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L p-approximation by Kantorovič operators

Аппроксимация вL p оп ераторами Канторови ча

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Abstract

Оператор Канторович а дляf∈L p(I), I=[0,1], определяе тся соотношением

$$P_n (f,x) = (n + 1)\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} x^k (1 - x)^{n - 1} \int\limits_{I_k } {f(t)dt,} $$

гдеI k=[k/(n}+1),(k+1)/(n+ 1)],n∈N. Доказывается, что есл ир>1 иf∈W 2 p (I), т.е.f абсол ютно непрерывна наI иf″∈L p(I), то

$$\left\| {P_n f - f} \right\|_p = O(n^{ - 1} ).$$

Далее, установлено, чт о еслиf∈L p(I),p>1 и ∥P n f-f∥р=О(n −1), тоf∈S, гдеS={f∶f аб-солютно непрерывна наI, x(1−x)f′(x)=∝ x0 h(t)dt, гдеh∈L p(I) и ∝ 10 h(t)dt=0}. Если жеf∈Lp(I),p>1, то из условия ∥P n(f)−f∥pL=o(n−1) вытекает, чтоf постоянна почти всюду.

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Authors and Affiliations

  1. Abteilung Mathematik, Universität Dortmund, Postfach 500 500, D-4600, Dortmund 50, Bundesrepublik Deutschland

    V. Maier

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  1. V. Maier
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I would like to thank Professor H. Berens for his suggestions concerning this paper.

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Maier, V. L p-approximation by Kantorovič operators. Analysis Mathematica 4, 289–295 (1978). https://doi.org/10.1007/BF02020576

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