Abstract
We discuss the qualitative behavior of the polynomials p*n(z) of best uniform approximation to a function f that is continuous on a compact set E of the z-plane, analytic in the interior of E, but not analytic at some point of the boundary of E. Particularly, we survey results on the asymptotic behavior of the zeros of the p*n(z) and the extreme points for the error f(z) - p*n(z). The theorems and examples presented support a "principle of contamination," which roughly states that the existence of one or more singularities of f on the boundary of E adversely affects the behavior over the whole boundary of E of a subsequence of the best approximants p*n(z).
The research of the author was supported, in part, by the National Science Foundation under grant DMS-862-0098.
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© 1988 Springer-Verlag
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Saff, E.B. (1988). A principle of contamination in best polynomial approximation. In: Gómez-Fernandez, J.A., Guerra-Vázquez, F., López-Lagomasino, G., Jiménez-Pozo, M.A. (eds) Approximation and Optimization. Lecture Notes in Mathematics, vol 1354. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0089584
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DOI: https://doi.org/10.1007/BFb0089584
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