Abstract:
We show that the least uniform rational deviations Rn(f) from the function f(x), continuous and convex on the interval [a,b], satisfy the condition Rn(f)=o(1/n) as n→∞, and that Rn(f)=O(1/n) uniformly for the continuous convex functions f whose absolute values are bounded by unity. These estimates are precise with respect to the rate of decrease of the right-hand sides.
Bibliography: 16 titles.
Citation:
V. A. Popov, P. P. Petrushev, “The exact order of the best approximation to convex functions by rational functions”, Math. USSR-Sb., 32:2 (1977), 245–251
\Bibitem{PopPet77}
\by V.~A.~Popov, P.~P.~Petrushev
\paper The exact order of the best approximation to convex functions by~rational functions
\jour Math. USSR-Sb.
\yr 1977
\vol 32
\issue 2
\pages 245--251
\mathnet{http://mi.mathnet.ru/eng/sm2808}
\crossref{https://doi.org/10.1070/SM1977v032n02ABEH002381}
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Linking options:
https://www.mathnet.ru/eng/sm2808
https://doi.org/10.1070/SM1977v032n02ABEH002381
https://www.mathnet.ru/eng/sm/v145/i2/p285
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V. N. Konovalov, “Impact of the shape of functions on the orders of piecewise polynomial and rational approximation”, Sb. Math., 196:5 (2005), 623–648
V. N. Rusak, I. V. Rybachenko, “The Properties of Functions and Approximation by Summation Rational Operators on the Real Axis”, Math. Notes, 76:1 (2004), 103–110
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P. P. Petrushev, “Uniform rational approximations of functions of class Vr”, Math. USSR-Sb., 36:3 (1980), 389–403
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