Approximation and entropy numbers of embeddings between approximation spaces

We consider general linear approximation spaces $X^b_q$ based on a quasi-Banach space $X$, and we analyse the degree of compactness of the embedding $X^b_q \hookrightarrow X$. Applications are given to periodic Besov spaces on the $d$-torus, including spaces of generalized and logarithmic smoothness. In particular, we obtain the exact asymptotic behavior of approximation and entropy numbers of embeddings of such Besov spaces in Lebesgue spaces and in Besov spaces of logarithmic smoothness.