Best interpolatory approximation in normed linear spaces (Q1915578)

Let \(M\) be an \(n\)-dimensional subspace of the normed space \(X\) and \(\{\varphi_1, \dots, \varphi_m\} \subseteq X^*\) be a given set of \(m\leq n\) linearly independent continuous functionals. For \(x\in X\), let \[ M(x)= \{y\in M: \varphi_i (y)= \varphi_i (x),\;i=1, 2, \dots, m\} \] be the set of elements in \(M\) interpolating \(x\), relative to the set \(\{\varphi_1, \dots, \varphi_m\}\). An element \(x_0\in M(x)\) is said to be a best approximation to \(x\) from \(M(x)\) (or with interpolatory constraints) if \(|x-x_0 |= d(x,M(x))\), i.e. \(x\in P_{M(x)} (x)\). The authors give a characterization of best approximation elements to \(x\) from \(M(x)\). This ``parametric approximation'' problem is reduced to another one more usual involving the fixed subspace \(M(0)\) of \(M\). More detailed results are obtained when \(X\) is a Hilbert space or when \(X\) is a normed space and \(M\) is a finite-dimensional interpolating subspace of \(X\) as it was defined in \textit{D. A. Ault}, \textit{F. R. Deutsch}, \textit{P. D. Morris}, \textit{J. E. Olson} [J. Approximation Theory 3, 164-182 (1970; Zbl 0193.09103)]. The pointwise Lipschitz continuity of the metric projection \(P_{M( \cdot)} (\cdot)\) is also proved.