Approximation of smooth functions on compact two-point homogeneous spaces

Gavin Brown; Feng Dai

paper

Approximation of smooth functions on compact two-point homogeneous spaces

arXiv:math/0510007

Abstract

Estimates of Kolmogorov $n$ -widths $d_{n} (B_{p}^{r}, L^{q})$ and linear $n$ -widths $\da_n(B_p^r, L^q)$, ( $1 \leq q \leq \infty$ ) of Sobolev's classes $B_{p}^{r}$ , ( $r > 0$ , $1 \leq p \leq \infty$ ) on compact two-point homogeneous spaces (CTPHS) are established. For part of $(p, q) \in [1, \infty] \times [1, \infty]$ , sharp orders of $d_{n} (B_{p}^{r}, L^{q})$ or $\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 $\da_n (B_p^r, L^q)$ for all the remaining $(p, q)$ . Our proof is based on positive cubature formulas and Marcinkiewicz-Zygmund type inequalities on CTPHS.