A converse result for approximation by a weighted Szász-Mirakyan operator
In a former paper [Acta Math. Hung. 111, No.~4, 325--345 (2006; Zbl 1174.41327)] the authors introduced some Szász-Mirakyan-type operators to approximate functions with singularities at the endpoints of the positive semiaxis and proved weighted error estimates involving a new weighted modulus of smoothness. In this article, they complete these results and prove a theorem which states the equivalence of the weighted error and the modulus of smoothness. This result is analogous to previous results for Bernstein-type operators obtained by \textit{Z. Ditzian and K. Ivanov} [J. Approximation Theory 56, No.~1, 72--90 (1989; Zbl 0692.41021)] and \textit{J. Szabados} [in: Mache, Detlef H. (ed.) et al., Trends and applications in constructive approximation. Papers of the 4th IBoMAT meeting, Witten-Bommerholz, Germany, February 15--19, 2004. Basel: Birkhäuser. ISNM. International Series of Numerical Mathematics 151, 247--258 (2005; Zbl 1080.41022)]
scientific article; zbMATH DE number 5355108