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.
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)
Compound linear and positive approximation operators; polynomial sequences of binomial type; integral representation of remainder.
M. Crăciun, On compound operators depending on s parameters, Rev. Anal. Numér. Théor. Approx., vol. 33 (2004) no. 1, pp.51-60.
Google Scholar Profile
 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.
 Craciun, M., Approximation operators constructed by means of Sheffer sequences, Rev. Anal. Numer. Theor. Approx., 30, no. 2, pp. 135–150, 2001.
 Craciun, M., On an approximating operator and its Lipschitz constant, Rev. Anal. Numer. Theor. Approx., 31, no. 1, pp. 55–60, 2002.
 Craciun, M., On compound operators constructed with binomial and Sheffer sequences, Rev. Anal. Numer. Theor. Approx., 32, no. 2, pp. 135–144, 2003.
 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.
 Lupas, L. and Lupas, A. Polynomials of binomial type and approximation operators, Studia Univ. Babes-Bolyai, Mathematica, 32, 4, pp. 61–69, 1987.
 Lupas, A., Approximation operators of binomial type, Proc. IDoMAT 98, International Series of Numerical Mathematics, ISNM vol. 132, Birkhauser Verlag, Basel, pp. 175–198, 1999.
 Manole, C., Expansions in series of generalized Appell polynomials with applications to the approximation of functions, PhD Thesis, “Babes-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—Bra¸sov, 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 poynomes binomiaux, Bul. Soc. Stiinte 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.
 Sablonniere, P. , Positive Bernstein-Sheffer Operators, J. Approx. Theory, 83, pp. 330–341, 1995.
 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. Babes-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, Birkh¨auser-Verlag, Basel, pp. 187–203, 1972.
 Stancu, D. D., Quadrature formulas constructed by using certain linear positive operators, Numerical Integration, Proc. Conf. Oberwolfach, 1981 ISNM vol. 57, Birkh¨auserVerlag, Basel, pp. 241–251, 1982.
 Stancu, D. D., Approximation of functions by means of a new generalized Bernstein operator, Calcolo, 20, no. 2, pp. 211–229, 1983.
 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. Babes-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. Numer. Theor. Approx., 30, no. 1, pp. 95 -105, 2001.
 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 S/α/m,r,s, 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. Numer. Theor. Approx., 27, no. 1, pp. 167–181, 1998.
 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, Bucuresti, (1) 51, no. 3, pp. 411–419, 1999.