aequationes mathematicae

, Volume 18, Issue 1–2, pp 206–217 | Cite as

Lipschitz-Nikolski\(\imath \) constants and asymptotic simultaneous approximation on theM n -operatorsconstants and asymptotic simultaneous approximation on theM n -operators

  • R. K. S. Rathore
Research Papers

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
$$(M_n f)(x) = (1 - x)^{n + 1} \sum\limits_{k = 0}^\infty {f\left( {\frac{k}{{n + k}}} \right)} \left( \begin{array}{l} n + k \\ k \\ \end{array} \right)x^k , n = 1,2,....$$
Among other results it is proved that for 0<α≤1
$$\mathop {\lim }\limits_{n \to \infty } n^{\alpha /2} \mathop {\sup }\limits_{f \in Lip_1 \alpha } \left| {(M_n f)(x) - f(x)} \right| = \frac{{\Gamma \left( {\frac{{\alpha + 1}}{2}} \right)}}{{\pi ^{1/2} }}\left\{ {2x(1 - x)^2 } \right\}^{\alpha /2} $$
and if for a functionf, the derivativeD m+2 f exist at a pointx∈(0, 1), then
$$\mathop {\lim }\limits_{n \to \infty } 2n[D^m (M_n f) - D^m f] = \Omega f,$$
where Ω is the linear differential operator given by
$$\Omega = x(1 - x)^2 D^{m + 2} + m(3x - 1)(x - 1)D^{m + 1} + m(m - 1)(3x - 2)D^m + m(m - 1)(m - 2)D^{m - 1} .$$

Keywords

Approximation Property Simultaneous Approximation Algebraic Polynomial Linear Differential Operator Linear Positive Operator 

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Copyright information

© Birkhäuser-Verlag 1978

Authors and Affiliations

  • R. K. S. Rathore
    • 1
  1. 1.Department of MathematicsIndian Institute of TechnologyKanpurIndia

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