The K-functional and saturated linear approximation processes

Abstract

It is shown that already the knowledge of the saturation class , {{mathfrak {F}}}(J_t, L^p) of a saturated convolution approximation process , {J_t} on , L^p({{mathbb {R}}}^n),, 1le p< infty , completely determines its norm approximation behavior. This is achieved by using the , K-functional , K(t,f;L^p,{{mathfrak {F}}}(J_t, L^p)) 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|^lambda -factor, , lambda >0. Proofs are mainly based on the Fourier transformation.