Approximations on compact symmetric spaces of rank 1

On an arbitrary Riemannian symmetric space $M$ of rank 1 the Nikol'skii classes $H_p^r(M)$ are defined by considering differences along geodesics. These spaces are described in terms of the best approximations by polynomials in spherical harmonics on $M$, that is, by linear combinations of the eigenfunctions of the Laplace–Beltrami operator on $M$. The results of Nikol'skii and Lizorkin on the approximation of functions on the sphere $S^n$ are generalized.