On a Korovkin-type theorem for simultaneous approximation

Abstract

Several years ago, Knoop and Pottinger [9] (see also 1201) proved a quantitative Korovkin-type theorem for some positive linear operators acting on the Banach space C’(K) of real-valued and r-times continuously differentiable functions on a compact interval K of the real axis. Recently, the same authors proved [lo] an analogous quantitative theorem for functions defined on an unbounded interval of the real axis. The purpose of the present paper is to generalize their first non-quantitative assertion for compact intervals 16, Korollar 2.23 by replacing the so-called almostconvexity property by the condition of convexity-preserving of two types of higher-order convex functions and by adding another condition :in terms of CebySev norms and derivatives of order Y. These two types of higher-order convex functions are some spline functions and some monosplines. Using a result of Bojanic and Roulier [2] (for a history of this result see the next section), one may give a system of Y (types of) higher-order convex functions which may be used in Knoop-Pottinger’s result but only when the operators L,, are applicabe also to functions in c’~ '(K), say. However, if L,, , n 3 1, act only on c“ ‘(K), just two more simpler types of higher-order convex functions will be indicated. These generalizations are motivated by the fact that even classical instances of positive linear operators such as those of Meyer-Konig and Zeller have the property of simultaneous approximation but do not entirely satisfy the conditions of Knoop’s and Pottinger’s result (see [ 13, 91). The present paper is organized in the following manner. The next section contains the notations and the basic concepts used in the sequel. In the third section we review the previous results on test function theorems on simultaneous approximation while in Section 4 the main results are stated and proved. In the last section we show that the main results, are more general than the result of Knoop and Pottinger, among other remarks. 223 002 l-9045/90 $3.00