RUS  ENG
Full version
JOURNALS // Uspekhi Matematicheskikh Nauk // Archive

Uspekhi Mat. Nauk, 2023 Volume 78, Issue 5(473), Pages 3–64 (Mi rm10115)

This article is cited in 3 papers

Density of quantized approximations

P. A. Borodinab, K. S. Shklyaevab

a Faculty of Mechanics and Mathematics, Moscow State University
b Moscow Center for Fundamental and Applied Mathematics

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 $R(M)=\{x_1+\dots+x_n\colon x_k\in M,\ n\in {\mathbb N}\}$ 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.

Keywords: approximation, additive semigroup, density, Banach space, simple partial fractions, shifts, integer coefficients.

UDC: 517

MSC: Primary 41A20, 41A29, 41A65; Secondary 30E10, 46B25

Received: 19.04.2023

DOI: 10.4213/rm10115


 English version:
Russian Mathematical Surveys, 2023, 78:5, 797–851

Bibliographic databases:


© Steklov Math. Inst. of RAS, 2025