Skip to main content
SpringerLink
Log in

Some Approximation Results on Compact Sets by (p, q)-Bernstein–Faber Polynomials, \(q>p>1\)

  • Research Paper
  • Published:
Iranian Journal of Science and Technology, Transactions A: Science Aims and scope Submit manuscript

Abstract

In the present article, we introduce the (p, q)-Bernstein–Faber polynomials for \(q>p>1\) attached to a compact set \(O \subset {\mathbb {C}}\) and to an analytic function on O. Here, we prove that the approximation results by these operators can hold on larger compact sets. Also we give explicit formulas of Voronovskaja-type for the same operators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Acar T (2016) \((p, q)\)-Generalization of Szász–Mirakyan operators. Math. Methods Appl. Sci. 39(10):2685–2695

    Article  MathSciNet  MATH  Google Scholar 

  • Acar T, Aral A, Mohiuddine SA (2016) On Kantorovich modification of \((p, q)\)-Baskakov operators. J. Inequal. Appl. 2016:98

    Article  MathSciNet  MATH  Google Scholar 

  • Acar T, Aral A, Mohiuddine SA (2018a) Approximation by bivariate \((p,q)\)-Bernstein Kantorovich operators. Iran. J. Sci. Technol.,Trans. A, Sci. 42:655–662

    Article  MathSciNet  MATH  Google Scholar 

  • Acar T, Aral A, Mohiuddine SA (2018b) On Kantorovich modification of \((p, q)\)-Bernstein operators. Iran. J. Sci. Technol. Trans. A, Sci. 42:1459–1464

    MATH  Google Scholar 

  • Acar T, Mohiuddine SA, Mursaleen M (2018c) Approximation by \((p, q)\)-Baskakov–Durrmeyer–Stancu operators. Complex Anal. Oper. Theory 12(6):1453–1468

    Article  MathSciNet  MATH  Google Scholar 

  • Anderson JM, Clunie J (1985) Isomorphisms of the disk algebra and inverse Faber sets. Math. Z. 188:545–558

    Article  MathSciNet  MATH  Google Scholar 

  • Curtiss JH (1971) Faber polynomials and Faber series. Amer. Math. Mon. 78(6):5677–5696

    Article  MathSciNet  MATH  Google Scholar 

  • Faber G (1903) Üfber polynomische Entwicklungen. Math. Ann. 57:398–408

    Google Scholar 

  • Gal SG (2009) Approximation by Complex Bernstein and Convolution-Type Operators. World Scientific Publishing Co, Singapore

    Book  MATH  Google Scholar 

  • Gal SG (2012) Differentiated generalized Voronovskaja’s theorem in compact disks. Results Math. 61(3):247–253

    MathSciNet  MATH  Google Scholar 

  • Gaier D (1987) Lectures on Complex Approximation. Birkhauser, Boston

    Book  MATH  Google Scholar 

  • Ilarslan HGI, Acar T (2018) Approximation by bivariate \((p,q)\)-Baskakov–Kantorovich operators. Geor. Math. J. 25(3):397–407

    Article  MathSciNet  MATH  Google Scholar 

  • Jebreen HB, Mursaleen M, Naaz A (2018) Approximation by quaternion (p, q)-Bernstein polynomials and Voronovskaja type result on compact disk. Adv. Differ. Equ. 2018:448

    Article  MathSciNet  MATH  Google Scholar 

  • Khan K, Lobiyal DK, Kilicman A (2018) A de Casteljau algorithm for Bernstein type polynomials based on \((p, q)\)-integers. Appl. Appl. Math. 13(2):997–1017

    MathSciNet  MATH  Google Scholar 

  • Maurya R, Sharma H, Gupta C (2018) Approximation properties of Kantorovich type modifications of \((p, q)\)-Meyer–Konig–Zeller operators. Constr. Math. Anal. 1(1):58–72

    Google Scholar 

  • Mohiuddine SA, Acar T, Alotaibi A (2017) Construction of a new family of Bernstein–Kantorovich operators. Math. Methods Appl. Sci. 40:7749–7759

