Direct and inverse results in variable Hilbert scales

Variable Hilbert scales are an important tool for the recent analysis of inverse problems in Hilbert spaces, as these constitute a way to describe smoothness of objects other than functions on domains. Previous analysis of such classes of Hilbert spaces focused on interpolation properties, which allows us to vary between such spaces. In the context of discretization of inverse problems, first results on approximation theoretic properties appeared. The present study is the first which aims at presenting such spaces in the context of approximation theory. The authors review and establish direct theorems and also provide inverse theorems, as such are common in approximation theory.