Best Approximation Rate of Constants by Simple Partial Fractions and Chebyshev Alternance

M. A. Komarov

Matematičeskie zametki, Tome 97 (2015) no. 5, pp. 718-732

Résumé

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 $n$.

Détail

Keywords: best approximation of constants, simple partial fraction, Chebyshev alternance.