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Complex Analysis and Operator Theory

, Volume 10, Issue 8, pp 1725–1740 | Cite as

Approximation by (pq)-Lorentz Polynomials on a Compact Disk

  • M. Mursaleen
  • Faisal Khan
  • Asif Khan
Article

Abstract

The (pq)-factors were introduced in order to generalize or unify several forms of q-oscillator algebras well known in the physics literature related to the representation theory of single parameter quantum algebras. This notion has been recently used in approximation by positive linear operators via (pq)-calculus which has emerged a very active area of research. In this paper, we introduce a new analogue of Lorentz polynomials based on (pq)-integers. We obtain quantitative estimate in the Voronovskaja’s type theorem and exact orders in simultaneous approximation by the complex (pq)-Lorentz polynomials of degree \(n\in \mathbb {N}\) (\(q>p>1)\), attached to analytic functions on compact disks of the complex plane. In this way, we put in evidence the overconvergence phenomenon for the (pq)-Lorentz polynomial, namely the extensions of approximation properties (with quantitative estimates) from real intervals to compact disks in the complex plane.

Keywords

(p, q)-Integer (p, q)-Lorentz polynomial Voronovskaja’s theorem Iterates Compact disk 

Mathematics Subject Classification

41A10 41A25 41A36 

Notes

Acknowledgments

Authors are very much thankful to the learned referee for his/ her valuable comments which improved the presentation of the paper.

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsAligarh Muslim UniversityAligarhIndia

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