Unicity in one-sided \(L_ 1\)-approximation and quadrature formulae
The following approximation problem is investigated. Let an n-dimensional subspace G of C(T) (T a compact metric space) which contains a strictly positive function and \(f\in C(T)\) be given. A function \(g_ f\in G\) with \(g_ f\leq f\) is called best one-sided \(L_ 1\)-approximation of f, if \(\| f-g_ f\|_ 1\leq \| f-g\|_ 1\) for all \(g\in G\) with \(g\leq f\). It is shown that there is a connection of global unicity in one-sided \(L_ 1\)-approximation and the non-existence of quadrature formulas. Every function \(f\in C(T)\) has a unique best one-sided \(L_ 1\)-approximation from G if and only if there does not exist a quadrature formula with at most n-1 points and positive weights which is exact for all \(g\in G\). Moreover, it is proved that if \(n\geq 2\) and T has at most n-1 components, then for each space G there exists such a quadrature formula. In this way a general non-unicity theorem on best one-sided \(L_ 1\)-approximations is obtained. Special cases of this theorem were proved earlier for \(T=[a,b]\) and additional assumptions on the space G by several authors. The above result does not hold if T has more than n-1 components.
scientific article; zbMATH DE number 3931656
scientific article