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Best Approximation Rate of Constants by Simple Partial Fractions and Chebyshev Alternance
M. A. Komarov Vladimir State University
Abstract:
We consider the problem of interpolation and best uniform approximation of constants
$c\ne 0$ by simple partial fractions
$\rho_n$ of order
$n$ on an interval
$[a,b]$. (All functions and numbers considered are real.) For the case in which
$n>4|c|(b-a)$, we prove that the interpolation problem is uniquely solvable, obtain upper and lower bounds, sharp in order
$n$, for the interpolation error on the set of all interpolation points, and show that the poles of the interpolating fraction lie outside the disk with diameter
$[a,b]$. We obtain an analog of Chebyshev's classical theorem on the minimum deviation of a monic polynomial of degree
$n$ from a constant. Namely, we show that, for
$n>4|c|(b-a)$, the best approximation fraction
$\rho_n^*$ for the constant
$c$ on
$[a,b]$ is unique and can be characterized by the Chebyshev alternance of
$n+1$ points for the difference
$\rho_n^*-c$. For the minimum deviations, we obtain an estimate sharp in order
$n$.
Keywords:
best approximation of constants, simple partial fraction, Chebyshev alternance.
UDC:
517.538 Received: 24.02.2014
Revised: 21.10.2014
DOI:
10.4213/mzm10470
Fulltext:
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