On strong uniqueness in linear Chebyshev approximation
Let Q be a compact subset of \({\mathbb{C}}\) and C(0) the set of all continuous functions f:\(Q\to {\mathbb{C}}\). A given function \(f\in C(Q)\) is uniformly approximated by elements of an n-dimensional Haar subspace V of C(Q). \textit{D. J. Newman} and \textit{H. Shapiro} [Duke Math. J. 30, 673-681 (1963; Zbl 0116.045)] have shown that the best approximation \(\tilde v\in V\) to f is in general not strongly unique. The main result of the present paper can be stated as follows: There exists an open and dense subset of C(Q) such that all functions of this subset have a strongly unique best approximation iff Q has at most n isolated points. Hence, in this case, strong uniqueness is a generic property.
scientific article; zbMATH DE number 3853673
scientific article