On compound operators depending on s parameters

Authors

  • Maria Crăciun Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy, Romania

DOI:

https://doi.org/10.33993/jnaat331-758

Keywords:

compound linear and positive approximation operators, polynomial sequences of binomial type, integral representation of remainder
Abstract views: 194

Abstract

In this note we introduce a compound operator depending on \(s\) parameters using binomial sequences. We compute the values of this operator on the test functions, we give a convergence theorem and a representation of the remainder in the corresponding approximation formula. We also mention some special cases of this operator.

Downloads

Download data is not yet available.

References

Agratini, O., On simultaneous approximation by Stancu-Bernstein operators, Approximation and Optimization, ICAOR Cluj-Napoca, vol. II, pp. 157-162, 1996.

Agratini, O., Binomial polynomials and their applications in approximation theory, Conf. Semin. Mat. Univ. Bari, 281, pp. 1-22, 2001.

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.

Brass, H., Eine Verallgemeinerung der Bernsteinschen 0peratoren, Abhandl. Math. Sem. Univ. Hamburg, 36, pp. 111-122, 1971, https://doi.org/10.1007/bf02995913 DOI: https://doi.org/10.1007/BF02995913

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 an approximating operator and its Lipschitz constant, Rev. Anal. Numér. Théor. Approx., 31, no. 1, pp. 55-60, 2002, http://ictp.acad.ro/jnaat/journal/article/view/2002-vol31-no1-art7

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

Di Bucchianico, A., Polynomials of convolution type, PhD thesis, University of Groningen, The Netherlands, 1991.

Di Bucchianico, A., Probabilistic and Analytical Aspects of the Umbral Calculus, CWI Tract 119, 1997.

Di Bucchianico, A. and Loeb, D.E., A selected survey of umbral calculus. Electron. J. Combin., 2, Dynamic Survey 3, 1995.

Lupaş, L. and Lupaş, A. Polynomials of binomial type and approximation operators, Studia Univ. Babeş-Bolyai, Mathematica, 32, 4, pp. 61-69, 1987.

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., Expansions in series of generalized Appell polynomials with applications to the approximation of functions, PhD Thesis, "Babeş-Bolyai" University, Cluj-Napoca, 1984 (in Romanian).

Manole, C., Approximation operators of binomial type, Univ. of Cluj-Napoca, Research Seminars, Seminar on numerical and statistical calculus, Preprint nr. 9, pp. 93-98, 1987.

Mihesan, V., Lipschitz constants for operators of binomial type of a Lipschitz continuous function. RoGer 2000-Braşov, pp. 81-87, Schr.reihe Fachbereichs Math. Gerhard Mercator Univ., 485, Gerhard-Mercator-Univ., Duisburg, 2000.

Mullin, R. and Rota, G.-C., On the foundations of combinatorial theory III, Theory of binomial enumeration, Graph Theory and its Applications, Academic Press, New York, 1970, pp. 167-213.

Popoviciu, T., Remarques sur les poynômes binomiaux, Bul. Soc. Ştiinte Cluj, 6, pp. 146-148, 1931.

Roman, S., The umbral calculus, Pure and Applied Mathematics, 111, Academic Press, Inc., New York, 1984, X+193 pp.

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

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 some new classes of positive linear operators, Numerische Methoden der Approximationstheorie, Proc. Conf. Oberwolfach 1971 ISNM vol. 16, Birkhäuser-Verlag, Basel, pp. 187-203, 1972, https://doi.org/10.1007/978-3-0348-5952-3_17 DOI: https://doi.org/10.1007/978-3-0348-5952-3_17

Stancu, D. D., Quadrature formulas constructed by using certain linear positive operators, Numerical Integration, Proc. Conf. Oberwolfach, 1981 ISNM vol. 57, Birkhäuser-Verlag, Basel, pp. 241-251, 1982, https://doi.org/10.1007/978-3-0348-6308-7_23 DOI: https://doi.org/10.1007/978-3-0348-6308-7_23

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, pp. 153-157, 1984.

Stancu, D. D., Bivariate approximation by some Bernstein-type operators, Proc. Colloq. Approx. Optim., Cluj-Napoca, pp. 25-34, 1984.

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., 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., On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu, Rev. Anal. Numér. Théor. Approx., 30, no. 1, pp. 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, 2002.

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 Occorsio, M. R., On approximation by binomial operators of Tiberiu Popoviciu type, Rev. Anal. Numér. Théor. Approx., 27, no. 1, pp. 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.

Stancu, D. D. and Vernescu, A., Approximation of bivariate functions by means of a class of operators of Tiberiu Popoviciu type, Mathematical Reports, Bucureşti, (1) 51, no. 3, pp. 411-419, 1999.

Downloads

Published

2004-02-01

How to Cite

Crăciun, M. (2004). On compound operators depending on s parameters. Rev. Anal. Numér. Théor. Approx., 33(1), 51–60. https://doi.org/10.33993/jnaat331-758

Issue

Section

Articles