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Poles and alternation points in real rational Chebyshev approximation. (English) Zbl 1057.41008

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.

Let $f\in C\left[-1,1\right]$ be real valued and let ${ℛ}_{n,m}$ denote the set of rational functions of the form $r=p/q$, where $q¬\equiv 0$ and $p$ and $q$ are polynomials of degrees at most $n$ and $m$ respectively. For each pair $\left(n,m\right)$ of nonnegative integers there then exists a unique function ${r}_{n,m}^{*}\in {ℛ}_{n,m}$ that satsfies

$\parallel f-{r}_{n,m}^{*}\parallel <\parallel f-r\parallel \phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}r\in {ℛ}_{n,m},r\ne {r}_{n,m}^{*},$

where $\parallel ·\parallel$ is the sup-norm on $C\left[-1,1\right]$. Write ${r}_{n,m}^{*}={p}_{n}^{*}/{q}_{m}^{*}$, with numerator and denominator having no common factors, and define the defect by

${d}_{n,m}=min\left\{n-deg{p}_{n}^{*},m-deg{q}_{m}^{*}\right\}·$

Then there exist $\delta =m+n+2-{d}_{n,m}$ points ${x}_{k}^{\left(n,m\right)}$ on the closed interval $\left[-1,1\right]$

$-1\le {x}_{1}^{\left(n,m\right)}<\cdots <{x}_{\delta }^{\left(n,m\right)}\le 1$

satisfying

${\lambda }_{n,m}{\left(-1\right)}^{k}\left(f-{r}_{n,m}^{*}\right)\left({x}_{k}^{\left(n,m\right)}\right)=\parallel f-{r}_{n,m}^{*}\parallel ,\phantom{\rule{1.em}{0ex}}1\le k\le \delta ,$

where ${\lambda }_{n,m}=±1$, fixed; this is called an alternation set (in general not unique). For the sequel, let for each pair $\left(n,m\right)$ an arbitrary, fixed alternation set for the best approximation to $f$ be given by

${A}_{n,m}={A}_{n,m}\left(f\right):={\left\{{x}_{k}^{\left(n,m\right)}\right\}}_{k=1}^{\delta }·$

In the paper of P. B. Borwein, A. Króo, R. Grothmann, 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 $\left[-1,1\right]$ whenever $n/m\to \kappa ,\phantom{\rule{4pt}{0ex}}\kappa >1$. 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 ${r}_{n,m}^{*}$ that lie outside an $\epsilon$-neighborhood of $\left[-1,1\right]$ is studied.

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 $n,m\left(n\right)$ satisfy

$m\left(n\right)\le n,m\left(n\right)\le m\left(n+1\right)\le m\left(n\right)+1·$

Then there exists a subsequence ${\Lambda }\subset \mathbf{\text{N}}$ such that

${\nu }_{n}-{\alpha }_{n}{\stackrel{^}{\tau }}_{n}-\left(1-{\alpha }_{n}\right)\mu \stackrel{}{\genfrac{}{}{0pt}{}{*}{\to }}0\phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}n\to \infty ,n\in {\Lambda }$

in the weak${}^{*}$ topology; here

${\alpha }_{n}=\frac{{\ell }_{n}}{\delta },$

and

${\nu }_{n}\left(A\right):=\frac{#\left\{{x}_{k}^{\left(n\right)}:{x}_{k}^{\left(n\right)}\in A\right\}}{\delta }$

the normalized counting measure. Furthermore if ${y}_{1},...,{y}_{\ell }$ are the zeros of the polynomial ${Q}_{n}\left(z\right)={q}_{m\left(n\right)}^{*}{q}_{m\left(n+1\right)}^{*}$ of degree $\ell$, then

${\tau }_{n}\left(A\right)=\frac{#\left\{{y}_{i}:{y}_{i}\in A\right\}}{{\ell }_{n}}$

(the normalized counting measure of all poles of ${r}_{n}^{*},{r}_{n+1}^{*}$), and the function $\stackrel{^}{\tau }$ is introduced using the logarithmic potential of $\nu$.

A clearly written and interesting paper.

##### MSC:
 41A20 Approximation by rational functions 41A50 Best approximation, Chebyshev systems