On compound operators constructed with binomial and Sheffer sequences

Maria Crăciun

doi:10.33993/jnaat322-742

Authors

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

DOI:

https://doi.org/10.33993/jnaat322-742

Keywords:

sequences of binomial type, Sheffer sequences, compound operators

Abstract views: 250

Abstract

In this note we consider a general compound approximation operator using binomial sequences and we give a representation for its corresponding remainder term. We also introduce a more general compound approximation operator using Sheffer sequences. We provide convergence theorems for both studied operators.

Downloads

Download data is not yet available.

References

Agratini, O., On a certain class of approximation operators, Pure Math. Appl., 11, no. 2, pp. 119-127, 2001.

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

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 approximation 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

Di Bucchianico, A., Probabilistic and analytical aspects of the umbral calculus, CWI Tract, 119, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1997.

Di Bucchianico, A. and Loeb, D., A selected survey of umbral calculus, Electron. J. Combin., 2, Dynamic Survey 3, 28 pp. (electronic), 1995.

Hildebrand, F. B., Introduction to numerical analysis, Second ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974.

Lupaş, A., Approximation operators of binomial type, Proc. IDoMAT 98, New developments in approximation theory, International Series of Numerical Mathematics, 132, Birkhäuser Verlag, pp. 175-198, 1999. DOI: https://doi.org/10.1007/978-3-0348-8696-3_12

Lupaş, L. and Lupaş, A., Polynomials of binomial type and approximation operators, Studia Univ. Babeş-Bolyai, Mathematica, 32, no. 4, pp. 40-44, 1987.

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

Manole, C., Approximation operators of binomial type, Research Seminar on Numerical and Statistical Calculus, Preprint no. 9, pp. 93-98, 1987.

Miheşan, 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.

Moldovan, G., Sur la convergence de certains opérateurs convolutifs positifs, C. R. Acad. Sci. Paris Sér. A, 272, pp. 1311-1313, 1971 (in French).

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

Rota, G.-C., with the collaboration of P. Doubilet, C. Greene, D. Kahaner, A. Odlyzko and R. Stanley, Finite operator calculus, Academic Press, New York-London, 1975.

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. and Mond, B., The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. USA, 60, pp. 1196-2000, 1968, https://doi.org/10.1073%2Fpnas.60.4.1196. DOI: https://doi.org/10.1073/pnas.60.4.1196

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., 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., Representation of remainders in approximation formulae by some discrete type linear positive operators, Proceedings of the Third International Conference on Functional Analysis and Approximation Theory, Vol. II (Acquafredda di Maratea, 1996), Rend. Circ. Mat. Palermo (2) Suppl. No. 52, Vol. II, pp. 781-791, 1998.

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 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, Proceedings of the Fourth International Conference on Functional Analysis and Approximation Theory (Acquafredda di Maratea, 2000), Rend. Circ. Mat. Palermo (2) 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.

Steffensen, J. F., Interpolation, Chelsea, New York, 1927.

Steffensen, J. F., The poweroid, an extension of the mathematical notion of power, Acta Math., 73, pp. 333-366, 1941, https://doi.org/10.1007/BF02392231. DOI: https://doi.org/10.1007/BF02392231