Approximation on compact symmetric spaces of rank one (Q1360756)

This paper concerns the generalization of \textit{S. M. Nikol'skij} and \textit{P. I. Lizorkin}'s results [Tr. Mat. Inst. Steklova 166, 186-200 (1984; Zbl 0582.41008); Mat. Zametki 41, 509-516 (1987; Zbl 0656.42002)]; Izv. Akad. Nauk SSSR, Ser. Mat. 51, 635-651 (1987; Zbl 0717.41061)] on the approximation of functions on a sphere to arbitrary compact symmetric spaces \(M\) of rank one. The main result proved in the paper is the following Theorem: (a) If \(f\in H^r_p (M)\), then \[ E_m(f)_p\leq C|f|_{H^r_p}/m^r, \quad m=1,2,3, \dots, \] where the constant \(C\) does not depend on \(f\) and \(m\). (b) Conversely, if \(E_m (f)_p \leq c_1/m^r\), \(m=1,2, \dots\), then \(f\in H^r_p(M)\), and \(|f|_{H^r_p(M)} \leq c_2 (|f|_p+ c_1)\). (For notations, one may refer to the original paper).