Representation, interpolation, and reiteration theorems for generalized approximation spaces

Abstract

We show that the representation theorem for classical approximation spaces can be generalized to spaces A(X,lq(ℬ))={f∈X:{En(f)}∈lq(ℬ)} in which the weighted lq-space lq(ℬ) can be (more or less) arbitrary. We use this theorem to show that generalized approximation spaces can be viewed as real interpolation spaces (defined with K-functionals or main-part K-functionals) between couples of quasi-normed spaces which satisfy certain Jackson and Bernstein-type inequalities. Especially, interpolation between an approximation space and the underlying quasi-normed space leads again to an approximation space. Together with a general reiteration theorem, which we also prove in the present paper, we obtain formulas for interpolation of two generalized approximation spaces.