Approximation of smooth functions on compact two-point homogeneous spaces

Estimates of Kolmogorov $n$-widths $d_n(B_p^r,L^q)$ and linear $n$-widths ${\text{da}}_n(B_p^r,L^q)$, ($1\le q\le \mathrm{\infty }$) of Sobolev’s classes $B_p^r$, ($r>0$, $1\le p\le \mathrm{\infty }$) on compact two-point homogeneous spaces (CTPHS) are established. For part of $(p,q)\in [1,\mathrm{\infty }]\times [1,\mathrm{\infty }]$, sharp orders of $d_n(B_p^r,L^q)$ or ${\text{da}}_n(B_p^r,L^q)$ were obtained by Bordin, Kushpel, Levesley and Tozoni in a recent paper `` J. Funct. Anal. 202 (2) (2003), 307–326’'. In this paper, we obtain the sharp orders of $d_n(B_p^r, L^q)$and$\da_n (B_p^r, L^q)$foralltheremaining$ (p,q)$. Our proof is based on positive cubature formulas and Marcinkiewicz-Zygmund type inequalities on CTPHS.