On a class of approximation operators


In this note we construct a class of approximation operators using polynomial sequences of binomial type. We compute the expression of these operators on the test functions and we give a convergence theorem. We give also quantitative evaluations of the order of approximation by using the moduli of continuity of first and second order and we establish an integral representation of the remainder term in the case when the approximation formula has the degree of exactness one. Some special cases, almost all of them being studied previously by D.D. Stancu, are mentioned.


Maria Crăciun
”Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy


positive approximation operators; polynomial sequences of binomial type; evaluation of remainder


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Maria Crăciun, On a class of approximation operators, Mathematical Sciences Research, 2005, pp. 292-303



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[1] Agratini, O., Binomial polynomials and their applications in approximation theory,  Conf. Semin. Mat. Univ. Bari, no. 281 (2001), 1-22
[2] Altomare, F. and Campiti, M.,  Korovkin-tupe approximation theory and its applications. Appendix A by Michael Pannenberg and Apendix B by Ferdinand Beckhoff. de Gruyter Studies in Mathematics, 17. Walter de Gruyter ^ Co., Berlin, 1994.
[3] Crăciun, M.,  Approximation operators constructed by means of Sheffer sequences,  Rev. Anal. Numer. Theor. Approx. 30, no.2, pp. 135-150, 2001.
[4] Crăciun, M., On an approximating operator and its Lipschitz constant, Rev. Anal. Numer. Theor. Approx., 31, no. 1, pp. 55-60, 2002
[5] Crăciun, M., On compound operators constructed with binomial and Sheffer sequences,  Rev. Anal. Numer. Theor. Approx., 32, no. 2, pp. 135-144, 2003
[6] Di Bucchianico, A., Probabilistic and Analytical Aspects of the Umbral Calculus, CWI Tract 119, 1997, 148 pp.
[7] Di Buchianico, A. and Loeb, D.E., A selected survey of umbral calculus. Electron. J. Combin. 2 (1995),  Dynamic Survey 3, 28 pp.
[8] Gonska, H.H. and Kovacheva, R.K.,  The second order modulus revisited: remarks, applications, problems,  Conferenze del Seminario di Matematica Univ. Bari 257 (1994), 1-32
[9] Gonska, H.H. and Meier, J., Quantitative theorems on approximation by Bernstein-Stancu operators, Calcolo 21 (1984), no. 4, 317-335
[10]  Lupaș, L. and Lupaș, A.,  Polinomials of  binomial type and approximation operators,  Studia Univ. Babeș-Bolyai, Mathematica, 32, 4 (1987), 61-69
[11]  Lupaș, A., Approximation operators of binomial type, Proc. IDoMAT 98, International Series of Numerical Mathematics, ISNM vol. 132, Birkhäuser Verlag, Basel, 1999, 175-198
[12] Manole, C., Expansions in series of generalized Appell polynomials with applications to the approximation of functions,  PhD Thesis, ”Babeș-Bolyai” University, Cluj-Napoca, 1984 (in Romanian).
[13] Manole, D., Approximation operators of binomial type,  Univ. of Cluj-Napoca, Research Seminars, Seminar on numerical and statistical calcujlus, Preprint nr. 9, 1987, 93-98
[14] Mihesan, V.,  Lipschitz constant for operators of binomial type of a Lipschitz continuous function.  RoGer 2000-Brasov, 81-87, Schr. reihe Fachbereichs Math. Gerhard Mercator Univ., 485, Gerhard-Mercator-Univ., Duisburg, 2000.
[15] 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, 167-213.
[16] Popoviciu, T., Remarques sur les poynomes binomiaux,  Bul. Soc. Știinte Cluj, 6 (1931), 146-148
[17] Roman, S., The theory of the umbral calculus,  J. Math. Anal. Appl. 87 (1982), no. 1, 58-115.
[18] Roman, S., The umbral calculus. Pure and Applied Mathematics, 111. Academic Press, Inc., New York, 1984.x+193 pp.
[19] Rota, G.C., Kahaner, D. and Odlyzko, A.,  Finite Operator Calculus,  J. Math. Anal. Appl. 42 (1973), 685-760
[20] Sab lonniere, P.,  Positive Bernstein-Sheffer Operators,  J. Approx. Theory 83 (1995), 330-341
[21] Schurer,  F.,  Linear positive operators in approximation theory.  Math. Inst. Techn. Univ. Delft, Report, 1962
[22] Shisha, O. and Mond, B.,  The degree of convergence of linear positive operators,  Proc. Nat. Acad. Sci. U.S.A., 60, pp. 1196-2000, 1968
[23] 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
[24] Stancu, D.D.,  Approximation of functions by means of a new generalized Bernstein operator,  Calcolo, 20, no. 2, pp. 211-229, 1983.
[25]  Stancu, D.D., Generalized Bernstein approximating operators,  Itinerant seminar on functional equations, approximation and convexity
(Cluj-Napoca, 1984), pp. 185-192,  Preprint, 84-6, Univ. ”Babeș-Bolyai”, Cluj-Napoca, 1984
[26] Stancu, D.D., Bivariate approximation by some Bernstein-type operators,  Proc. Colloq. Approx. Optim., Cluj-Napoca, pp. 25-34, 1984
[27] Stancu, D.D., Representation of remainders in approximation formulae by some discrete type linear positive operators, Redinconti del Circolo Matematico di Palermo, Suppl. 52 (1998), pp. 781-791
[28] Stancu, D.D., A note on the remainder in a polynomial  approximation formula. Studia Univ. Babeș-Bolyai Math., 41, no. 2, pp. 95-101, 1996
[29] Stancu, D.D., Approximation properties of a class of multiparameter positive linear operators.  Approximation and optimization, Vol. I (Cluj-Napoca, 1996), 107-120, Transilvania, Cluj-Napoca, 1997.
[30] Stancu, D.D.,  On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu, Rev. Anal. Numer. Theor. Ap;prox. 30 (2001), no. 1, 95-105
[31] Stancu, D.D., On approximation of functions by means of compund poweroid operators,  Mathematical Analysis and Approximation Theory, Proceedings of ROGER 2002-Sibiu, pp. 259-272
[32] Stancu, D.D., and Drane, J.W., Approximation of functions by means of the poweroid operators  S_{m,r,s,}^{α}, Trends in approximation theory (Nashville, TN, 2000), pp. 401-405, Innov. Appl. Math., Vanderbilt Univ. Press. Nashville, TN, 2001
[33] Stancu, D.D.,  and Occorsio, M.R., On approximation by binomial operators of Tiberiu Popoviciu type, Rev. Anal. Numer. Theor. Approx. 27 (1998), no. 1, 167-181
[34] Stancu, D.D., and Vernescu, A., Approximation of bivariate functions by means of a class of operators of Tiberiu Popoviciu type,  Mathematical Reports, București, (1) 51 (1999), no. 3, 411-419
[35] Stancu, D.D. and Simoncelli, A. C., Compound poweroid operators of approximation,  Redinconti del Circolo Matematico di Palermo, Suppl. 68 (2002), pp. 845-854
[36] Vlaic, G.,  Approximation properties of a class of bivariate operators of D.D. Stancu,  Studia Universitatis Babeș-Bolyai, Mathematica, 42, no. 2, pp. 109-115, 1997


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