    Article  MathSciNet  MATH  Google Scholar 

  • Mohiuddine SA, Acar T, Alotaibi A (2018a) Durrmeyer type \((p, q)\)-Baskakov operators preserving linear functions. J. Math. Inequal. 12:961–973

    Article  MathSciNet  MATH  Google Scholar 

  • Mohiuddine SA, Acar T, Alghamdi MA (2018b) Genuine modified Bernstein–Durrmeyer operators. J. Inequal. Appl. 2018:104

    Article  MathSciNet  Google Scholar 

  • Mursaleen M, Ahasan M (2018) The Dunkl generalization of Stancu type \(q\)-Szász–Mirakjan–Kantorovich operators and some approximation results. Carpathian J. Math. 34(3):363–370

    MathSciNet  Google Scholar 

  • Mursaleen M, Nasiruzzaman M (2018) Approximation of modified Jakimovski–Leviatan-beta type operators. Constr. Math. Anal. 1(2):88–98

    Google Scholar 

  • Mursaleen M, Ansari KJ, Khan A (2015) On \((p, q)\)-analogue of Bernstein operators. Appl. Math. Comput. 266:874–882 (Erratum: Appl. Math. Comput., 278, 70–71 (2016))

    MathSciNet  MATH  Google Scholar 

  • Mursaleen M, Khan F, Khan A (2016) Approximation by \((p, q)\)-Lorentz polynomials on a compact disk. Complex Anal. Oper. Theory 10(8):1725–1740

    Article  MathSciNet  MATH  Google Scholar 

  • Mursaleen M, Nasiruzzaman M, Khan F, Khan A (2017) On \((p, q)\)-analogue of divided difference and Bernstein operators. J. Nonlinear Funct. Anal. 2017(25):1–13

    Google Scholar 

  • Mursaleen M, Rahman S, Alkhaldi AH (2018) Convergence of iterates of \(q\)-Bernstein and \((p, q)\)-Bernstein operators and the Kelisky–Rivlin type theorem. Filomat 32(12):4351–4364

    Article  MathSciNet  Google Scholar 

  • Mursaleen M, Naaz A, Khan A (2019) Improved approximation and error estimations by King type \((p, q)\)-Szász–Mirakjan–Kantorovich operators. Appl. Math. Comput. 348:175–185

    MathSciNet  Google Scholar 

  • Ostrovska S (2003) q-Bernstein polynomials and their iterates. J. Approx. Theory 123:232–255

    Article  MathSciNet  MATH  Google Scholar 

  • Ostrovska S (2005) On the \(q\)-Bernstein polynomials and their iterates. Adv. Stud. Contemp. Math. 11:193–204

    MathSciNet  MATH  Google Scholar 

  • Phillips GM (1997) Bernstein polynomials based on the \(q\)-integers. Ann. Numer. Math. 4:511–518

    MathSciNet  MATH  Google Scholar 

  • Phillips GM (2000) A generalization of the Bernstein polynomials based on the q-integers. ANZIAMJ 42:79–86

    Article  MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Özger F, Mohiuddine SA (2019) Construction of Stancu-type Bernstein operators based on Bézier bases with shape parameter \(\lambda \). Symmetry 11(3):316

    Article  Google Scholar 

  • Suetin PK (1998) Series of Faber Polynomials. Gordon and Breach, Amsterdam

    MATH  Google Scholar 

  • Wang H, Wu XZ (2008) Saturation of convergence for \(q\)-Bernstein polynomials in the case \(q \ge 1\). J. Math. Anal. Appl. 337:744–750

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

    Ambreen Naaz & M. Mursaleen

Authors
  1. Ambreen Naaz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Mursaleen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Mursaleen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Naaz, A., Mursaleen, M. Some Approximation Results on Compact Sets by (p, q)-Bernstein–Faber Polynomials, \(q>p>1\). Iran J Sci Technol Trans Sci 43, 2585–2593 (2019). https://doi.org/10.1007/s40995-019-00750-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40995-019-00750-0

Keywords

Search

Navigation