Skip to main content
SpringerLink
Log in

RETRACTED ARTICLE: On eigenstructure of q-Bernstein operators

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

This article was retracted on 15 May 2021

This article has been updated

Abstract

The quantum analogue of Bernstein operators \({\mathcal {B}}_{m,q}\) reproduce the linear polynomials which are therefore eigenfunctions corresponding to the eigenvalue \(1,~\forall ~ q>0\). In this article the rest of eigenstructure of q-Bernstein operators and the distinct behaviour of zeros of eigenfunctions for cases (i) \(1>q>0\), and (ii) \(q>1\) are discussed. Graphical analysis for some eigenfunctions and their roots are presented with the help of MATLAB. Also, matrix representation for diagonalisation of q-Bernstein operators is discussed.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Availability of data and material

Not applicable

Change history

References

  1. Ando, T.: Totally positive matrices. Linear Algebra Appl. 90, 165–219 (1987)

    Article  MathSciNet  Google Scholar 

  2. Asnaoui, K.E., Ouhda, M., Aksasse, B., Ouanan, M.: Homological and combinatorial methods in algebra. SAA, Springer Proceedings in Mathematics and Statistics, vol. 228. Springer, Cham (2018)

    Google Scholar 

  3. Bernstein, S.N.: Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités. Commun. Soc. Math. Charkow Sér. 13, 1–2 (1912)

    Google Scholar 

  4. Chui, C.K., Li, X.: Realization of neural networks with one hidden layer. Series in approximatipn and decompositions, multivariate approximation, pp. 77–89 (1993)

  5. Cooper, S., Waldron, S.: The eigenstructure of the Bernstein operator. J. Approx. Theory 105, 133–162 (2000)

    Article  MathSciNet  Google Scholar 

  6. Finch, D., Hickmann, K.S.: Transmission eigenvalues and thermoacoustic tomography. arxiv:1306.5481

  7. Karlin, S.: Total Positivity. Stanford University Press, Stanford (1968)

    MATH  Google Scholar 

  8. Lupaş, A.: A \(q\)-analogue of the Bernstein operator. Sem. Numer. Stat. Calculus 9, 85–92 (1987)

    MathSciNet  MATH  Google Scholar 

  9. Mast, T.D., Nachman, A.I., Waag, R.C.: Focusing and imaging using eigenfunctions of the scattering operator. J. Acoust Soc. Am. 102(2), 715–725 (1997)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Özger, F., Srivastava, H.M., Mohiuddine, S.A.: Approximation of functions by a new class of generalized Bernstein-Schurer operators. Rev. R. Acad. Cienc. Exactas Fís Nat. Ser. A Math. RACSAM 114, 173 (2020)

    Article  MathSciNet  Google Scholar 

  12. Mohiuddine, S.A., Acar, T., Alotaibi, A.: Construction of a new family of Bernstein-Kantorovich operators. Math. Meth. Appl. Sci. 40, 7749–7759 (2017)

    Article  MathSciNet  Google Scholar 

  13. Mohiuddine, S.A., Özger, F.: Approximation of functions by Stancu variant of Bernstein-Kantorovich operators based on shape parameter \(\alpha \). Rev. R. Acad. Cienc. Exactas Fís Nat. Ser. A Math. RACSAM 114, 70 (2020)

    Article  MathSciNet  Google Scholar 

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

  15. Mursaleen, M., Ansari, K.J., Khan, A.: Some approximation results by \((p,q)\)-analogue of Bernstein-Stancu operators, Appl. Math. Comput., 264, 392–402 (2015) [Corrigendum: Appl. Math. Comput, 269 (2015) 744-746]

  16. Naaz, A., Mursaleen, M.: Some approximation results on compact sets by \((p, q)\)-Bernstein-Faber polynomials, \(q>p>1\). Iran. J. Sci. Tech. Trans. Sci. 43(5), 2585–2593 (2019)

    Article  MathSciNet  Google Scholar 

  17. Rahman, S., Mursaleen, M., Acu, A.M.: Approximation properties of \(\lambda \)-Bernstein-Kantorovich operators with shifted knots. Math. Meth. Appl. Sci. 42(11), 4042–4053 (2019)

    Article  MathSciNet  Google Scholar 

  18. Phillips, G.M.: Bernstein polynomials based on the \(q\)-integers. Annals Numer. Math. 4, 511–518 (1997)

    MathSciNet  MATH  Google Scholar 

  19. Pinkus, A.: Spectral properties of totally positive kernels and matrices. In: Gasca, M., Micchelli, C.A. (eds.) Total positivity and its applications, pp. 477–511. Kluwer Acadamic Publication, Dordrecht (1996)

    Chapter  Google Scholar 

  20. Rajan, K., Abbott, L.F.: Eigenvalue spectra of random matrices for neural networks. Phys. Rev. Lett. 97, 188104 (2006)

    Article  Google Scholar 

  21. Verma, S., Krishna, J.P.: Image compression and linear algebra. www.cmi.ac.in/ksutar/NLA2013/imagecompression.pdf

  22. Ward, M.: A calculus of sequences. Am. J. Math. 58, 255–266 (1936)

    Article  MathSciNet  Google Scholar 

  23. Weierstrass, K.: Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu, Berlin 2(633–639), 789–805 (1885)

    MATH  Google Scholar 

Download references

Funding

Not applicable

Author information

Authors and Affiliations

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

    Ambreen Naaz & M. Mursaleen

  2. Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan

    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

Contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to M. Mursaleen.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article has been retracted. Please see the retraction notice for more detail: https://dx.doi.org/10.1007/s13324-021-00546-9

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Naaz, A., Mursaleen, M. RETRACTED ARTICLE: On eigenstructure of q-Bernstein operators. Anal.Math.Phys. 11, 6 (2021). https://doi.org/10.1007/s13324-020-00443-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13324-020-00443-7

Keywords

Mathematics Subject Classification

Search

Navigation