We present direct and strong converse theorems for a general sequence of positive linear operators satisfying some functional equations. The results can be applied to some extensions of Baskakov and Szász–Mirakyan operators.
[1] Becker, M. 1978 Global approximation theorems for Szász–Mirakjan and Baskakov operators in polynomial weight spaces Indiana Math. J. 27 127–142 .
[2] Bustamante, J. 2008 Estimates of positive linear operators in terms of second-order moduli J. Math. Anal. Appl. 345 203–212 .
[3] Ditzian, Z. 1994 Direct estimate for Bernstein polynomials J. Approx. Theory 79 165–166 .
[4] Ditzian, Z. Ivanov, K. G. 1993 Strong converse inequalities J. Anal. Math. 61 61–111 .
[5] Ditzian, Z. Totik, V. 1987 Moduli of Smoothness Springer New York .
[6] Felten, M. 1998 Local and global approximation theorems for positive linear operators J. Approx. Theory 94 396–419 .
[7] Finta, Z. 2005 On converse approximation theorems J. Math. Anal. Appl. 312 159–180 .
[8] Guo, S. Tong, H. Zhang, G. 2002 Stechkin–Marchaud-type inequalities for Baskakov polynomials J. Approx. Theory 114 33–47 .
[9] Guo, S. Qi, Q. 2003 Strong converse inequalities for Baskakov operators J. Approx. Theory 124 219–231 .
[10] Haase, M. 2007 Convexity inequalities for positive operators Positivity 11 57–68 .
[11] Impens, Ch. Gavrea, I. 2002 A Leibniz differentiation formula for positive operators J. Math. Anal. Appl. 271 175–181 .
[12] Knoop, H. B. Zhou, X. L. 1994 The lower estimate for linear positive operators (II) Resultate Math. 25 315–330.
[13] Lan, Qi Qiu 2002 The strong converse inequalities for generalized Baskakov-type operators Pure Applied Math. 18 49 317–321 in Chinese.
[14] Sikkema, P. C. 1970 On some linear positive operators Indag. Math. 32 327–337.
[15] Totik, V. 1994 Strong converse inequalities J. Approx. Theory 76 369–375 .
[16] Totik, V. 1994 Approximation by Bernstein polynomials Amer. J. Math. 116 995–1018 .
[17] Volkov, Yu. I. 1978 Certain positive linear operators Math. Notes 23 363–368 .