On Kantorovich type operators and Schoenberg integral splines (Q1081043)

In a previous paper [Chin. Ann. Math. 2, 243-256 (1981; Zbl 0466.41007)] the author has studied the asymptotic behavior of generalized Bernstein- Kantorovich polynomials. He now investigates the same problem for a spline operator \(T_{n,k}\) of Bernstein-Schoenberg-Kantorovich type, studied thoroughly by \textit{M. W. Müller} [Fourier analysis and approximation theory, Vol. II, Budapest 1976, Colloq. Math. Soc. János Bolyai 19, 565-570 (1978; Zbl 0398.41007); J. Approximation Theory 21, 385-393 (1977; Zbl 0371.41006)] and partly by \textit{K. Scherer} [Constructive theory of functions, Varna 1970, 277-286 (1972; Zbl 0232.41002)]. The main contribution of the author consists in a generalization of a result of Müller (Theorem 3 from the second paper cited above) concerning the order of approximation of a function \(f\in L_ p(I)\) (1\(\leq p\leq \infty\), \(I=[0,1])\) by \(T_{n,k}f\), using the integral modulus of continuity. \{Reviewer's remark: The reference to paper [1] from the bibliography is incorrect, because I. J. Schoenberg introduced his Berstein-type spline operator in 1967 in [12] and not in 1959 in [1]. Such a confusion has occurred in several other papers, including the MR review of the second Müller paper cited above, since in the fundamental paper of \textit{M. J. Marsden} [J. Approximation Theory 3, 7-49 (1970; Zbl 0192.421)] all the numbers, between [11]-[23], of the items cited in the text are wrongly decreased by one with respect to those in the bibliography.\}