Some problems in the theory of approximation of functions on compact homogeneous manifolds

Problems in the theory of approximation of functions on an arbitrary compact rank-one symmetric space $M$ in the metric of $L_p$, $1\le p\le\infty$, are investigated. The approximating functions are generalized spherical polynomials, that is, linear combinations of eigenfunctions of the Beltrami-Laplace operator on $M$. Analogues of the direct Jackson theorems are proved for the modulus of smoothness (of arbitrary order) constructed by using the operator of spherical averaging. It is established that the modulus of smoothness and the $K$-functional constructed from the Sobolev-type space corresponding to the Beltrami-Laplace differential operator are equivalent.
Bibliography: 35 titles.