The K-functional and saturated linear approximation processes

It is shown that already the knowledge of the saturation class F(Jₜ,Lᵖ) of a saturated convolution approximation process {Jₜ} on Lᵖ(ℝⁿ),1≤p<∞, completely determines its norm approximation behavior. This is achieved by using the K-functional K(t,f;Lᵖ,F(Jₜ,Lᵖ)) as a comparison scale, which relates on the one hand the approximation process and on the other appropriate moduli of smoothness. This implies that simultaneously one gets the so-called direct and inverse theorems. There are open problems if the saturation order is slightly perturbed, e.g., by a |log t|^λ-factor, λ>0. Proofs are mainly based on the Fourier transformation.