A unified treatment of tractability for approximation problems defined on Hilbert spaces

Abstract

A large literature specifies conditions under which the information complexity for a sequence of numerical problems defined for dimensions $1, 2, \ldots$ grows at a moderate rate, i.e., the sequence of problems is tractable. Here, we focus on the situation where the space of available information consists of all linear functionals and the problems are defined as linear operator mappings between Hilbert spaces. We unify the proofs of known tractability results and generalize a number of existing results. These generalizations are expressed as five theorems that provide equivalent conditions for (strong) tractability in terms of sums of functions of the singular values of the solution operators.