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| Entropy numbers measure the compactness of operators(or the set) in a qualitative way. The asymptotic decay at infinityof the sequence of entropy numbers of a compact operator T describes the degree of compactness of T. They have manyapplication in the theory of function spaces and spectral theory, signal and image processing, probabilitytheory, learning theory,etc.. In this paper, the asymptoticbehavior of the entropy numbers of Besov classes$BB_{p, heta}^{Omega}(mathbb{S}^{d-1})$ of generalized smoothnesson the sphere in $L_q(ss)$ for $1leq p, q, hetaleqinfty$ isinvestigated, and their asymptotic orders are gotten. The exactorders of entropy numbers of Sobolev classes$BW_p^r(mathbb{S}^{d-1})$ in $L_q(mathbb{S}^{d-1})$ when $p$and/or $q$ is equal to $1$ or $infty$ are also obtained. Thisprovides the last piece as far as exact orders of entropy numbers of$BW_p^r(mathbb{S}^{d-1})$ in $L_q(mathbb{S}^{d-1})$ are concerned. |
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| Keywords:Approximation of functions; entropynumbers; Besov classes of generalized smoothness; Sobolev classes; discretization theorem |
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