Geodesic
Approximation of functions on the sphere
Izvestiya. Mathematics , Tome 30 (1988) no. 3, pp. 599-614.

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The authors consider classes $H_p^r(\sigma)$ of functions $f$ on a sphere $\sigma$, whose smoothness is determined by the properties of differences along geodesics (duly averaged) in the metric of $L_p(\sigma)$. An integral representation of a function $f \in L_p(\sigma)$ is obtained in terms of the differences mentioned. On this basis direct and inverse theorems on approximation of functions $f \in H_p^r(\sigma)$ be polynomials in spherical harmonics are established. These theorems completely characterize the class $H_p^r(\sigma)$. Bibliography: 9 titles. 
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S. M. Nikol'skii; P. I. Lizorkin. Approximation of functions on the sphere. Izvestiya. Mathematics , Tome 30 (1988) no. 3, pp. 599-614. http://geodesic.mathdoc.fr/item/IM2_1988_30_3_a7/

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