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Abstract and Applied Analysis
Volume 2011, Article ID 670509, 11 pages
http://dx.doi.org/10.1155/2011/670509
Research Article

Asymptotic Behaviour of the Iterates of Positive Linear Operators

Department of Mathematics, Technical University of Cluj-Napoca, Street Memorandumului 28, 400114 Cluj-Napoca, Romania

Received 4 December 2010; Accepted 20 March 2011

Academic Editor: Pavel Drábek

Copyright © 2011 Ioan Gavrea and Mircea Ivan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. P. Kelisky and T. J. Rivlin, “Iterates of Bernstein polynomials,” Pacific Journal of Mathematics, vol. 21, pp. 511–520, 1967. View at Google Scholar · View at Zentralblatt MATH
  2. S. Karlin and Z. Ziegler, “Iteration of positive approximation operators,” Journal of Approximation Theory, vol. 3, pp. 310–339, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and Its Applications, vol. 17 of de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, Germany, 1994, Appendix A by Michael Pannenberg and Appendix B by Ferdinand Beckhoff.
  4. H. H. Gonska and X. L. Zhou, “Approximation theorems for the iterated Boolean sums of Bernstein operators,” Journal of Computational and Applied Mathematics, vol. 53, no. 1, pp. 21–31, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. J. A. Adell, F. G. Badía, and J. de la Cal, “On the iterates of some Bernstein-type operators,” Journal of Mathematical Analysis and Applications, vol. 209, no. 2, pp. 529–541, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. H. Oruç and N. Tuncer, “On the convergence and iterates of q-Bernstein polynomials,” Journal of Approximation Theory, vol. 117, no. 2, pp. 301–313, 2002. View at Publisher · View at Google Scholar
  7. S. Ostrovska, “q-Bernstein polynomials and their iterates,” Journal of Approximation Theory, vol. 123, no. 2, pp. 232–255, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. J. P. King, “Positive linear operators which preserve x2,” Acta Mathematica Hungarica, vol. 99, no. 3, pp. 203–208, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. I. A. Rus, “Iterates of Bernstein operators, via contraction principle,” Journal of Mathematical Analysis and Applications, vol. 292, no. 1, pp. 259–261, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. U. Itai, “On the eigenstructure of the Bernstein kernel,” Electronic Transactions on Numerical Analysis, vol. 25, pp. 431–438, 2006. View at Google Scholar · View at Zentralblatt MATH
  11. H. Gonska, D. Kacsó, and P. Piţul, “The degree of convergence of over-iterated positive linear operators,” Journal of Applied Functional Analysis, vol. 1, no. 4, pp. 403–423, 2006. View at Google Scholar · View at Zentralblatt MATH
  12. H. Gonska and I. Raşa, “The limiting semigroup of the Bernstein iterates: degree of convergence,” Acta Mathematica Hungarica, vol. 111, no. 1-2, pp. 119–130, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. H. Gonska, P. Piţul, and I. Raşa, “Over-iterates of Bernstein-Stancu operators,” Calcolo, vol. 44, no. 2, pp. 117–125, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. H.-J. Wenz, “On the limits of (linear combinations of) iterates of linear operators,” Journal of Approximation Theory, vol. 89, no. 2, pp. 219–237, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. O. Agratini, “On the iterates of a class of summation-type linear positive operators,” Computers & Mathematics with Applications, vol. 55, no. 6, pp. 1178–1180, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. F. Galaz Fontes and F. J. Solís, “Iterating the Cesàro operators,” Proceedings of the American Mathematical Society, vol. 136, no. 6, pp. 2147–2153, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. U. Abel and M. Ivan, “Over-iterates of Bernstein's operators: a short and elementary proof,” American Mathematical Monthly, vol. 116, no. 6, pp. 535–538, 2009. View at Publisher · View at Google Scholar
  18. C. Badea, “Bernstein polynomials and operator theory,” Results in Mathematics, vol. 53, no. 3-4, pp. 229–236, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. I. Rasa, “Asymptotic behaviour of certain semigroups generated by differential operators,” Jaen Journal on Approximation, vol. 1, no. 1, pp. 27–36, 2009. View at Google Scholar · View at Zentralblatt MATH
  20. I. Gavrea and M. Ivan, “On the iterates of positive linear operators preserving the affine functions,” Journal of Mathematical Analysis and Applications, vol. 372, no. 2, pp. 366–368, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. I. Rasa, “C0 semigroups and iterates of positive linear operators: asymptotic behaviour,” Rendiconti del Circolo Matematico di Palermo, vol. 2, supplement 82, pp. 123–142, 2010. View at Google Scholar
  22. I. Gavrea and M. Ivan, “On the iterates of positive linear operators,” Journal of Approximation Theory. In press. View at Publisher · View at Google Scholar
  23. H. Bauer, “Šilovscher Rand und Dirichletsches Problem,” Université de Grenoble. Annales de l'Institut Fourier, vol. 11, pp. 89–136, 1961. View at Google Scholar · View at Zentralblatt MATH
  24. R. R. Phelps, Lectures on Choquet's Theorem, D. Van Nostrand, Princeton, NJ, USA, 1966.
  25. W. Meyer-König and K. Zeller, “Bernsteinsche potenzreihen,” Studia Mathematica, vol. 19, pp. 89–94, 1960. View at Google Scholar · View at Zentralblatt MATH
  26. E. W. Cheney and A. Sharma, “Bernstein power series,” Canadian Journal of Mathematics, vol. 16, pp. 241–252, 1964. View at Google Scholar · View at Zentralblatt MATH
  27. M. Becker and R. J. Nessel, “A global approximation theorem for Meyer-König and Zeller operators,” Mathematische Zeitschrift, vol. 160, no. 3, pp. 195–206, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. P. C. Sikkema, “On the asymptotic approximation with operators of Meyer-König and Zeller,” Indagationes Mathematicae, vol. 32, pp. 428–440, 1970. View at Google Scholar · View at Zentralblatt MATH
  29. C. P. May, “Saturation and inverse theorems for combinations of a class of exponential-type operators,” Canadian Journal of Mathematics, vol. 28, no. 6, pp. 1224–1250, 1976. View at Google Scholar · View at Zentralblatt MATH
  30. J. M. Aldaz, O. Kounchev, and H. Render, “Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces,” Numerische Mathematik, vol. 114, no. 1, pp. 1–25, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. D. D. Stancu, “Approximation of functions by a new class of linear polynomial operators,” Revue Roumaine de Mathématiques Pures et Appliquées, vol. 13, pp. 1173–1194, 1968. View at Google Scholar · View at Zentralblatt MATH
  32. E. W. Cheney and A. Sharma, “On a generalization of Bernstein polynomials,” Rivista di Matematica della Universita' Degli Studi di Parma, vol. 5, pp. 77–84, 1964. View at Google Scholar · View at Zentralblatt MATH