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Strongly unique best approximation in Banach spaces. (English) Zbl 0615.41027
The main results of the paper are strong unicity theorems for \(L_ p\) (p\(\geq 2)\) spaces and for abstract spline approximation. Let (S,\(\Sigma\),\(\mu)\) be a positive measure space of all \(\mu\)-measurable real valued functions y on S such that \(\| y\| =\| y\|_ p=(\int_{S}| y(s)|^ p\mu (ds))^{1/p}<\infty\) \((2\leq p<\infty)\) and suppose that X is a linear closed subspace of \(L_ p\). Then for a function y in X there exists a unique function z in X so that \(\| y-z\|^ p\leq \| y-x\|^ p-2^{2-p}\| z-x\|^ p\) for all x in X.
Reviewer: S.Aljančić

MSC:
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A52 Uniqueness of best approximation
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