Commutativity and spectral properties of genuine Baskakov-Durrmeyer type operators and their \(k\)th order Kantorovich modification

Margareta Heilmann

doi:10.33993/jnaat442-1078

Authors

Margareta Heilmann University of Wuppertal, Germany

DOI:

https://doi.org/10.33993/jnaat442-1078

Keywords:

Positive linear operators, Durrmeyer type operators, Kantorovich type modification, commutativity, differential operators, spectral properties.

Abstract views: 356

Abstract

In this paper we present an overview of commutativity results and different methods for the proofs for Baskakov-Durrmeyer type operators and associated differential operators. We discuss the spectral properties and generalize all results to \(k\)th order Kantorovich modifications and corresponding Durrmeyer type variants of Bleimann, Butzer and Hahn operators and Meyer-Konig and Zeller operators.

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References

Abel, U., An identity for a general class of approximation operators, J. Approx. Theory, 142, pp. 20–35, 2006, http://doi.org/10.1016/j.jat.2006.03.005 DOI: https://doi.org/10.1016/j.jat.2006.03.005

Abel, U., Gupta, V. and Ivan, M., The complete asymptotic expansion for a general Durrmeyer variant of the Meyer-Konig and Zeller operators Math. Comput. Modelling 40, no. 7-8, pp. 867–875, 2004, http://doi.org/10.1016/j.mcm.2004.10.016 DOI: https://doi.org/10.1016/j.mcm.2004.10.016

Abel, U. and Ivan, M., Durrmeyer variants of the Bleimann, Butzer and Hahn operators, Mathematical analysis and approximation theory, Burg, Sibiu, pp. 1–8, 2002.

Abel, U. and Ivan, M., Enhanced asymptotic approximation ans approximation of truncated functions by linear operators, Constructive Theory of Functions, Proceedings of the International Conference on Constructive Theory of Functions, Varna, June 2 - June 6, 2005, B. D. Bojanov (Ed.), Prof. Marin Drinov Academic Publishing House, pp. 1–10, 2006.

Baskakov V. A., An instance of a sequence of positive linear operators in the space of continuous functions, Doklady Akademii Nauk SSSR, 113:2, pp. 249–251, 1957.

Baumann, K., Heilmann, M. and Ras¸a, I., Further results for k-th order Kantorovich modification of linking Baskakov type operators, Results Math., 2015, http://doi.org/10.1007/s00025-015-0511-x DOI: https://doi.org/10.1007/s00025-015-0511-x

Berdysheva, E., Studying Baskakov-Durrmeyer operators and quasi-interpolants via special functions, J. Approx. Theory, 149, no. 2, pp. 131–150, 2007, http://doi.org/10.1016/j.jat.2007.04.009 DOI: https://doi.org/10.1016/j.jat.2007.04.009

Berdysheva, E., Jetter, K. and St¨ockler, J. New polynomial preserving operators on simplices: direct results, J. Approx. Theory 131, no. 1, pp. 59–73, 2004, http://doi.org/10.1016/j.jat.2004.09.004 DOI: https://doi.org/10.1016/j.jat.2004.09.004

Berdysheva, E., Jetter, K. and Stockler, J., Bernstein-Durrmeyer type quasiinterpolants on intervals, Approximation Theory: a Volume dedicated to Borislav Bojanov, Prof. M. Drinov Acad. Publ. House, Sofia, pp. 32–42,2004.

Berens, H. and Xu, Y. On Bernstein-Durrmeyer polynomials with Jacobi weights, Approximation theory and functional analysis (College Station, TX, 1990), Academic Press, Boston, MA, pp. 25–46,1991.

Berens, H. and Xu, Y., On Bernstein-Durrmeyer polynomials with Jacobi-weights: the cases p = 1 and p = 1, Approximation Interpolation and Summability (Ramat Aviv, 1990/Ramat Gan, 1990), Israel Math. Conf. Proc., 4, Bar-Ilan Univ., Ramat Gan, pp. 51–62,1991.

Chen, W. On the modified Durrmeyer-Bernstein operator, (handwritten, Chinese, 3 pages), Report of the Fifth Chinese Conference on Approximation Theory, Zhen Zhou, China (1987).

Derriennic, M.-M., Sur l’approximation de fonctions integrables sur [0, 1] par des polynomes de Bernstein modifies, J. Approx. Theory 31, no. 4, pp. 325–343, 1981, http://doi.org/10.1016/0021-9045(81)90101-5 DOI: https://doi.org/10.1016/0021-9045(81)90101-5

Derriennic, M.-M., On multivariate approximation by Bernstein-type polynomials, J. Approx. Theory 45, no. 2, pp. 155-166, 1985, http://doi.org/10.1016/0021-9045(85)90043-7 DOI: https://doi.org/10.1016/0021-9045(85)90043-7

Ditzian, Z., Multidimensional Jacobi-type Bernstein-Durrmeyer operators, Acta Sci. Math. (Szeged) 60, no. 1-2, pp. 225–243, 1995.

