On a compound approximation operator of D.D. Stancu type

Abstract

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.

Authors

Maria Crăciun

Maria Craciun
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Keywords

Compound linear and positive approximation operators, representation of
remainder.
Compound linear and positive approximation operators, representation of remainder.

References

[1] 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.
[2] Cao, F. Modulus of continuity, K-functional and Stancu operator on a simplex, Indian J. Pure
Appl. Math., 35, no. 12, 1343–1364, 2004.
[3] Craciun, M. ˘ , Approximation operators constructed by means of Sheffer sequences, Rev. Anal.
Num´er. Th´eor. Approx., 30, no. 2, pp. 135–150, 2001
40 Maria Cr˘aciun 8
[4] Craciun, M., ˘ On compound operators constructed with binomial and Sheffer sequences, Rev.
Anal. Num´er. Th´eor. Approx., 32, no. 2, pp. 135–144, 2003.
[5] Craciun, M., ˘ On compound operators depending on s parameters, Rev. Anal. Num´er. Th´eor.
Approx., 33, no. 1, pp. 51–60, 2004.
[6] 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.
[7] Lupas¸, A., Approximation operators of binomial type, Proc. IDoMAT 98, International Series
of Numerical Mathematics, ISNM vol. 132, Birkh¨auser Verlag, Basel, pp. 175–198, 1999.
[8] 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.
[9] Popoviciu, T., Remarques sur les polynˆomes binomiaux, Bul. Soc. S¸tiinte Cluj, 6, 146–148,
1931.
[10] Popoviciu, T., Sur le reste dans certaines formules lineaires d’approximation de l’analyse,
Mathematica, Cluj, 1(24), 95–142, 1959.
[11] Rota, G.C., Kahaner, D. and Odlyzko, A., Finite Operator Calculus, J. Math. Anal. Appl.
42, pp. 685–760, 1973.
[12] Sablonniere, P. ` , Positive Bernstein-Sheffer Operators, J. Approx. Theory, 83, pp. 330–341,
1995.
[13] 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.
[14] 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.
[15] 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.
[16] Stancu, D.D., Approximation properties of a class of linear positive operators, Studia Univ.
Babe¸s-Bolyai, Cluj, 15, pp. 31–38, 1970.
[17] Stancu, D.D., Approximation of functions by means of a new generalized Bernstein operator,
Calcolo, 20, no. 2, pp. 211–229, 1983.
[18] Stancu, D.D., A note on a multiparameter Bernstein-type approximating operator, Mathematica
(Cluj), 26(49), no. 2, 153–157, 1984.
[19] Stancu, D.D., A note on the remainder in a polynomial approximation formula, Studia Univ.
Babe¸s-Bolyai Math., 41, no. 2, pp. 95–101, 1996.
[20] 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.
[21] 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.
[22] Stancu, D.D., On the approximation of functions by means of the operators of binomial type
of Tiberiu Popoviciu, Rev. Anal. Num´er. Th´eor. Approx., 30, no. 1, 95–105, 2001.
[23] 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.
[24] Stancu, D.D., and Drane, J.W., Approximation of functions by means of the poweroid operators
S
αm,r,s, Trends in approximation theory (Nashville, TN, 2000), pp. 401–405, Innov. Appl.
Math., Vanderbilt Univ. Press, Nashville, TN, 2001.
[25] Stancu, D.D. and Giurgescu, P., On the evaluation of remainders in some linear approximation
formulas, RoGer 2000—Bra¸sov, 141–147, Schrreihe Fachbereichs Math. Gerhard Mercator
Univ., 485, Gerhard-Mercator-Univ., Duisburg, 2000.
[26] Stancu, D.D. and Occorsio, M.R., On approximation by binomial operators of Tiberiu Popoviciu
type, Rev. Anal. Num´er. Th´eor. Approx. 27, no. 1, 167–181, 1998.
[27] Stancu, D.D. and Simoncelli, A. C., Compound poweroid operators of approximation, Rendiconti
del Circolo Matematico di Palermo, Suppl. 68, pp. 845–854, 2002.

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Cite this paper as:
M. Crăciun, On a compound approximation operator of D.D. Stancu type, Rev. Anal. Numér. Théor. Approx., 35 (2006), 33-40.

M. Crăciun, On a compound approximation operator of D.D. Stancu type, Rev. Anal. Numér. Théor. Approx., 35 (2006), 33-40.

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Rev. Anal. Numér. Théor. Approx.

Publisher Name

Academia Republicii S.R.

DOI

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1222-9024

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[1] 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.
[2] Cao, F. Modulus of continuity, K-functional and Stancu operator on a simplex, Indian J. Pure Appl. Math., 35, no. 12, 1343–1364, 2004.
[3] Craciun, M., Approximation operators constructed by means of Sheffer sequences, Rev. Anal. Numer. Theor. Approx., 30, no. 2, pp. 135–150, 2001.
[4] Craciun, M., On compound operators constructed with binomial and Sheffer sequences, Rev. Anal. Numer. Theor. Approx., 32, no. 2, pp. 135–144, 2003.
[5] Craciun, M., On compound operators depending on s parameters, Rev. Anal. Numer. Theor. Approx., 33, no. 1, pp. 51–60, 2004.
[6] 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.
[7] 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.
[8] 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.
[9] Popoviciu, T., Remarques sur les polynomes binomiaux, Bul. Soc. S¸tiinte Cluj, 6, 146–148, 1931.
[10] Popoviciu, T., Sur le reste dans certaines formules lineaires d’approximation de l’analyse, Mathematica, Cluj, 1(24), 95–142, 1959.
[11] Rota, G.C., Kahaner, D. and Odlyzko, A.,Finite Operator Calculus, J. Math. Anal. Appl. 42, pp. 685–760, 1973.
[12] Sablonniere, P.,Positive Bernstein-Sheffer Operators, J. Approx. Theory, 83, pp. 330–341, 1995.
[13] 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.
[14] 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.
[15] 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.
[16] Stancu, D.D.,Approximation properties of a class of linear positive operators, Studia Univ. Babes-Bolyai, Cluj, 15, pp. 31–38, 1970.
[17] Stancu, D.D.,Approximation of functions by means of a new generalized Bernstein operator, Calcolo, 20, no. 2, pp. 211–229, 1983.
[18] Stancu, D.D.,A note on a multiparameter Bernstein-type approximating operator, Mathematica (Cluj), 26(49), no. 2, 153–157, 1984.
[19] 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.
[20] 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.
[21] 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.
[22] 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, 95–105, 2001.
[23] 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.
[24] Stancu, D.D., and Drane, J.W.,Approximation of functions by means of the poweroid operators Sα m,r,s, Trends in approximation theory (Nashville, TN, 2000), pp. 401–405, Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 2001.
[25] Stancu, D.D. and Giurgescu, P., On the evaluation of remainders in some linear approximation formulas, RoGer 2000—Bra¸sov, 141–147, Schrreihe Fachbereichs Math. Gerhard Mercator Univ., 485, Gerhard-Mercator-Univ., Duisburg, 2000.
[26] Stancu, D.D. and Occorsio, M.R.,On approximation by binomial operators of Tiberiu Popoviciu type, Rev. Anal. Numer. Theor. Approx. 27, no. 1, 167–181, 1998.
[27] Stancu, D.D. and Simoncelli, A. C.,Compound poweroid operators of approximation, Rendiconti del Circolo Matematico di Palermo, Suppl. 68, pp. 845–854, 2002.

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