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Interpolation and best simultaneous approximation. (English) Zbl 1207.41009

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 [A. S. B. Holland and B. N. Sahney, Theory Approx., Proc. Conf. Calgary 1975, 332–337 (1976; Zbl 0357.41016); H. H. Cuenya, F. E. Levis and M. D. Lorenzo, Numer. Funct. Anal. Optim. 30, No. 3–4, 245–258 (2009; Zbl 1179.41017); H. H. Cuenya and C. N. Rodriguez, Note Mat. 28, No. 2, 153–162 (2008; Zbl 1210.41004)] from the References.

MSC:

41A28 Simultaneous approximation
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