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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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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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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