ISSN:
1420-8903
Keywords:
Primary 41A25, 41A35
;
Secondary 41A60, 41A40
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract This note is a study of approximation of classes of functions and asymptotic simultaneous approximation of functions by theM n-operators of Meyer-König and Zeller which are defined by $$(M_n f)(x) = (1 - x)^{n + 1} \sum\limits_{k = 0}^\infty {f\left( {\frac{k}{{n + k}}} \right)\left( {\begin{array}{*{20}c} {n + k} \\ k \\ \end{array} } \right)} x^k ,n = 1,2, \ldots .$$ Among other results it is proved that for 0〈α≤1 $$\mathop {\lim }\limits_{n \to \infty } n^{\alpha /2} \mathop {\sup }\limits_{f \in Lip1\alpha } |(M_n f)(x) - f(x)| = \frac{{\Gamma \left( {\frac{{a + 1}}{2}} \right)}}{{\pi ^{1/2} }}\{ 2x(1 - x)^2 \} ^{\alpha /2} $$ and if for a functionf, the derivativeD m+2 f exists at a pointx ∈ (0, 1), then $$\mathop {\lim }\limits_{n \to \infty } 2n[D^m (M_n f) - D^m f] = \Omega f,$$ whereΩ is the linear differential operator given by $$\Omega = x(1 - x)^2 D^{m + 2} + m(3x - 1)(x - 1)D^{m + 1} + m(m - 1)(3x - 2)D^m + m(m - 1)(m - 2)D^{m - 1} .$$
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01844075
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