The Second Ditzian-Totik modulus revisited: refined estimates for positive linear operators

Heinz H. Gonska; Gancho T. Tachev

doi:10.33993/jnaat321-733

Authors

Heinz H. Gonska University of Duisburg, Germany
Gancho T. Tachev University of Architecture, Sofia, Bulgaria

DOI:

https://doi.org/10.33993/jnaat321-733

Keywords:

Bernstein operator, Ditzian-Totik moduli of smoothness, best constants, piecewise linear interpolation, Bernstein-Stancu operator, Gavrea operator

Abstract views: 237

Abstract

Direct theorems for approximation by positive linear operators in terms of the second order Ditzian-Totik modulus of smoothness are proved. Special emphasis is on the magnitude of the absolute constants. New results are obtained for Bernstein operators, for piecewise linear interpolation, for general Bernstein-Stancu operators and for those of Gavrea.

Downloads

Download data is not yet available.

References

Brass, H., Eine Verallgemeinerung der Bernsteinschen Operatoren, Abh. Math. Sem. Univ. Hamburg, 36, pp. 111-122, 1971. DOI: https://doi.org/10.1007/BF02995913

Brudnyı, Yu. A., On a certain method for approximation of bounded functions given on a segment, in: "Studies Contemporary Problems Constructive Theory of Functions", Proc. Second All-Union Conf. Baku, 1962; ed. by I. I. Ibragimov, Izdat. Akad. Nauk Azerbaidzan. SSR, Baku, pp. 40-45, 1965.

Burkhill, H., Cesàro-Perron almost periodic functions, Proc. London Math. Soc. (Series 3), 2, pp. 150-174, 1952, https://doi.org/10.1112/plms/s3-2.1.150. DOI: https://doi.org/10.1112/plms/s3-2.1.150

Cao, J.-D., On linear approximation methods, Acta Sci. Natur. Univ. Fudan, 9, pp. 43--52, 1964 (in Chinese).

DeVore, R. A. and Lorentz, G. G., Constructive Approximation, Springer, Berlin, 1993. DOI: https://doi.org/10.1007/978-3-662-02888-9

Ditzian, Z., A global inverse theorem for combinations of Bernstein polynomials, J. Approx. Theory, 26, pp. 277-292, 1979, http://www.dx.doi.org/10.1016/0021-9045(79)90065-0 DOI: https://doi.org/10.1016/0021-9045(79)90065-0

Ditzian, Z., On interpolation of L_{p}[a,b] and weighted Sobolev spaces, Pacific J. Math., 90, pp. 307-323, 1980, http://www.dx.doi.org/10.2140/pjm.1980.90.307 DOI: https://doi.org/10.2140/pjm.1980.90.307

Ditzian, Z. and Ivanov, K. G., Strong converse inequalities, J. Anal. Math., 61, pp. 61-111, 1993, http://www.dx.doi.org/10.1007/BF02788839 DOI: https://doi.org/10.1007/BF02788839

Ditzian, Z. and Totik, V., Moduli of Smoothness, Springer, New York, 1987, http://www.dx.doi.org/10.1007/978-1-4612-4778-4 DOI: https://doi.org/10.1007/978-1-4612-4778-4

Ditzian, Z. and Zhou, X.-L., Kantorovich-Bernstein polynomials, Constr. Approx., 6, pp. 421-435, 1980, http://www.dx.doi.org/10.1007/BF01888273 DOI: https://doi.org/10.1007/BF01888273

Dzafarov, A. S., Averaged moduli of continuity and some of their connections with best approximation, Dokl. Akad. Nauk SSSR, 236, no. 2, pp. 288-291, 1977 (in Russian).

Felten, M., Direct and inverse estimates for Bernstein polynomials, Constr. Approx., 14, pp. 459-468, 1998, http://www.dx.doi.org/10.1007/s003659900084 DOI: https://doi.org/10.1007/s003659900084

Gavrea, I., The approximation of the continuous functions by means of some linear positive operators, Resultate Math., 30, pp. 55-66, 1996, http://www.dx.doi.org/10.1007/BF03322180 DOI: https://doi.org/10.1007/BF03322180

Gavrea, I., Gonska, H. and Kacsó, D., Variation on Butzer's problem: characterization of the solutions, Computers Math. Appl., 34, pp. 51-64, 1997, http://www.dx.doi.org/10.1016/S0898-1221(97)00188-0 DOI: https://doi.org/10.1016/S0898-1221(97)00188-0

Gonska, H., The second order modulus again: some (still?) open problems, Schriftenreihe des Fachbereichs Mathematik, Univ. Duisburg, Germany, SM-DU-426, 1998.

Gonska, H. and Kovacheva, R., The second order modulus revisited: remarks, applications, problems, Confer. Sem. Mat. Univ. Bari, 257, pp. 1-32, 1994.

Gonska, H. and Meier, J., A bibliography on approximation of functions by Bernstein-type operators (1955-1982), in: "Approximation Theory IV", Proc. Int. Sympos. College Station 1983, ed. by C. K. Chui et al., Acad. Press, New York, pp. 739-785, 1983.

