On an approximation operator and its Lipschitz constant

In this note we consider an approximation operator of Kantorovich type in which expression appears a basic sequence for a delta operator and a Sheffer sequence for the same delta operator.

We give a convergence theorem for this operator and we find its Lipschitz constant.

Maria Craciun
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

approximation operators of Kantorovich type; Sheffer sequences; Lipschitz constants.

https://ictp.acad.ro/craciun/papers/craciun-2002-ANTA.pdf

M. Crăciun, On an approximating operator and its Lipschitz constant, Rev. Anal. Numér. Théor. Approx., vol. 31 (2002), no. 1, 55-60.

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[1] Agratini, O., On a certain class of approximation operators, Pure Math. Appl., 11, pp. 119–127, 2000.

[2] Brown, B. M., Elliot, D. and Paget, D. F., Lipschitz constants for the Bernstein polynomials of a Lipschitz continuous function, J. Approx. Theory, 49, pp. 196–199, 1987.

[3] Craciun, M., Approximation operators constructed by means of Sheffer sequences, Rev. Anal. Numer. Theor. Approx., 30, 2001, pp. 135–150.

[4] Lupas, L. and Lupas, A., Polynomials of binomial type and approximation operators, Studia Univ. Babe¸s-Bolyai, Mathematica, 32, pp. 61–69, 1987.

[5] Manole, C., Approximation operators of binomial type, Univ. Cluj-Napoca, Research Seminar on Numerical and Statistical Calculus, Preprint no. 9, pp. 93–98, 1987.

[6] Mihesan, V., Approximation of continuous functions by means of linear positive operators, Ph.D. Thesis, Cluj-Napoca, 1997 (in Romanian).

[7] Moldovan, G., Discrete convolutions and linear positive operators, Ann. Univ. Sci. Budapest R. E¨otv¨os, 15, pp. 31–44, 1972.

[8] Popoviciu, T., Remarques sur les polynomes binomiaux, Bull. Soc. Math. Cluj, 6, pp. 146–148, 1931.

[9] Rota, G.-C., Kahaner, D. and Odlyzko, A., On the foundations of combinatorial theory. VIII. Finite operator calculus, J. Math. Anal. Appl., 42, pp. 684–760, 1973.

[10] Sablonniere, P., Positive Bernstein-Sheffer operators, J. Approx. Theory, 83, pp. 330–341, 1995.

[11] Stancu, D. D., Approximation of functions by a new class of linear positive operators, Rev. Roum. Math. Pures Appl., 13, pp. 1173–1194, 1968.

[12] 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, pp. 95–105, 2001.

[13] Stancu, D. D. and Occorsio, M. R., On approximation by binomial operators of Tiberiu Popoviciu type, Rev. Anal. Numer. Theor. Approx., 27, pp. 167–181, 1998.