Bernstein and Gelfand widths of certain classes of analytic functions. II

Abstract

The Gelfand widths of the unit ball of H2(v) We emphasize that no conditions are imposed on v, except G(v) # 0 and the condition of finiteness. If G(v) = 0, then the order of decay of the sequence {d"(Ja)} changes. Now, let tt be an arbitrary finite Borel measure on T~, and let G(#) be the geometric mean of t* with respect to the normalized Lebesgue measure on T~. We consider the embedding operator Thus, this paper is aimed at lifting all irrelevant conditions on the measure. In passing, we discuss a Bernstein-type inequality for polynomials and for certain spaces of rational functions. At the end, we discuss the limit case in Theorem 2, namely, the embedding J : H ~ --+ L~(T, #). Proof of Theorem t.Though the lower estimate is an easy consequence of the results of [1], we recall its proof for the sake of Completeness. We introduce the embedding operator *We recall that G(v) = exp(f tn(dv/dm)drn). T j0 __+