Skip to main content
SpringerLink
Log in

On the Variation-Diminishing Property

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In the present paper we give a new approach in proving the common variation-diminishing property for operators of a certain form, and we apply this method in order to show that some well-known positive linear operators have this property. The introduction includes several historical remarks; in it also a first attempt is made to draw clear lines between the various meanings of ”variation-diminuition” which were employed in the past.

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

  1. P.J. Barry, R.N. Goldman: Recursive polynomial curve schemes and Computer Aided Geometric Design, Constr. Approx. 6 (1990), 65–96.

    Article  MathSciNet  MATH  Google Scholar 

  2. H. Berens, R.A. DeVore: A characterization of Bernstein polynomials. In: ”Approximation Theory III” (Proc. Int. Sympos. Austin 1980; ed. by E. W. Cheney), 213–219. New York: Acad. Press 1980.

    Google Scholar 

  3. H. Berens, Y. Xu: On Bernstein-Durrmeyer polynomials with Jacobi weights. In: ”Approximation Theory and Functional Analysis” (ed. by C.K. Chui) 25–46. Boston: Acad. Press 1991.

    Google Scholar 

  4. C. de Boor, R.A. DeVore: A geometric proof of total positivity for spline interpolation, Math. Comp. 172 (1985), 497–504.

    Article  MathSciNet  Google Scholar 

  5. B. Delia Vecchia: On the approximation of functions by means of the operators of D.D. Stancu, Studia Univ. Babeş-Bolyai, Mathematica 37 (1992), 3–36.

    MATH  Google Scholar 

  6. R.A. DeVore, G.G. Lorentz: Constructive Approximation. Berlin: Springer-Verlag1993.

    Book  Google Scholar 

  7. R.N. Goldman: Polya’s urn model and Computer Aided Geometric Design, SIAM J. Algebraic Discrete Meth. 6 (1985), 1–28.

    Article  MATH  Google Scholar 

  8. H.H. Gonska, J. Meier: Quantitative theorems on approximation by Bernstein-Stancu operators, Calcolo 21 (1984), 317–335.

    Article  MathSciNet  MATH  Google Scholar 

  9. G. Farin: Curves and Surfaces for Computer Aided Geometric Design. New York: Acad. Press1988.

    Google Scholar 

  10. A. Lupaş: Die Folge der Betaoperatoren. Dissertation, Univ. Stuttgart, 1972.

  11. D.H. Mache: Gewichtete Simultanapproximation in der Lp-Metrik durch das Verfahren der Kantorovič Operatoren. Dissertation, Univ. Dortmund, 1991.

  12. D.H. Mache: A link between Bernstein polynomials and Durrmeyer polynomials with Jacobi weights. In: ”Approximation Theory VIII” (ed. by C.K. Chui and L.L. Schnmaker), World Scientific Publishing Co. (1995), 403-410.

  13. J. Meier: Zur Approximation durch gewöhnliche und modifizierte Bernstein-Operatoren unter besonderer Berücksichtigung quantitativer Aussagen und asymptotischer Formeln. Staatsexamensarbeit, Univ. Duisburg 1982.

  14. R. Păltănea: Sur un opérateur polynomial defini sur l’ensemble des fonctions intégrables. In: ”Itinerant Seminar on Functional Equations, Approximation and Convexity” (Cluj-Napoca, 1983), 101-106. Preprint 83-2, Univ. Babeş-Bolyai, Cluj-Napoca, 1983.

  15. G. Pólya, I.J. Schoenberg: Remarks on de la Vallée Poussin means and convex conformai maps of the circle, Pacific J. Math. 8 (1958), 295–334.

    Article  MathSciNet  MATH  Google Scholar 

  16. T. Popoviciu: Sur l’approximation des fonctions convexes d’ordre supérieur, Mathematica (Cluj) 10 (1934), 49–54.

    Google Scholar 

  17. I.J. Schoenberg: On variation-diminishing approximation methods. In: ”On Numerical Approximation” (Proc. Sympos. Conducted by the MRC; ed. by R.E. Langer), 249–274. Madison, Wisconsin: Univ. of Wisconsin Press 1959.

    Google Scholar 

  18. I.J. Schoenberg: On spline functions. In: ”Inequalities” (Proc. Sympos. Wright-Patterson Air Force Base Ohio 1965; ed. by O. Shisha), 255–291. New York: Acad. Press 1967.

    Google Scholar 

  19. D.D. Stancu: Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl. 13 (1968), 1173–1194.

    MathSciNet  MATH  Google Scholar 

  20. D.D. Stancu: Approximation of functions by means of some new classes of positive linear operators. In: ”Numerische Methoden der Approximationstheorie” (Proc. Conf. Math. Res. Inst. Oberwolfach 1971; ed. by L. Collatz, G. Meinardus), 187–203. Basel: Birkhäuser 1972.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Technical University of Cluj-Napoca, RO - 3400, Cluj-Napoca, Romania

    Ioan Gavrea & Daniela P. Kacsó

  2. Department of Mathematics, University of Duisburg, D - 47048, Duisburg, Germany

    Heinz H. Gonska

Authors
  1. Ioan Gavrea
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Heinz H. Gonska
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daniela P. Kacsó
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to Professor Dr. D.D. Stancu on the occasion of his 70th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gavrea, I., Gonska, H.H. & Kacsó, D.P. On the Variation-Diminishing Property. Results. Math. 33, 96–105 (1998). https://doi.org/10.1007/BF03322074

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03322074

1991 Mathematics Subject Classification

En]Keywords

Search

Navigation