×

Pointwise estimate for linear combinations of Bernstein-Kantorovich operators. (English) Zbl 1028.47012

The authors provide an equivalence theorem for certain linear combinations of Bernstein-Kantorovich operators which characterizes the pointwise order of approximation of a continuous function \(f\in C[0,1]\) in terms of the \(t\)-order of the Ditzian-Totik-type modulus of smoothness \(\omega^{2 r}_{\varphi^\lambda}(t)\). The theorem unifies some order of approximation results characterized in terms of classical and Ditzian-Totik moduli of smoothness.

MSC:

47A58 Linear operator approximation theory
41A10 Approximation by polynomials
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ditzian, Z.; Totik, V., Moduli of Smoothness (1987), Springer-Verlag: Springer-Verlag New York · Zbl 0666.41001
[2] Ditzian, Z., Direct estimate for Bernstein polynomials, J. Approx. Theory, 79, 165-166 (1994) · Zbl 0814.41005
[3] Ditzian, Z.; Jiang, D., Approximation of function by polynomials in \(C\)[−1,1], Canad. J. Math., 44, 924-940 (1992) · Zbl 0798.41012
[4] Ditzian, Z., Rate of approximation of linear processes, Acta Sci. Math., 48, 103-128 (1985) · Zbl 0608.41011
[5] Guo, S.; Li, C.; Sun, Y.; Yang, G.; Yue, S., Pointwise estimate for Szász-type operators, J. Approx. Theory, 94, 160-171 (1998) · Zbl 0911.41013
[6] Guo, S.; Yue, S.; Li, C.; Yang, G.; Sun, Y., A pointwise approximation theorem for linear combinations of Bernstein polynomials, Abstr. and Appl. Anal., 1, 397-406 (1996) · Zbl 0939.41011
[7] Guo, S.; Qi, Q., Pointwise estimates for Bernstein-type operators, Studia Sci. Math. Hungar., 35, 237-246 (1999) · Zbl 0931.41006
[8] Kantorovich, L., Sur certains developpements suivant les polynomes de la forme de S. Bernstein, I.II, C. R. Acad. Sci. URSS, 563-568 (1930) · JFM 57.1393.02
[9] Zhou, D. X., On multivariate Kantorovich operators in \(L_p\), Anal. Math., 19, 85-100 (1993)
[10] Guo, S.; Liu, L.; Song, Z., Steckin-Marchaud-type inequalities in connection with Bernstein-Kantorovich polynomials, Northeast. Math. J., 16, 319-328 (2000) · Zbl 1014.41010
[11] Xie, L., Uniform approximation by combinations of Bernstein polynomials, Approx. Theory Appl., 11, 36-51 (1995) · Zbl 0841.41022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.