Density of quantized approximations

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)=₁+…+xₙ xₖ∈ M, n∈ 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.