光滑函数在紧致两点齐性空间上的逼近
Approximation of smooth functions on compact two-point homogeneous spaces
摘要：
建立了Sobolev类$B_p^r$，（$r>0$，$1\leq p\leq \infty$）在紧两点齐性空间（CTPHS）上的Kolmogorov $n$-宽度$d_n(B_p^r, L^q)$和线性$n$ -宽度$\da_n(B_p^r, L^q)$，（$1\leq q\leq \infty$）的估计.对于$(p, q)\in[1,\infty]\times[1,\infty]$的一部分，Bordin，Kushpel，Levesley和Tozoni在最近的一篇论文“J. Funct.”中获得了$d_n(B_p^r, L^q)$或$\da_n (B_p^r, L^q) $的精确顺序。Anal. 202（2）（2003），307- 326“。在本文中，我们得到了所有剩余的$ (p,q)$的$d_n(B_p^r, L^q)$和$\da_n (B_p^r, L^q)$的锐阶。我们的证明是基于CTPHS上的正体积公式和Marcinkiewicz-Zygmund型不等式。
英文摘要：
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.
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