Best monotone approximations in \(L_ 1[0,1]\)
The author proves the following theorems. Theorem 1. If \(\tilde f \)(resp. \(\underset \tilde{} f)\) is constant on (a,b)C\(\Omega\), then there exists a disjoint sequence (ai,bi): \(i\geq 1\) of subintervals of (a,b) such that \(\underset \tilde{} f\) (resp. f) is constant on each (ai,bi) and \(\mu [U_{i\geq 1}(ai,bi)]=b-a\). Theorem 2. If \(g\in \mu_ 1(f/M)\) and K is defined as in lemma 3, then g is constant on each component of [K\(\neq 0]\). Theorem 3. If \(f\in L_ 1\), then \(g\in \mu_ i(f/M)\) if and only if \(g\in M\), \(f\leq g\leq \tilde f\), g is constant on each component of V and for almost every x in \(\Omega\)-V, \(g(x)=f(x)\). Theorem 4. Let \(f\in L_ 1A\) g in \(\mu_ 1(f/M)\) is an extreme point of \(\mu_ 1(f/M)\) if and only if, for any \(n\geq 1\), \(g_ n\in C1\{g_ m:g_ m=\tilde f_ m\) or \(g_ m=\underset \tilde{} f_ m\}.\) In support of his theorems the author proves 3 lemmas and two corollaries.
scientific article; zbMATH DE number 3967002