Weighted rational minimax families and defect

Let $I=[a,b]$ with $a<b$, let $m,n\in {\mathbb{N}}_0$, and let $f,w\in C(I,\mathbb{R})$ with $w(x)>0$ for every $x\in I$. A normalized reduced pair is a pair of real polynomials $p,q$ such that $q>0$ on $I$, $\mathrm{deg}p\le m$, $\mathrm{deg}q\le n$, $p$ and $q$ have no nonconstant common divisor, and $\mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f},2}:={\left({\sum }_{j=0}^n{q}_j^2\right)}^{1\mathrm{/}2}=1\text{when }q(x)={\sum }_{j=0}^n{q}_j{x}^j\mathrm{.}$ The zero rational function is represented by $(p,q)=(0,1)$. By Reduced real rational representations are unique up to scale, each rational function admissible under these conditions has exactly one normalized reduced pair. Define ${\mathcal{R}}_{m,n}^{+}(I):=\{p\mathrm{/}q:(p,q)\text{ is a normalized reduced pair}\}$ and $E_{m,n}(f,w;I):={\inf }_{r\in {\mathcal{R}}_{m,n}^{+}(I)}\mathrm{\parallel }w(f-r){\mathrm{\parallel }}_{\mathrm{\infty },I}\mathrm{.}$ The family is nonempty because it contains $r=0$. For each admissible $r=p\mathrm{/}q$, $q>0$ on $I$ makes $r$ continuous by Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, so $w(f-r)$ is continuous and bounded on compact $I$. The infimum is therefore finite, bounded below by zero, and exists as a real number by completeness of $\mathbb{R}$.

For $r=p\mathrm{/}q\mathrm{\ne }0$ in reduced form, put $d_{m,n}(r):=\min \{m-\mathrm{deg}p,n-\mathrm{deg}q\},N_{m,n}(r):=m+n-d_{m,n}(r)\mathrm{.}$ For $r=0$, define $d_{m,n}(0):=n$ and hence $N_{m,n}(0):=m$. The integer $d_{m,n}(r)$ is the defect. A candidate is normal when $d_{m,n}(r)=0$ and defective when $d_{m,n}(r)>0$. The alternation count associated with the candidate is $N_{m,n}(r)+2$; it equals $m+n+2$ only in the normal case.