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On the stability of the operator of \(\varepsilon\)-projection onto the set of splines in the space \(C[0,1]\). (English. Russian original) Zbl 1079.41007

Izv. Math. 67, No. 1, 91-119 (2003); translation from Izv. Ross. Akad. Nauk Ser. Mat. 67, No. 1, 99-130 (2003).
Let \(L\) be a subset of a real Banach space \((X,\|\cdot\|_X)\) and \(x\in X\). The (multi-valued) map \(x\to P_{L,\varepsilon}(x)= \{z\in L:\| x-z\|_X\leq \text{dist}(x,L)+ \varepsilon\}\), \(\varepsilon> 0\) is called the operator of almost best approximation or \(\varepsilon\)-projection of \(x\). For \(Y,Z\subset X\), let \(C(Y,Z)\) denote the set of all maps from \(Y\) to \(Z\) that are continuous in the sense of the norm \(\|\cdot\|_X\). A map \(G: x\to z\in P_{L,\varepsilon}\) is called an \(\varepsilon\)-selection for the operator of almost best approximation. If, in addition \(G\in C(X,L)\), then \(G\) is called a continuous \(\varepsilon\)-selection to \(L\).
In this paper, the author discusses the problem of the existence of a continuous selection for the metric projection \((P_{L,0})\) to the set of \(n\)-link piecewise-linear functions in the space \(([0,1]\) with the uniform norm. It is shown that there is a continuous selection \((\varepsilon= 0)\) if and only if \(n=1\) or \(n=2\) and that there is a continuous \(\varepsilon\)-selection to \(L\subset C[0,1]\) if \(L\) belongs to a certain class of sets that contains, in particular, the set of algebraic rational fractions and the set of piecewise-linear functions. An example is constructed showing that sometimes there is no \(\varepsilon\) for a set of splines of degree \(d> 1\).

MSC:

41A15 Spline approximation
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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