functions in \(X\) and \(A\) is a bounded linear operator form \(X\) into \(\ell_ 1(n)\). The second corresponds to the case \(X=L_ 2(a,b)\), \(C\) is the cone of increasing functions in \(X\) and \(A\) is as above. The third case disposes of the cone of convex functions. Finally they deal with the minimization problem described above in the case that \(C\) is an arbitrary closed convex set.
scientific article; zbMATH DE number 97529
scientific article