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JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb., 2009 Volume 200, Number 8, Pages 25–44 (Mi sm7466)

This article is cited in 16 papers

Approximation by simple partial fractions on the semi-axis

P. A. Borodin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: This paper investigates the simple partial fractions (that is, the logarithmic derivatives of polynomials) all of whose poles lie within the angular domain $\Lambda_\gamma=\{z:\arg z\in(\gamma,2\pi-\gamma)\}$, for any $\gamma\in[0,\pi/2]$. It is shown that they are contained in a proper half-space of the space $L_p({\mathbb R}_+)$ for any $p\in(1,p_0)$ (in particular, they are not dense in this space) and conversely, they are dense in $L_p({\mathbb R}_+)$ for any $p\geqslant p_0$, where $p_0=(2\pi-2\gamma)/(\pi-2\gamma)$. The distances from the poles of a simple partial fraction $r$ to the semi-axis ${\mathbb R}_+$ are estimated in terms of the degree of the fraction $r$ and its norm in $L_2({\mathbb R}_+)$. The approximation properties of sets of simple partial fractions of degree at most $n$ are investigated, as well as properties of the least deviations $\rho_n(f)$ from these sets for the functions $f\in L_2({\mathbb R}_+)$.
Bibliography: 14 titles.

Keywords: approximation, simple partial fraction, integral metrics.

UDC: 517.538.5

MSC: 30E10, 41A20

Received: 24.10.2008 and 01.04.2009

DOI: 10.4213/sm7466


 English version:
Sbornik: Mathematics, 2009, 200:8, 1127–1148

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