Entropy numbers of Besov classes of generalized smoothness on the sphere

We investigate the asymptotic behavior of the entropy numbers of Besov classes $$BB_{p,\theta }^\Omega \left( {\mathbb{S}^{d - 1} } \right)$$ of generalized smoothness on the sphere in $$L_q \left( {\mathbb{S}^{d - 1} } \right)$$ for 1 ≤ p, q, θ ≤ ∞, and get their asymptotic orders. We also obtain the exact orders of entropy numbers of Sobolev classes $$BW_p^r \left( {\mathbb{S}^{d - 1} } \right)$$ in $$L_q \left( {\mathbb{S}^{d - 1} } \right)$$ when p and/or q is equal to 1 or ∞. This provides the last piece as far as exact orders of entropy numbers of $$BW_p^r \left( {\mathbb{S}^{d - 1} } \right)$$ in $$L_q \left( {\mathbb{S}^{d - 1} } \right)$$ are concerned.

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