On a compound approximation operator of D.D. Stancu type
DOI:
https://doi.org/10.33993/jnaat351-1008Keywords:
compound linear and positive approximation operators, representation of remainderAbstract
In this note we consider a linear and positive compound approximation operator of D.D. Stancu type depending of several parameters; we give the expressions of this operator on the test functions, the conditions under which this operator converges to a given continuous function, an estimate of the order of approximation using the moduli of continuity and an integral representation of the remainder. Also, by using Stancu's method we find an expression for the remainder using divided differences of second order for a special case of this operator.Downloads
References
Altomare, F. and Campiti, M., Korovkin-type approximation theory and its applications. Appendix A by Michael Pannenberg and Appendix B by Ferdinand Beckhoff. de Gruyter Studies in Mathematics, 17. Walter de Gruyter & Co., Berlin, 1994.
Cao, F. Modulus of continuity, K-functional and Stancu operator on a simplex, Indian J. Pure Appl. Math., 35, no. 12, 1343-1364, 2004.
Crăciun, M., Approximation operators constructed by means of Sheffer sequences, Rev. Anal. Numér. Théor. Approx., 30, no. 2, pp. 135-150, 2001, http://ictp.acad.ro/jnaat/journal/article/view/2001-vol30-no2-art3
Crăciun, M., On compound operators constructed with binomial and Sheffer sequences, Rev. Anal. Numér. Théor. Approx., 32, no. 2, pp. 135-144, 2003, http://ictp.acad.ro/jnaat/journal/article/view/2003-vol32-no2-art2
Crăciun, M., On compound operators depending on s parameters, Rev. Anal. Numér. Théor. Approx., 33, no. 1, pp. 51-60, 2004, http://ictp.acad.ro/jnaat/journal/article/view/2004-vol33-no1-art6
Gonska, H.H. and Kovacheva, R.K., The second order modulus revisited: remarks, applications, problems, Conf. Semin. Mat. Univ. Bari, 257, pp. 1-32, 1994.
Lupaş, A., Approximation operators of binomial type, Proc. IDoMAT 98, International Series of Numerical Mathematics, ISNM vol. 132, Birkhäuser Verlag, Basel, pp. 175-198, 1999, https://doi.org/10.1007/978-3-0348-8696-3_12 DOI: https://doi.org/10.1007/978-3-0348-8696-3_12
Manole, C., Approximation operators of binomial type, Univ. of Cluj-Napoca, Research Seminars, Seminar on numerical and statistical calculus, Preprint nr. 9, 1987, 93-98.
Popoviciu, T., Remarques sur les polynômes binomiaux, Bul. Soc. Ştiinte Cluj, 6, 146-148, 1931.
Popoviciu, T., Sur le reste dans certaines formules lineaires d'approximation de l'analyse, Mathematica, Cluj, 1(24), 95-142, 1959.
Rota, G.C., Kahaner, D. and Odlyzko, A., Finite Operator Calculus, J. Math. Anal. Appl. 42, pp. 685-760, 1973, https://doi.org/10.1016/0022-247x(73)90172-8 DOI: https://doi.org/10.1016/0022-247X(73)90172-8
Sablonnière, P., Positive Bernstein-Sheffer Operators, J. Approx. Theory, 83, pp. 330-341, 1995, https://doi.org/10.1006/jath.1995.1124 DOI: https://doi.org/10.1006/jath.1995.1124
Shisha, O., Mond, B., The degree of convergence of linear and positive operators, Proc. Nat. Acad. Sci. U.S.A., 60, pp. 1196-1200, 1968, https://doi.org/10.1073/pnas.60.4.1196 DOI: https://doi.org/10.1073/pnas.60.4.1196
Stancu, D.D., Approximation of functions by a new class of linear positive operators, Rev. Roum. Math. Pures et Appl., 13, pp. 1173-1194, 1968.
Stancu, D.D., Use of probabilistic methods in the theory of uniform approximation of continuous functions, Rev. Roumaine Math. Pures Appl., 14 pp. 673-691, 1969.
Stancu, D.D., Approximation properties of a class of linear positive operators, Studia Univ. Babeş-Bolyai, Cluj, 15, pp. 31-38, 1970.
Stancu, D.D., Approximation of functions by means of a new generalized Bernstein operator, Calcolo, 20, no. 2, pp. 211-229, 1983, https://doi.org/10.1007/bf02575593 DOI: https://doi.org/10.1007/BF02575593
Stancu, D.D., A note on a multiparameter Bernstein-type approximating operator, Mathematica (Cluj), 26(49), no. 2, 153-157, 1984.
Stancu, D.D., A note on the remainder in a polynomial approximation formula, Studia Univ. Babeş-Bolyai Math., 41, no. 2, pp. 95-101, 1996.
Stancu, D.D., The remainder in the approximation by a generalized Bernstein operator: a representation by a convex combination of second-order divided differences, Calcolo, 35, 53-62, 1998, https://doi.org/10.1007/s100920050008 DOI: https://doi.org/10.1007/s100920050008
Stancu, D.D., Representation of remainders in approximation formulae by some discrete type linear positive operators, Rendiconti del Circolo Matematico di Palermo, Suppl., 52, pp. 781-791, 1998.
Stancu, D.D., On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu, Rev. Anal. Numér. Théor. Approx., 30, no. 1, 95-105, 2001, http://ictp.acad.ro/jnaat/journal/article/view/2001-vol30-no1-art13
Stancu, D. D., On approximation of functions by means of compound poweroid operators, Mathematical Analysis and Approximation Theory, Proceedings of ROGER 2002-Sibiu, pp. 259-272.
Stancu, D.D., and Drane, J.W., Approximation of functions by means of the poweroid operators Sm,r,s,α, Trends in approximation theory (Nashville, TN, 2000), pp. 401-405, Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 2001.
Stancu, D.D. and Giurgescu, P., On the evaluation of remainders in some linear approximation formulas, RoGer 2000-Braşov, 141-147, Schrreihe Fachbereichs Math. Gerhard Mercator Univ., 485, Gerhard-Mercator-Univ., Duisburg, 2000.
Stancu, D.D. and Occorsio, M.R., On approximation by binomial operators of Tiberiu Popoviciu type, Rev. Anal. Numér. Théor. Approx. 27, no. 1, 167-181, 1998, http://ictp.acad.ro/jnaat/journal/article/view/1998-vol27-no1-art17
Stancu, D.D. and Simoncelli, A. C., Compound poweroid operators of approximation, Rendiconti del Circolo Matematico di Palermo, Suppl. 68, pp. 845-854, 2002.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2015 Journal of Numerical Analysis and Approximation Theory
This work is licensed under a Creative Commons Attribution 4.0 International License.
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
