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Results in Mathematics

, 74:39 | Cite as

An Error Estimate for a Bernstein–Stancu Operator with Negative Parameter

  • Mihai Nicolae PascuEmail author
  • Nicolae Radu Pascu
  • Florenţa Tripşa
Article
  • 12 Downloads

Abstract

We recently showed (Pascu et al., in: Proceedings of the Romanian Academy. Series A. Mathematics, physics, technical sciences, information science, 2019) that by choosing a negative value of the parameter of the classical Bernstein–Stancu operator, the resulting operator is still a positive, linear, uniform approximation operator, and it improves the error estimates for the classical Bernstein operator. However, in the case of the error estimate in terms of the modulus of continuity, the constant involved was slightly larger than that for the corresponding inequality for the classical Bernstein operator. In the present paper we prove an inequality for the rising factorial (of independent interest), and we use it in order to show that the constant appearing in the error estimate (in terms of the modulus of continuity) for the new operator is in fact smaller than Sikkema’s optimal constant for the Bernstein operator.

Keywords

Bernstein operator Bernstein–Stancu operator Pólya–Eggenberger distribution positive linear operator approximation theory 

Mathematics Subject Classification

41A36 41A25 41A20 

Notes

Acknowledgments

The first author kindly acknowledges the support by a Grant of the Romanian National Authority for Scientific Research, CNCS—UEFISCDI, Project Number PNII-ID-PCCE-2011-2-0015.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceTransilvania University of BraşovBraşovRomania
  2. 2.Department of MathematicsKennesaw State UniversityMariettaUSA

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