Quantitative theorems on approximation by Bernstein-Stancu operators |
| |
Authors: | H H Gonska J Meier |
| |
Affiliation: | 1. Department of Mathematics and Computer Science, Drexel University, 19104, Philadelphia, PA, USA 2. Department of Mathematics, University of Duisburg, D-4100, Duisburg, 1, West Germany
|
| |
Abstract: | In 1972 D. D. Stancu introduced a generalization \(L_{mp} ^{< \alpha \beta \gamma > }\) of the classical Bernstein operators given by the formula $$L_{mp}< \alpha \beta \gamma > (f,x) = \sum\limits_{k = 0}^{m + p} {\left( {\begin{array}{*{20}c} {m + p} \\ k \\ \end{array} } \right)} \frac{{x^{(k, - \alpha )} \cdot (1 - x)^{(m + p - k, - \alpha )} }}{{1^{(m + p, - \alpha )} }}f\left( {\frac{{k + \beta }}{{m + \gamma }}} \right)$$ . Special cases of these operators had been investigated before by quite a number of authors and have been under investigation since then. The aim of the present paper is to prove general results for all positiveL mp <αβγ> 's as far as direct theorems involving different kinds of moduli of continuity are concerned. When applied to special cases considered previously, all our corollaries of the general theorems will be as good as or yield improvements of the known results. All estimates involving the second order modulus of continuity are new. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|