Lower bounds of cowidths and widths of multiplier operators

Abstract

The main objective of this article is to present new results on optimal reconstruction of function classes on probability spaces (Ω,A,ν) in the standard Lq spaces. We consider the problem of optimal reconstruction in the sense of the respective cowidths of standard function classes ΛUp generated by multiplier or pseudo differential operators Λ:Lp→Lq, 1≤p,q≤∞. Our approach is based on the estimates of volumes of John-Löwner ellipsoids and expectations of norms induced by orthonormal systems on (Ω,A,ν). It is shown that the results obtained are order sharp in many cases. In particular, we obtain sharp orders of entropy of Sobolev classes W∞γ, γ>0 in L1 and n-widths of ΛUp in Lq, 1<q≤p<∞ in the case of two-point homogeneous spaces and torus.