Approximation of smooth functions on compact two-point homogeneous spaces

Abstract

Estimates of Kolmogorov n-widths d n ( B p r , L q ) and linear n-widths δ n ( B p r , L q ) , ( 1 ⩽ q ⩽ ∞ ) of Sobolev's classes B p r , ( r > 0 , 1 ⩽ p ⩽ ∞ ) on compact two-point homogeneous spaces (CTPHS) are established. For part of ( p , q ) ∈ [ 1 , ∞ ] × [ 1 , ∞ ] , sharp orders of d n ( B p r , L q ) or δ n ( B p r , L q ) were obtained by Bordin et al. (J. Funct. Anal. 202(2) (2003) 307). In this paper, we obtain the sharp orders of d n ( B p r , L q ) and δ 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.