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Exact estimates for partially monotone approximation

Точные оценки частич но-монотонной аппрок симации

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Abstract

Пустьf(x) — функция, неп рерывная и кусочно-мо нотонная на [−1,1], и пустьω(f, δ) — модуль непрерывнос ти этой функции, as—чис ло участков монотоннос ти f, т. е. число (наи-мень шее) таких интервалов (x i,x i+ 1) (i=0, 1, ...,s−1; хв=−1,x s,=1), чтоf(x) монотонна на каждом из них. Доказано, что для кажд огоn=0,1,... найдется такой алгебраический мног очленР n(х) степенип, что на [− 1,1] signf′(x) signР′ n(x) ≧ 0, ¦f(x)−Р n(x)¦ ≦ C(s)ω (f, 1/n+1, где величинаC(s) зависи т только отs. Таким образом, в теоре ме Джексона приближа ющие многочлены могут быт ь выб-раны так, что они «наследую т» свойства монотонн ости приближаемой функци я.

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  1. Institute of Mathematics and Mechanics with computing center, P. O. B. 373, 1000, Sofia, Bulgaria

    G. L. Iliev

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  1. G. L. Iliev
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Iliev, G.L. Exact estimates for partially monotone approximation. Analysis Mathematica 4, 181–197 (1978). https://doi.org/10.1007/BF01908988

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