On linear positive operators of the Jackson type

In this paper we consider the approximation of functions f(x) E Cn by means of sequences of linear positive operators {L n, ¿ } (n = 1, 2, . .) of the Jackson type. Section 1 is introductory; some well-known definitions and theorems are formulated and some lemmas are proved which are important for the rest of the paper. The explicit form of the operators considered is given in 2. This has been done b MATSUOKA [131 for slightly more general operators in a rather complicated way. Error estimates in terms of the modulus of continuity of 1(x) are the core of sections 3 and 4 ; 3 is based on a theorem of KOROYKIN [7], whereas 4 is a continuation of papers by WANG XING-HUA [21] and VAN NIEKERK 14. In 5 asymptotic approximation formulae are derived for the operators L n, ¿ ~, (¿=2, fixed) and functions f(x) having a kthl derivative (k = 2) at a fixed point x E[-p,p ]. The case k 2 is covered by MATSUOKA [13]. Section 6, finally, deals with the degree of approximation b~ the operators L n, ¿ ~, for functions f(x) belonging to certain classes of Zygmund. Part of this section is contained in PETROV [16], who considers a more general class of functions.