Continuity envelopes of spaces of generalised smoothness, entropy and approximation numbers

We study continuity envelopes in spaces of generalised smoothness B-pq((s,Psi)) and F-pq((s,Psi)) and give some new characterisations for spaces B-pq((s,Psi)). The results are applied to obtain sharp asymptotic estimates for approximation numbers of compact embeddings of type id: B-pq((s1,Psi)) (U) --> B-infinityinfinity(s2) (U), where n/(p) over bar < s(1) -s(2) < n/(p) over bar+ 1 and U stands for the unit ball in R-n. In case of entropy numbers we can prove two-sided estimates. (C) 2004 Elsevier Inc. All rights reserved.