About some approach to the theory of Nikolski\v{i}–Besov spaces on homogeneous manifolds

Let $M$ be a compact symmetric space of rank 1. We have defined the Nikolski\v{i}–Besov function classes $B_{p,\theta}^r(M)$, $r>0$, $1\leq\theta\leq\infty$, $1\leq p\leq\infty$, and we have obtained a constructive description of these classes in terms of the best approximations of functions $f\in L_p(M)$ by the spherical polynomials on $M$.