Durrmeyer type operators on a simplex

Year 2021, , 215 - 228, 01.06.2021

Radu Paltanea

https://doi.org/10.33205/cma.862942

Cited By: 6

Abstract

The paper contains the definition and certain approximation properties of a sequence of Durrmeyer-type operators on a simplex, which preserve affine functions and make a link between the multidimensional "genuine" Durrmeyer operators and the multidimensional Bernstein operators.

Keywords

Multidimensional linear positive operators, Durrmeyer type operators, Bernstein operators on a simplex, limit operators, estimates with moduli of continuity

References

• T. Acar, A. Aral and I. Raşa: Modified Bernstein-Durrmeyer operators, Gen. Math., 22 (1) (2014), 27-41.
• F. Altomare, M. Campiti: Korovkin-type approximation theory and its applications, Walter de Gruyter, Berlin-New York (1994).
• A. Attalienti: Generalized Bernstein-Durrmeyer operators and the associated limit semigroup, J. Approx. Theory, 99 (1999), 289-309.
• E. Berdysheva, K. Jetter: Multivariate Bernstein–Durrmeyer operators with arbitrary weight functions, J. Approx. Theory, 162 (2010), 576-598.
• H. Berens, Y. Xu: On Bernstein-Durrmeyer polynomials with Jacobi weights, Approximation Theory and Functional Analysis, (College Station, TX, 1990), 25–46, Academic Press, Boston (1991).
• W. Z. Chen: On the modified Bernstein-Durrmeyer operator, In Report of the Fifth Chinese Conference on Approximation Theory, Zhen Zhou, China (1987).
• M. M. Derriennic: Sur l’approximation de fonctions intégrables sur [0, 1] par des polynômes de Bernstein modifies, J. Approx. Theory, 31 (1981), 325–343.
• M. M. Derriennic: On multivariate approximation by Bernstein-type polynomials, J. Approx. Theory, 45 (2) (1985), 155–166.
• Z. Ditzian: Multidimensional Jacobi-type Bernstein–Durrmeyer operators, Acta Sci. Math., (Szeged) 60 (1995), 225–243.
• S. Durrmeyer: Une formule d’inversion de la transforme de Laplace: Application a la theorie de moments, Dissertation, These de 3e cycle, Faculté de Sci. de Univ. Paris, (1967).
• I. Gavrea: The approximation of the continuous functions by means of some linear positive operators, Result. Math., 30 (1-2) (1996), 55-66.
• H. Gonska, R. Pâltânea: Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions, Czech. Math. J., 60 (135) (2010), 783–799.
• H. Gonska, I. Raşa and E.- D. Stânilâ: The eigenstructure of operators linking the Bernstein and the genuine BernsteinDurrmeyer operators, Mediterr. J. Math., 11 (2014), 561-576.
• T. N. T. Goodman, A. Sharma: A modified Bernstein-Schoenberg operator, Proc. of the Conference on Constructive Theory of Functions, Varna (1987) (ed. by Bl. Sendov et al.). Sofia: Publ. House Bulg. Acad. of Sci., (1988), 166-173.
• V. Gupta, G. Tachev: Approximation with positive linear operators and linear combinations, Springer, (2017).
• B. Li: Approximation by a class of modified Bernstein-Durrmeyer operators, Approx. Th. Appl., 10 (1994), 32-44.
• A. Lupa¸s: Die Folge der Betaoperatoren, Dissertation, Universität Stuttgart (1972).
• D. H. Mache, D.X. Zhou: Characterization theorems for the approximation by a family of operators, J. Approx. Theory, 84 (1996), 145–161.
• P. E. Parvanov, B. D. Popov: The limit case of Bernstein’s operators with Jacobi weights, Math. Balkanica (N.S.), 8 (1994), 165–177.
• R. Pâltânea: Sur un operateur polynômial défini sur l’ensemble des fonctions intégrables, Babes Bolyai Univ., Fac. Math., Res. Semin., 2 (1983), 101–106.
• R. Pâlt3anea: Une propriété d’extrémalité des valeurs propres des opérateurs polynômiaux de Durrmeyer généralisés, L’Analyse Numér. et la Theor. de l’Approx., 15 (1986), 57–64.
• R. Pâltânea: On a limit operator. Proc. of the "Tiberiu Popoviciu" Itinerant Seminar of Functional Equations, Approximation and Convexity, (ed. by E. Popoviciu), Srima Press, Cluj-Napoca (2001), 169-179.
• R. Pâltânea: Approximation Theory Using Positive Linear Operators, Birkhäuser, Boston (2004).
• R. Pâltânea: A class of Durrmeyer type operators preserving linear functions, Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity, 5 (2007), 109–117.
• R. Pâltânea: Generalized Bernstein-Durrmeyer operators on a simplex, Gen. Math., 20 (5) (2012), 71-82.
• T. Sauer: The genuine Bernstein–Durrmeyer operator on a simplex, Results. Math., 26 (1–2) (1994), 99–130.
• L. Song: Some approximation theorems for modified Bernstein-Durrmeyer operators, Approx. Th. Appl., 10 (1994), 1-12.
• S. Waldron: A generalised beta integral and the limit of the Bernstein–Durrmeyer operator with Jacobi weights, J. Approx. Theory, 122 (2003), 141-150.