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This article is cited in 16 scientific papers (total 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.
Received: 24.10.2008 and 01.04.2009
Citation:
P. A. Borodin, “Approximation by simple partial fractions on the semi-axis”, Mat. Sb., 200:8 (2009), 25–44; Sb. Math., 200:8 (2009), 1127–1148
Linking options:
https://www.mathnet.ru/eng/sm7466https://doi.org/10.1070/SM2009v200n08ABEH004031 https://www.mathnet.ru/eng/sm/v200/i8/p25
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Abstract page: | 1045 | Russian version PDF: | 250 | English version PDF: | 12 | References: | 70 | First page: | 31 |
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