Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

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

  • 27 Accesses

  • 6 Citations

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

$$\Omega = x(1 - x)^2 D^{m + 2} + m(3x - 1)(x - 1)D^{m + 1} + m(m - 1)(3x - 2)D^m + m(m - 1)(m - 2)D^{m - 1} .$$

This is a preview of subscription content, log in to check access.

References

  1. [1]

    Cheney, E. W. andSHARMA, A.,Bernstein power series. Canad. J. Math.16 (1964), 251–252.

  2. [2]

    Lupas, A. andMuller, M. W.,Approximation properties of the M n -operators. Aequationes Math.5 (1970), 19–37.

  3. [3]

    Meyer-Konig, W. andZeller, K.,Bernsteinsche Potenzreihen. Studia Math.19 (1960), 89–94.

  4. [4]

    Müller, M. W.,Uber die Ordnung der Approximation dutch die Folge der Operatoren yon Meyer-Kö⊁g and Zeller and dutch die Folge deren erster Ableitungen. Bull. Inst. Politehn. Iasi14 (1968), 83–90.

  5. [6]

    RATHORE, R. K. S.,Approximation of unbounded functions with linear positive operators. Doctoral Thesis, Technische Hogeschool Delft, 1974.

  6. [7]

    Rathore, R. K. S.,Lipschitz-Nikolskii constants of the gamma operators of Müller. Math. Z.141 (1975), 193–198.

  7. [8]

    Rathore, R. K. S.,On (L, p)-summability of a multiply differentiated Fourier series. Indag. Math.38 (1976), 217–230.

  8. [9]

    Sikkema, P. C.,On the asymptotic approximation with operators of Meyer-König and Zeller. Indag. Math.32 (1970), 428–440.

  9. [10]

    Watanabe, S. andSuzuki, Y.,Approximation of functions by generalized Meyer-König and Zeller operators. Bull. of Yamagata Univ. Natur. Sci.7(2) (1969), 123–127.

Download references

Author information

Correspondence to R. K. S. Rathore.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rathore, R.K.S. Lipschitz-Nikolski\(\imath \) constants and asymptotic simultaneous approximation on theM n -operatorsconstants and asymptotic simultaneous approximation on theM n -operators. Aeq. Math. 18, 206–217 (1978). https://doi.org/10.1007/BF01844075

Download citation

Keywords

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