Ditzian, Z. and Ivanv, K., Bernstein-type operators and their derivatives, J. Approx. Theory 56 , no. 1, pp. 72–90, 1989, http://doi.org/10.1016/0021-9045(89)90134-2 DOI: https://doi.org/10.1016/0021-9045(89)90134-2

Durrmeyer, J. L., Une formule d’inversion de la transforme de Laplace: applications a la theorie des moments, These de 3e cycle, Faculte des Sciences de l’Universite de Paris, 1967.

Goodman, T. N. T. and Sharma, A., A modified Bernstein-Schoenberg operator, Constructive theory of functions (Varna, 1987), Publ. House Bulgar. Acad. Sci., Sofia, pp. 166–173,1988.

Goodman, T. N. T. and Sharma, A., A Bernstein type operator on the simplex, Math. Balkanica (N.S.) 5 , no. 2, pp. 129–145, 1991.

Heilmann, M., Approximation auf [0,1) durch das Verfahren der Operatoren vom Baskakov-Durrmeyer Typ, Dissertation, Universitat Dortmund, 1987.

Heilmann, M., Commutativity of operators from Baskakov-Durrmeyer type Constructive theory of functions (Varna, 1987), Publ. House Bulgar. Acad. Sci., Sofia, pp. 197–206, 1988.

Heilmann, M., Direct and converse results for operators of Baskakov-Durrmeyer type, Approx. Theory Appl. 5 , no. 1, pp. 105–127, 1989.

Heilmann, M., Erhöhung der Konvergenzgeschwindigkeit bei der Approximation von Funktionen mit Hilfe von Linearkombinationen spezieller positiver linearer Operatoren, Habilitationschrift Universität Dortmund, 1992.

Heilmann, M., Eigenfunctions of Durrmeyer-type modifications of Meyer-Konig and Zeller operators, J. Approx. Theory 125 , no. 1, pp. 63–73, 2003, http://doi.org/10.1016/j.jat.2003.09.006 DOI: https://doi.org/10.1016/j.jat.2003.09.006

Heilmann, M., Rodriguez-type representation for the eigenfunctions of Durrmeyer-type operators, Results Math. 44, no. 1-2, pp. 97–105, 2003, http://doi.org/10.1007/bf03322916 DOI: https://doi.org/10.1007/BF03322916

Heilmann, M., Commutativity of Durrmeyer-type modifications of Meyer-Konig and Zeller and Baskakov-operators, Constructive theory of functions, DARBA, Sofia, pp. 295–301, 2003.

Heilmann, M. and Rasa, I., k-th order Kantorovich type modification of the operators U_n, J. Appl. Funct. Anal. 9, no. 3-4, pp. 320–334, 2014.

Heilmann, M. and Rasa, I., k-th order Kantorovich modification of linking Baskakov type operators, Recent Trends in Mathematical Analysis and its Applications, Rorkee, India, December 2014, (ed. P. N. Agrawal et al.), Springer Proceedings in Mathematics & Statistics, Vol. 143, pp. 229-242, 2015, http://doi.org/10.1007/978-81-322-2485-3_18 DOI: https://doi.org/10.1007/978-81-322-2485-3_18

Heilmann, M. and Tachev, G., Commutativity, direct and strong converse results for Phillips operators, East J. Approx. 17, no. 3, pp. 299–317, 2011.

Heilmann, M. and Wagner, M., The genuine Bernstein-Durrmeyer operators and quasi-interpolants, Results Math. 62 , no. 3-4, pp. 319–335, 2012, http://doi.org/10.1007/s00025-012-0247-9 DOI: https://doi.org/10.1007/s00025-012-0247-9

Lupas, A., Die Folge der Betaoperatoren, Dissertation, Universität Stuttgart 1972.

Mazhar, S. M. and Totik, V., Approximation by modified Szasz operators, Acta Sci. Math. (Szeged) 49 , no. 1-4, pp. 257–269, 1985.

Paltanea, R., Sur un operateur polynomial defini sur l’ensemble des fonctions integrables, “Babes-Bolyai” Univ., Fac. Math., Res. Semin. 2, pp. 101–106, 1983.

Phillips, R. S., An inversion formula for Laplace transforms and semi-groups of linear operators, Ann. of Math. (2) 59, pp. 325–356, 1954. DOI: https://doi.org/10.2307/1969697

Sahai, A. and Prasad, G., On simultaneous approximation by modified Lupas operators, J. Approx. Theory 45 , no. 2, pp. 122–128, 1985, http://doi.org/10.1016/0021-9045(85)90039-5 DOI: https://doi.org/10.1016/0021-9045(85)90039-5

Wagner, M., Quasi-Interpolanten zu genuinen Baskakov-Durrmeyer-Typ Operatoren, Disssertation Bergische Universität Wuppertal, 2013.