Saturation of positive linear operators
Toshihiko Nishishiraho · Tohoku Mathematical Journal · 1976
Introduction.Roughly speaking, the phenomenon of saturation of approximation is that there exists an "optimal" order of approximation, called "saturation order," such that better approximation occurs only in trivial cases.This is exactly defined as follows (cf.P. L. Butzer and R. J. Nessel [2]):Let B be a Banach space with norm || ||, and let (L t ) be a net of bounded linear operators of B into itself, converging strongly to the identity operator, which will be called a strong approximation process on B. Denote by T[B; (Li)] the closed linear subspace of B, consisting of all f in B for which L^f) = / for all i.Suppose that there exists a net (φ t ) of positive real numbers, converging to zero such that every f in B for which \\L t (f) -f\\ = o(φ t ) belongs to T[B; (L,)], and there exists a g in B but not in T[B; (L t )] such that || L t (g) -g\\ = O(φ t ).Then the strong approximation process (L t ) is said to be saturated in B with order (φ t ).The set T[B; (L { )] and the net (φi) are called the trivial class of (L^ and the saturation order of (L t ), respectively.Furthermore, the set S[B; (Li)] consisting of all/in B f or which \\L,(f) -f\\ = O(φi) is called the saturation class of (L t ).The saturation problem may actually consist of two different questions: firstly, the question of whether saturation holds, that is, the establishment of the existence of the saturation order of a given strong approximation process (L % ) on B; secondly, the characterization of the saturation class S[B; (L t )].The purpose of this paper is to establish a result concerning the first problem of saturation of positive linear operators on C(X), the Banach space of all real-valued continuous functions on a compact Hausdorff space X with sup-norm || ||.The applications will be made to the Bernstein-Schnabl functions constructed by M. W. Grossman [3].The arguments of this paper can be based on the author [4].Throughout this paper, &~ will be a subset of C(X), separating the points of X. 1 will denote the unit function on X. A saturation theorem in C(X). The main result of this paper