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Zbl 1057.41008
Blatt, Hans-Peter; Grothmann, René; Kovacheva, Ralitza
Poles and alternation points in real rational Chebyshev approximation. (English)
[J] Comput. Methods Funct. Theory 3, No. 1-2, 165-177 (2003). ISSN 1617-9447; ISSN 2195-3724/e

The concept of alternation sets is well known in the theory of best polynomial approximation and the extension to real rational approximation has been studied extensively. \par Let $f\in C[-1,1]$ be real valued and let ${\cal R}_{n,m}$ denote the set of rational functions of the form $r=p/q$, where $q\not\equiv 0$ and $p$ and $q$ are polynomials of degrees at most $n$ and $m$ respectively. For each pair $(n,m)$ of nonnegative integers there then exists a unique function $r^{\ast}_{n,m}\in{\cal R}_{n,m}$ that satsfies $$ \| f-r^{\ast}_{n,m}\| <\| f-r\| \text { for all }r\in{\cal R}_{n,m}, r\not= r^{\ast}_{n,m}, $$ where $\| \cdot \| $ is the sup-norm on $C[-1,1]$. Write $r^{\ast}_{n,m}=p^{\ast}_n/ q^{\ast}_m$, with numerator and denominator having no common factors, and define the {\it defect\/} by $$ d_{n,m}=\min\{n-\deg p^{\ast}_n, m-\deg q^{\ast}_m\}. $$ Then there exist $\delta=m+n+2-d_{n,m}$ points $x_k^{(n,m)}$ on the closed interval $[-1,1]$ $$ -1\leq x_1^{(n,m)}<\cdots <x_{\delta}^{(n,m)}\leq 1 $$ satisfying $$ \lambda_{n,m}(-1)^k(f-r^{\ast}_{n,m}) (x_k^{(n,m)})=\| f-r^{\ast}_{n,m}\| ,\quad 1\leq k\leq \delta, $$ where $\lambda_{n,m}=\pm 1$, fixed; this is called an alternation set (in general not unique). For the sequel, let for each pair $(n,m)$ an arbitrary, fixed alternation set for the best approximation to $f$ be given by $$ A_{n,m}=A_{n,m}(f):= \{x_k^{(n,m)}\}_{k=1}^{\delta}. $$ \par In the paper of {\it P. B. Borwein}, {\it A. Króo}, {\it R. Grothmann}, {\it E. B. Saff}, Proc. Am. Math. Soc. 105, No.4, 881-888 (1989; Zbl 0688.41018)], it is shown that a subsequence of alternation sets is dense in $[-1,1]$ whenever $n/m\rightarrow\kappa,\ \kappa>1$. In ``Behavior of alternation points in best rational approximation'', [Acta Appl. Math. 33, No.2-3, 195-210 (1993; Zbl 0801.41021)], the connection between denseness and the number of poles of $r^{\ast}_{n,m}$ that lie outside an $\varepsilon$-neighborhood of $[-1,1]$ is studied. \par In the paper under review the main result now is a new theorem that implies (amongst others) many of the results found in the papers quoted above. Let $f$ be a nonrational function and let $n,m(n)$ satisfy $$ m(n)\leq n,m(n)\leq m(n+1)\leq m(n)+1. $$ Then there exists a subsequence $\Lambda\subset\text {\bf N}$ such that $$ \nu_n-\alpha_n\widehat{\tau}_n-(1-\alpha_n) \mu\overset\to{\ast\atop \rightarrow} 0 \text { as }n\rightarrow\infty,n\in\Lambda $$ in the weak$^{\ast}$ topology; here $$ \alpha_n={\ell_n\over \delta}, $$ and $$ \nu_n(A):={\#\{x_k^{(n)} : x_k^{(n)}\in A\}\over \delta} $$ the normalized counting measure. Furthermore if $y_1,\ldots,y_{\ell}$ are the zeros of the polynomial $Q_n(z)=q^{\ast}_{m(n)} q^{\ast}_{m(n+1)}$ of degree $\ell$, then $$ \tau_n(A)={\#\{y_i : y_i\in A\} \over \ell_n} $$ (the normalized counting measure of all poles of $r^{\ast}_n,r^{\ast}_{n+1}$), and the function $\widehat{\tau}$ is introduced using the logarithmic potential of $\nu$. \par A clearly written and interesting paper.
[Marcel G. de Bruin (Delft)]

MSC 2000:: *41A20 Approximation by rational functions
41A50 Best approximation

Keywords: rational approximation; best approximation; alternation points

Citations: Zbl 0688.41018; Zbl 0801.41021

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