Interpolation and best simultaneous approximation
The paper deals with best simultaneous approximation (b.s.a.) to \(k\) continuous functions on the interval \([a,b]\) from a finite subspace of \(C[a,b]\). The authors establish the limit of the b.s.a., which is important to provides qualitative and approximation analytic information concerning the the b.s.a. on small regions, a problem which is difficult to solve from a strictly numerical treatment. The main results of the paper are contained in Section 2 (Interpolating of best simultaneous approximation), where the properties of the b.s.a. function are established. Applying these results, in Section 3, the b.s.a. in small regions is studied. Reviewer's remark: The paper contains important generalizations of the results in [\textit{A. S. B. Holland} and \textit{B. N. Sahney}, Theory Approx., Proc. Conf. Calgary 1975, 332--337 (1976; Zbl 0357.41016); \textit{H. H. Cuenya, F. E. Levis} and \textit{M. D. Lorenzo}, Numer. Funct. Anal. Optim. 30, No.~3--4, 245--258 (2009; Zbl 1179.41017); \textit{H. H. Cuenya} and \textit{C. N. Rodriguez}, Note Mat. 28, No.~2, 153--162 (2008; Zbl 1210.41004)] from the References.
scientific article; zbMATH DE number 5817250
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