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Uniform approximation by positive operators on infinite intervals

Равномерная аппрокс имация положительны ми операторами на беско нечных интервалах

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Abstract

В работе устанавливае тся оценка (*) |L n (ƒ−ƒ| ≦Kω ϕ (ƒ;α n) для положительных оп ераторов, определенн ых на конечном или бесконе чном интервале (a,b), гдеL n(1,χ)≡1,L n((t−χ)2;χ)≦Kϕ 2(χ)α 2 n (x∈(a,b)) ;и\(\omega _\varphi (f;\delta ) = \mathop {\sup }\limits_{0 \leqq h \leqq \delta ,x \pm h\varphi (x) \in (a,b)} \left| {f(x - h\varphi (x)) - 2f(x) + f(x + h\varphi (x))} \right|\) модуль гладкостиƒ, св язанный с ϕ (функцияϕ удовлетворяет некот орым условиям регуля рности). С помощью (*) для некотор ых {L n } получена характеристика тех ф ункцийƒ, для которыхL n (ƒ)−ƒ=o(1) равном ерно на (a, b). Наконец, рассматриваются слу чай насыщения и случай так называем ой неоптимальной апп роксимации.

Результаты применяю тся к операторам Саса —Миракяна, Баскакова, Мейер-Кëни га и Целлера, гамма и бета операторам, а также к н екоторым операторам типа свер тки.

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  1. Bolyai Institute, University Szeged, Aradi Vértanúk Tere 1, 6720, Szeged, Hungary

    V. Totik

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Totik, V. Uniform approximation by positive operators on infinite intervals. Analysis Mathematica 10, 163–182 (1984). https://doi.org/10.1007/BF02350525

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