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On the rate of convergence of Bezier variant of Szász-Durrmeyer operators. (English) Zbl 1062.41024
Let \(p_{n,k}(x):=e^{-nx}(nx)^k/k!\), and \(J_{n,k}(x):=\sum_{j=k}^\infty p_{n,j}(x)\), \(n,k\geq0\). The authors set \(Q^{(\alpha)}_{n,k}(x):=J^\alpha_{n,k}(x)-J^\alpha_{n,k+1}(x)\), \(n,k\geq0\), and introduce the following Durrmeyer variant of the Szász approximation operators on \([0,\infty)\), associated with a function \(f\in L[0,\infty)\). Namely, \[ M_{n,\alpha}(f,x):=n\sum_{k=0}^\infty Q^{(\alpha)}_{n,k}(x)\int_0^\infty p_{n,k}(t)f(t)\,dt,\quad n=0,1,2,\dots. \] If \(\alpha=1\), then the operators reduce to those defined by S. M. Mazhar and V. Totik [Acta Sci. Math. (Szeged), 49, 257–269 (1985; Zbl 0611.41013)]. Under the assumptions that \(f\) is of bounded variation in every finite subinterval, and satisfies a growth condition \(| f(x)| \leq Ke^{\beta x}\), the authors obtain pointwise estimates on how close \(M_{n,\alpha}(f,x)\) is to the average \((f(x+)+\alpha f(x-))/(\alpha+1)\).

MSC:
41A36 Approximation by positive operators
41A25 Rate of convergence, degree of approximation
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[1] Chang, F., On The Rate of Convergence of the Szasz-Mirakyan Operators for Functions of Bounded Variation, J. Approx. Theory, 40 (1984), 226–241. · Zbl 0532.41026
[2] Gupta, V. and Agrawal, P. N., An Estimate on the Rate of Convergence for Modified Szasz-Mirakyan Operators of Functions of Bounded Variation. Publ. Inst. Math. (Beograd)(N.S.), 49: 63 (1991), 97–103. · Zbl 0738.41025
[3] Gupta, V. and Pant, R. P., Rate of Convergence for the Modified Szasz-Mirakyan Operators on Functions of Bounded Variation, J. Math. Anal. Appl. 233 (1999), 476–483. · Zbl 0931.41012
[4] Gupta, V., Gupta, P. and Rogalski, M., Improved Rate of Convergence for the Modified Szasz-Mirakyan Operators. Approx. Theory and Its Appl., 16: 3 (2000), 94–99. · Zbl 1030.41014
[5] Gupta, V. and Agrawal, P. N., Inverse Theorem in Simultaneous Approximation by Szasz Durrmeyer Operators, J. Indian Math. Soc., 6: 31 (1996), 27–41.
[6] Gupta, V., Gupta, P. and Agrawal, P. N., Saturation Theorem in Simultaneous Approximation by Szasz Durrmeyer Operators, J. Indian Math. Soc., 67(1–4) (2000). · Zbl 1104.41009
[7] Kasana, H. S., Prasad, G., Agrawal, P. N. and Sahai, A., Modified Szasz Operators, Proceedings International Conference on Mathematical Analysis and Its Applications, Kuwait (1985), 29–41, Pergamon Press, Oxford.
[8] Mazhar, S. M. and Totik, V., Approximation by Modified Szasz Operators, Acta Sci. Math. (Szeged), 49 (1985), 257–269. · Zbl 0611.41013
[9] Sahai, A. and Prasad, G., On the Rate of Convergence for Modified Szasz-Mirakyan Operators, on Functions of Bounded Variation, Publ. Inst. Math. (Beograd) (N.S.), 53: 67 (1993), 73–80. · Zbl 0795.41019
[10] Zeng, X. M., On the Rate of Convergence of the Generalized Szasz Type Operators for Functions of Bounded Variation, J. Math. Anal. Appl., 226 (1998), 309–325. · Zbl 0915.41016
[11] Zeng, X. M. and Chen, W., On the Rate of Convergence of the Generalized Durrmeyer Type Operators for Functions of Bounded Variation, J. Approx. Theory, 102 (2000), 1–12. · Zbl 0956.41013
[12] Zeng, X. M. and Zhao J. N., Exact Bounds for Some Basic Functions of Approximation Operators, J. Inequal. Appl., 6 (2001), 563–575. · Zbl 0991.41016
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