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One-sided \(L^ 1\)-approximation. (English) Zbl 0595.41023

The problem of uniqueness of best one-sided \(L^ 1\)-approximations to continuous functions from a finite-dimensional subspace is considered. Two main results obtained are embodied in theorems with proofs which are established after introducing a number of prepositions, lemmas and remarks. For (n\(\geq 2)\) the unicity space \(U_ n\) containing a function strictly positive on (0,1) it is shown that there exists a function \(f\in C[0,1]\) having more than one best one-sided L’-approximation. Further such \(U_ n\) with the property that each \(f\in C[0,1]\) has a unique best one-sided L’(w)-approximation from \(U_ n\) (with respect to every strictly positive continuous weight function w) are characterized.
Reviewer: P.Achuthan

MSC:

41A50 Best approximation, Chebyshev systems
49K35 Optimality conditions for minimax problems
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