RUS  ENG
Full version
JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb., 2016 Volume 207, Number 3, Pages 19–30 (Mi sm8500)

This article is cited in 18 papers

Approximation by simple partial fractions with constraints on the poles. II

P. A. Borodin

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: It is shown that if a compact set $K$ not separating the plane $\mathbb C$ lies in the union $\widehat{E}\setminus E$ of the bounded components of the complement of another compact set $E$, then the simple partial fractions (the logarithmic derivatives of polynomials) with poles in $E$ are dense in the space $AC(K)$ of functions that are continuous on $K$ and analytic in its interior. It is also shown that if a compact set $K$ with connected complement lies in the complement $\mathbb C\setminus\overline{D}$ of the closure of a doubly connected domain $D\subset \overline{\mathbb C}$ with bounded connected components of the boundary $E^+$ and $E^-$, then the differences $r_1-r_2$ of the simple partial fractions such that $r_1$ has its poles in $E^+$ and $r_2$ has its poles in $E^-$ are dense in the space $AC(K)$.
Bibliography: 9 titles.

Keywords: simple partial fractions, uniform approximation, restriction on the poles, neutral distribution, condenser.

UDC: 517.538.5

MSC: 41A20, 30E10

Received: 02.03.2015

DOI: 10.4213/sm8500


 English version:
Sbornik: Mathematics, 2016, 207:3, 331–341

Bibliographic databases:


© Steklov Math. Inst. of RAS, 2025