Approximation by simple partial fractions on the semi-axis

P. A. Borodin

Sbornik. Mathematics, Tome 200 (2009) no. 8, pp. 1127-1148

Résumé

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.

Détail

Keywords: approximation, simple partial fraction, integral metrics.