On Geometric Series of Positive Linear Operators

Year 2019, , 49 - 56, 01.06.2019

Radu Paltanea

https://doi.org/10.33205/cma.506015

Cited By: 2

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] U. Abel, Geometric series of Bernstein-Durrmeyer operators, East J. on Approx. Vol. 15, No. 4 (2009) 439–450.
• [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] 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] 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] J. Bustamante, Bernstein Operators and Their Properties, Birkhäuser, 2017.
• [6] R. A. DeVore, G. G. Lorentz, Constructive approximation, Springer, Berlin, 1993.
• [7] H Gonska, R. Paltanea, General Voronovskaja and asymptotic theorems in simultaneous approximation, Mediterranean J. Math. Vol. 7 (2010) 37-49.
• [8] V. Gupta, G. Tachev, Approximation with Positive Linear Operators and Linear Combinations, Springer, 2017.
• [9] G. G. Lorenz, Bernstein polynomials, Univ. Toronto Press, 1953.
• [10] R. Paltanea, Approximation Theory Using Positive Linear Operators, Birkhäuser, Boston, 2004.
• [11] R. Paltanea, The power series of Bernstein operators, Automation Computers Applied Mathematics Vol. 15, No. 1 2006, 7-14.
• [12] I. Raşa, Power series of Bernstein operators and approximation resolvents Mediterr. J. Math. Vol. 9 (2012) 635-644.
• [13] I. A. Rus, Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl. Vol 292, No. 1 (2004) 259-261.