n ‐Term Approximation by Controllable Step Functions

Controllable step functions on a compact metric space ( X , d ) are defined on partitions of X into subsets whose size is subject to an entropy condition. The paper deals with the class A ( X ) of all bounded real‐valued functions f ∈ M ( X ) which can be approximated uniformly by controllable step functions. We show that every function f ∈ A ( X ) is a controllable step function itself or can be approximated by a sequence of controllable step functions on an ascending chain K = ( n ) ∞ n =1 of controllable partitions n . For functions f chain‐approximable in this sense a discrete version of the formula representing the corresponding approximation quantities â n ( f ) can be derived. Furthermore, chain‐approximable functions turn out to be quasi‐continuous. On the m ‐dimensional cube [–1, 1] m with the maximum metric d ∞ chain‐approximable functions are even Riemann integrable. In the case m = 1 all quasi‐continuous regulated functions prove to be chain‐approximable.