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On the best approximation properties of \(C^\infty\)-smooth functions on an interval of the real axis (to the phenomenon of unsaturated numerical methods). (Russian, English) Zbl 1102.65014

Sib. Mat. Zh. 46, No. 3, 483-489 (2005); translation in Sib. Math. J. 46, No. 3, 373-385 (2005).
Summary: K. I. Babenko [Sov. Math., Dokl. 16, 263–267 (1975; Zbl 0343.65044)] announced his discovery of conceptually new unsaturated numerical methods. They are distinguished by the absence of the principal error term, which results in their ability to adjust automatically to all natural correctness classes of problems (the phenomenon of unsaturated numerical methods). We show that the phenomenon of unsaturation of a numerical method on an interval is a consequence, although exceptionally subtle, of the well-developed theory of polynomial approximation to continuous functions. By the way, K. I. Babenko always insisted on that.

MSC:

65D15 Algorithms for approximation of functions
41A30 Approximation by other special function classes
41A40 Saturation in approximation theory
41A50 Best approximation, Chebyshev systems

Citations:

Zbl 0343.65044
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