Belykh, V. N. 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. Cited in 8 Documents MSC: 65D15 Algorithms for approximation of functions 41A30 Approximation by other special function classes 41A40 Saturation in approximation theory 41A50 Best approximation, Chebyshev systems Keywords:unsaturated numerical method; exponential convergence; overconvergence; polynomial approximation; continuous functions Citations:Zbl 0343.65044 PDFBibTeX XMLCite \textit{V. N. Belykh}, Sib. Mat. Zh. 46, No. 3, 483--489 (2005; Zbl 1102.65014); translation in Sib. Math. J. 46, No. 3, 373--385 (2005) Full Text: EuDML EMIS