On the composite Bernstein type quadrature formula

Dan Bărbosu; Dan Miclăuş

doi:10.33993/jnaat391-915

Authors

Dan Bărbosu North University of Baia Mare, Romania
Dan Miclăuş North University of Baia Mare, Romania

DOI:

https://doi.org/10.33993/jnaat391-915

Keywords:

Bernstein operator, Bernstein approximation formula, Bernstein quadrature formula, divided differences, remainder term

Abstract views: 344

Abstract

Considering a given function \(f\in C[0,1]\), the interval \([0,1]\) is divided in \(m\) equally spaced subintervals \(\left[\tfrac{k-1}{m},\tfrac{k}{m}\right]\), \(k=\overline{1,m}\). On each of such type of interval the Bernstein approximation formula is applied and a corresponding Bernstein type quadrature formula is obtained. Making the sum of mentioned quadrature formulas, the composite Bernstein type quadrature formula is obtained.

Downloads

Download data is not yet available.

References

Agratini, O., Approximation by linear operators, Presa Universitară Clujeană, 2000, (in Romanian), 2, pp. 27-31, 2007.

Bernstein, S.N., Démonstration du theorème de Weierstrass fondée sur le calcul de probabilités, Commun. Soc. Math. Kharkow, 13, no. 2, pp. 1-2, 1912-1913.

Popoviciu, T., Sur le rest dans certains formules lineaires d'approximation de l'analyse, Mathematica I, 24, pp. 95-142, 1959.

Stancu, D. D., Quadrature formulas constructed by using certain linear positive operators, Numerical Integration (Proc. Conf. Math. Res. Inst. Oberwolffach) (Basel) (G. Hammerlin, ed.), Birkhäuser, pp. 241-251, 1982. DOI: https://doi.org/10.1007/978-3-0348-6308-7_23

Stancu, D. D. and Vernescu, A., On some remarkable positive polynomial operators of approximation, Rev. Anal. Numér. Th eor. Approx., 28, pp. 85-95, 1999, http://ictp.acad.ro/jnaat/journal/article/view/1999-vol28-no1-art8

Stancu, D. D., Coman, Gh. and Blaga, P., Numerical Analysis and Approximation Theory, II, Presa Universitară Clujeană, 2002 (in Romanian).