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Extensions of linear operators from hyperplanes of \(l_ \infty^{(n)}\). (English) Zbl 0831.41014

Summary: Let \(Y\subset l_\infty^{(n)}\) be a hyperplane and let \(A\in {\mathcal L} (Y)\) be given. Denote \({\mathcal A}= \{L\in {\mathcal L} (l_\infty^{(n)}), Y\): \(L|Y= A\}\) and \(\lambda_A= \inf \{|L|\): \(L\in {\mathcal A}\}\). In this paper the problem of calculating of the constant \(\lambda_A\) is studied. We present a complete characterization of those \(A\in {\mathcal L} (Y)\) for which \(\lambda_A= |A|\). Next we consider the case \(\lambda_A> |A|\). Finally some computer examples will be presented.

MSC:

41A35 Approximation by operators (in particular, by integral operators)
41A52 Uniqueness of best approximation
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A55 Approximate quadratures
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