A variation of rational \(L_ 1\) approximation
Consider the norm on the space of continuous functions functions on the unit interval, introduced by \textit{A. Pinkus} and \textit{O. Shisha} [ibid. 35, 148-168 (1982; Zbl 0532.41032)]: \[ ||| f||| =\sup_{0\leq a<b\leq 1}\{| \int^{b}_{a}f(x)dx|:\;f(x)\neq 0\text{ on } (a,b)\}. \] It is a variation on the usual \(L_ 1\)-norm and was studied in the setting of best polynomial approximation. The authors of the paper under review extend the problem to best rational approximation: given \(f\in C[0,1]\), find \(r_ 0\in R^ n_ m\) (the set of all irreducible rational functions p/q with deg \(p\leq n\), deg \(q\leq m\) and \(q>0\) on [0,1]) such that \(||| f-r_ 0||| =\inf \{||| f-r|||:\;r\in R^ n_ m\}.\) The main results of the paper are: \(\bullet\) for \(m\leq 1\) every \(f\in C[0,1]\) has at least one best approximant, \(\bullet\) for \(m\geq 2\) there exists a function \(f\in C[0,1]\) without a best approximant from \(R^ n_ m,\) \(\bullet\) characterization of \(r_ 0=p_ 0/q_ 0\) by the existence of at least \(s+1\) so called alternating extremal intervals \((s=\max \{\deg p_ 0+m\), deg \(q_ 0+n\}+2),\) \(\bullet\) every \(f\in C[0,1]\) has at most one best approximant from \(R^ n_ m\) (if it has one, it is strongly unique), \(\bullet\) the operator T: \(f_ 0\to r_ 0\in R^ n_ m\), associating the best approximant to a function \(f\in C[0,1]\), is continuous with respect to the ordinary Chebyshev-norm at all elements \(f_ 0\) such that \((\deg p_ 0-n)(\deg q_ 0-m)=0,\) \(\bullet\) the operator T is discontinuous with respect to \(||| \cdot |||\) everywhere in C[0,1]. The style of the paper is very compact.
scientific article; zbMATH DE number 4205060