Embeddings of block-radial functions -- approximation properties and nuclearity

Let $R_{\gamma }{B}_{p,q}^s(\mathbb{R}d)$ be a subspace of the Besov space $B_{p,q}^s({\mathbb{R}}^d)$ that consists of block-radial (multi-radial) functions. We study an asymptotic behaviour of approximation numbers of compact embeddings $id:R_{\gamma }{B}_{p_1,q_1}^{s_1}({\mathbb{R}}^d)\to R_{\gamma }{B}_{p_2,q_2}^{s_2}({\mathbb{R}}^d)$. Moreover we find the sufficient and necessary condition for nuclearity of the above embeddings. Analogous results are proved for fractional Sobolev spaces $R_{\gamma }{H}_p^s({\mathbb{R}}^d)$.