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A criterion for the best uniform approximation by simple partial fractions in terms of alternance. II
M. A. Komarov Vladimir State University
Abstract:
In the problem of approximating real functions
$f$
by simple partial fractions of order
${\le}\,n$ on closed intervals
$K=[c-\varrho,c+\varrho]\subset\mathbb{R}$, we obtain a criterion
for the best uniform approximation which is similar to Chebyshev's
alternance theorem and considerably generalizes previous results:
under the same condition $z_j^*\notin B(c,\varrho)=
\{z\colon|z-c|\le\varrho\}$ on the poles
$z_j^*$ of the fraction
$\rho^*(n,f,K;x)$ of best approximation, we omit the restriction
$k=n$ on the order
$k$ of this fraction. In the case of approximation
of odd functions on
$[-\varrho,\varrho]$, we obtain a similar
criterion under much weaker restrictions on the position
of the poles
$z_j^*$: the disc
$B(0,\varrho)$ is replaced by the
domain bounded by a lemniscate contained in this disc. We give some
applications of this result. The main theorems are extended to the case
of weighted approximation. We give a lower bound for the distance
from
$\mathbb{R}^+$ to the set of poles of all simple partial fractions
of order
${\le}\,n$ which are normalized with weight
$2\sqrt x$ on
$\mathbb{R}^+$
(a weighted analogue of Gorin's problem on the semi-axis).
Keywords:
simple partial fraction, approximation, alternance, uniqueness, disc,
odd function, lemniscate.
UDC:
517.538
MSC: 41A20,
41A50 Received: 15.03.2016
Revised: 05.05.2016
DOI:
10.4213/im8548
Fulltext:
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