On trigonometric n-widths and their generalization

is called the n-width of Q in X (in the sense of Kolmogorov). The left infimum in (1) is taken over all n-dimensional linear subspaces r,, c X. One can obtain various modifications of this definition by taking the inlimum over special classes of i;, . For instance, in X = L,[O, 27r] one may consider subspaces r,, spanned by any tz of the functions {exp(ik . )}, k E Z. If the infimum in (1) is taken over all such subspaces, the corresponding n-width, introduced by Ismagilov [I], is called the trigonometric n-width, d,T(R, L,). It is obvious that d,T > d,, but if the class 0 is translation-invariant, one would expect that dc = d,. And indeed, in all cases for which dz has been estimated, d,T d,(n + 03). In a more general setting, let G be a compact Abelian group with the invariant measure ,u, p(G) = 1. In X = L, = L,(G, p) we consider the subspaces r, of the form