On Geometric Series of Positive Linear Operators

Abstract

We study the existence and the norm of operators obtained as power series of linear positive operators with particularization to Bernstein operators. We also obtain a Voronovskaja-kind theorem.

Keywords

• Positive linear operators,Geometric series of operators,Bernstein operators,Voronovskaja theorem

References

1. [1] U. Abel, Geometric series of Bernstein-Durrmeyer operators, East J. on Approx. Vol. 15, No. 4 (2009) 439–450.
2. [2] U. Abel, M. Ivan, R. Paltanea, Geometric series of Bernstein operators revisited, J. Math. Anal. Appl. Vol. 400. No. 1 (2013) 22-24.
3. [3] U. Abel, M. Ivan, R. Paltanea, Geometric series of positive linear operators and the inverse Voronovskaya theorem on a compact interval, J. Approx. Theory Vol. 184 (2014), 163-175.
4. [4] F. Altomare, S. Diomede, Asymptotic formulae for positive linear operators: direct and converse results, Jaen J. Approx. Vol. 2, No. 2 (2010) 255–287.
5. [5] J. Bustamante, Bernstein Operators and Their Properties, Birkhäuser, 2017.
6. [6] R. A. DeVore, G. G. Lorentz, Constructive approximation, Springer, Berlin, 1993.
7. [7] H Gonska, R. Paltanea, General Voronovskaja and asymptotic theorems in simultaneous approximation, Mediterranean J. Math. Vol. 7 (2010) 37-49.
8. [8] V. Gupta, G. Tachev, Approximation with Positive Linear Operators and Linear Combinations, Springer, 2017.

9. [9] G. G. Lorenz, Bernstein polynomials, Univ. Toronto Press, 1953.
10. [10] R. Paltanea, Approximation Theory Using Positive Linear Operators, Birkhäuser, Boston, 2004.
11. [11] R. Paltanea, The power series of Bernstein operators, Automation Computers Applied Mathematics Vol. 15, No. 1 2006, 7-14.
12. [12] I. Raşa, Power series of Bernstein operators and approximation resolvents Mediterr. J. Math. Vol. 9 (2012) 635-644.
13. [13] I. A. Rus, Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl. Vol 292, No. 1 (2004) 259-261.