Approximation operators constructed by means of Sheffer sequences

Maria Crăciun

doi:10.33993/jnaat302-692

Authors

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

DOI:

https://doi.org/10.33993/jnaat302-692

Keywords:

positive linear operators, umbral calculus, delta operator, invertible shift invariant operator, binomial operators of T. Popoviciu, Sheffer sequences

Abstract views: 241

Abstract

In this paper we introduce a class of positive linear operators by using the "umbral calculus", and we study some approximation properties of it. Let \( Q \) be a delta operator, and \(S\) an invertible shift invariant operator. For \(f\in C[0,1]\) we define \begin{equation*} (L_{n}^{Q,S}f)(x)= {\textstyle\frac{1}{s_{n}(1)}} \sum\limits_{k=0}^{n}{\textstyle\binom{n}{k}}p_{k}(x)s_{n-k}(1-x)f\left( \tfrac{k}{n} \right), \end{equation*} where \((p_{n})_{n\geq0}\) is a binomial sequence which is the basic sequence for \(Q,\) and \((s_{n})_{n\geq0}\) is a Sheffer set, \( s_{n}=S^{-1}p_{n} \). These operators generalize the binomial operators of T. Popoviciu.

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References

Cheney, E. W. and Sharma, A., On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma, 5, pp. 77-82, 1964.

Gonska, H. H. and Kovacheva, R. K., The second order modulus revisited: remarks, applications, problems, Conferenze del Seminario di Matematica Univ. Bari, 257, pp. 1-32, 1994.

Lupaş, L. and Lupaş, A., Polynomials of binomial type and approximation operators, Studia Univ. Babes-Bolyai, Mathematica, 32, pp. 61-69, 1987.

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

Manole, C., Developments in series of generalized Appell polynomials, with applications to the approximation of functions, Ph.D. Thesis, Cluj-Napoca, Romania, 1984 (in Romanian).

Moldovan, G., Generalizations of the S. N. Bernstein operators, Ph.D. Thesis, Cluj-Napoca, Romania, 1971 (in Romanian).

Moldovan, G., Discrete convolutions and positive operators I, Annales Univ. Sci. Budapest R. Eötvös, 15, pp. 31-34, 1972.

Mullin, R. and Rota, G. C., On the foundations of combinatorial theory III, theory of binomial enumeration, in: B. Harris, ed., Graph Theory and Its Applications, Academic Press, New York, pp. 167-213, 1970.

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

Roman, S., Operational formulas, Linear and Multilinear Algebra, 12, pp. 1-20, 1982, https://doi.org/10.1080/03081088208817466 DOI: https://doi.org/10.1080/03081088208817466

Rota, G. C., Kahaner, D. and Odlyzko, A., Finite operator calculus, J. Math. Anal. Appl., 42, pp. 685-760, 1973, https://doi.org/10.1016/0022-247x(73)90172-8 DOI: https://doi.org/10.1016/0022-247X(73)90172-8

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. U.S.A., 60, pp. 1196-1200, 1968, https://doi.org/10.1073/pnas.60.4.1196 DOI: https://doi.org/10.1073/pnas.60.4.1196

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

Stancu, D. D., On a generalization of the Bernstein polynomials, Studia Univ. Babeş--Bolyai, Cluj, 14, pp. 31-45, 1969.

Stancu, D. D., Approximation properties of a class of linear positive operators, Studia Univ. Babeş-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 16, Birkhäuser-Verlag, Basel, pp. 187-203, 1972, https://doi.org/10.1007/978-3-0348-5952-3_17 DOI: https://doi.org/10.1007/978-3-0348-5952-3_17

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.

Stancu, D. D. and Cismaşiu, C., On an approximating linear positive operator of Cheney--Sharma, Rev. Anal. Numér. Théor. Approx., 26, pp. 221-227, 1997.