Entropy and Widths of Multiplier Operators on Two-Point Homogeneous Spaces

Using multiplier operators we introduce a thin scale of function spaces on symmetric two-point homogeneous manifolds. Different spaces of smooth functions including sets of finite, infinite and analytic smoothness are considered. Sharp in sense of order estimates of respective entropy numbers and $n$-widths are established for a general class of multiplier operators. Various applications of these resultsare considered for different multiplier operators. In particular, sharp order estimates of entropy and widths of Sobolev's classes are found. A range of sharp order estimates for entropy and widths isestablished for sets of finitely and infinitely smooth and analytic functions on two-point homogeneous manifolds. The results we derive are apparently new even in the one dimensional case.