Rational approximation in uniform and weighted \(L^ 1\) norm
The authors give an estimate for the constant of best approximation of functions from a class of functions having non-uniform smoothness behaviour on \([-1,1]\) in the following sense: \(f \in C^ p ([-1,1])\), \(p \geq 1\), \(f \in C^ q ([c,d])\), \(p < q < \infty\), for every closed segment \([c,d]\) in \([-1,1] \backslash \{a\}\), \(| a | \leq 1\), \(\lim_{x \to a, | x | \leq 1} (a-x)^ k\), \(f^{(p+k)} (x)\) is finite for \(k = 1, \dots, q - p\). Furthermore they consider a class of functions having an integrable singularity in \(a \in [-1,1]\) and that in addition belong to \(C^ q ([c,d])\) for every closed segment \([c,d] \subset [-1,1] \backslash \{a\}\). For this class rational approximation in \(L^ 1\)-norm with respect to a weight function having an algebraic singularity in \([-1,1]\) is studied.
scientific article; zbMATH DE number 721013
scientific article