Density of quantized approximations

Abstract

The paper contains a review of known results and proofs of new results on conditions on a set $M$ in a Banach space $X$ that are necessary or sufficient for the additive semigroup $R(M)=\{x_1+…+x_n\colon x_k\in M, n\in {\mathbb N}\}$ to be dense in $X$. We prove, in particular, that if $M$ is a rectifiable curve in a uniformly smooth real space $X$, and $M$ does not lie entirely in any closed half-space, then $R(M)$ is dense in $X$. We present known and new results on the approximation by simple partial fractions (logarithmic derivatives of polynomials) in various spaces of functions of a complex variable. Meanwhile, some well-known theorems, in particular, Korevaar's theorem, are derived from new general results on the density of a semigroup. We also study approximation by sums of shifts of one function, which are a natural generalization of simple partial fractions. Bibliography: 79 titles.