Average complexity for linear operators over bounded domains

AbstractSuppose one wants to compare worst case and average complexities for approximation of a linear operator. In order to get a fair comparison the complexities have to be obtained for the same domain of the linear operator. In previous papers, average complexity was studied when the domain was the entire space. To avoid trivial results, worst case complexity has been studied for bounded domains, and in particular, for balls of finite radius. In this paper we study the average complexity for approximation of linear operators whose domain is a ball of finite radius q. We prove that the average complexities even for modest q and for q = +∞ are closely related. This and existing results enable us to compare the worst case and average complexities for balls of finite radius. We also analyze the average complexity for the normalized and relative errors. The paper is illustrated by integration of functions of one variable and by approximation of functions of d variables which are equipped with a Wiener measure