Gonska, H. and Meier, J., A bibliography on approximation of functions by Bernstein-type operators (Supplement 1986), in: "Approximation Theory V", Proc. Int. Sympos. College Station 1986, ed. by C. K.Chui et al., Acad. Press, New York, pp. 621-654, 1986.

Ivanov, K. G., On Bernstein Polynomials, C. R. Acad. Bulg., 35, pp. 893--896, 1982. DOI: https://doi.org/10.1016/0021-9681(82)90120-5

Ivanov, K. G., A constructive characterization of the best algebraic approximation in Lp[-1,1], in: "Constructive Function Theory '81", Proc. Int. Conf. Varna 1981, ed. by Bl. Sendov et al., Sofia, Publishing House of the Bulgarian Academy of Sciences, pp. 357-367, 1983.

Ivanov, K. G., A characterization of weighted Peetre K-functionals, J. Approx. Theory, 56, pp. 185-211, 1989, https://doi.org/10.1016/0021-9045(89)90109-3. DOI: https://doi.org/10.1016/0021-9045(89)90109-3

Kacsó, D. P., Discrete Jackson-type operators via a Boolean sum approach, J. Comp. Anal. Appl., 3 no. 4, pp. 399-413, https://doi.org/10.1023/A:1012054425159. DOI: https://doi.org/10.1023/A:1012054425159

Knoop, H. B. and Zhou, X.-L., The lower estimate for linear positive operators, Schriftenreihe des Fachbereichs Mathematik, Univ. Duisburg, Germany, SM-DU-201, 1992.

Knoop, H. B. and Zhou, X.-L., The lower estimate for linear positive operators (II), Results Math., 25, pp. 315-330, 1994, https://doi.org/10.1007/BF03323413. DOI: https://doi.org/10.1007/BF03323413

Păltănea, R., L'estimation de l'approximation des fonctions continues par les opérateurs de Brass, Itinerant Seminar of Functional Equations, Approximation and Convexity (Cluj-Napoca), pp. 123-126, 1984, Preprint 84-6, Univ. Babeş-Bolyai, Cluj-Napoca, 1984.

Păltănea, R., L'estimation de l'approximation des derivées d'ordre r par les polynomes de Brass, Itinerant Seminar on Functional Equations, Approximation and Convexity (Cluj-Napoca), pp. 207--210, 1986, Univ. Babeş-Bolyai, Cluj-Napoca, 1986.

Păltănea, R., A generalization of the operators of H. Brass, Manuscript, 1993.

Păltănea, R., Best constants in estimates with second order moduli of continuity, in: "Approximation Theory", Proc. Int. Dortmund Meeting IDoMAT 95, ed. by M. W. Müller et al., Akademie-Verlag, Berlin, pp. 251-275, 1995.

Păltănea, R., On the degree of approximation by Bernstein operators, in: "RoGer 98", Proc. 3rd Romanian-German Seminar on Approximation Theory, Sibiu 1998, ed. by A. Lupaş et al., Universitatea "Lucian Blaga", Sibiu, pp. 53-56, 1998.

Popoviciu, T., Curs de Analiză Matematică, Partea III-a: Continuitate, Cluj-Napoca: Babeş-Bolyai University, 1974 (in Romanian).

Potapov, M. K., On approximation by Jacobi polynomials, Vestnik Moskov. Univ., Ser. I Mat. Mekh., no. 5, pp. 70-82, 1977 (in Russian).

Stancu, D. D., Approximation of functions by means of some new classes of positive linear operators, in: "Numerische Methoden der Approximationstheorie", Vol. 1, Proc. Conf. Math. Res. Inst. Oberwolfach 1971, ed. by L. Collatz and G. Meinardus, Birkhäuser, Basel, pp. 187-203, 1972. DOI: https://doi.org/10.1007/978-3-0348-5952-3_17

Stancu, D. D., Generalized Bernstein approximating operators, Itinerant Seminar on Functional Equations, Approximation and Convexity, University of Cluj-Napoca, pp. 185-192, 1984.

Stancu, D. D., Probabilistic approach to a class of generalized Bernstein approximating operators, Rev. Anal. Numér. Théor. Approx, 14, pp. 83--89, 1984.

Stancu, D. D., Approximation properties of a class of multiparameter positive linear operators, in: "Approximation and Optimization", Proc. Int. Conf. on Approximation and Optimization (Romania)-ICAOR; ed. by D. D. Stancu et al., Transilvania Press, Cluj-Napoca, vol. 1, pp. 203-218, 1997.

Totik, V., Approximation by Bernstein polynomials, American J. Math., 116, pp. 995-1018, 1994, https://doi.org/10.2307/2375007. DOI: https://doi.org/10.2307/2375007

Zhou, X.-L., Approximation of integrable functions by Baskakov-type operators, Master Thesis, Hangzhou University, 1981 (in Chinese).

Zhou, X.-L., On Bernstein polynomials, Acta Math. Sinica, 28, pp. 848-855, 1985 (in Chinese).