Properties of Korovkin type in \(L^ p(a,b)\)
Let \(\Phi_ i(t)=t^ i\), \(i=0,1,2\). The author shows that a sequence of selfadjoint positive linear operators converges to the identity on \(L^ p(a,b)\), \(p\geq 1\), if and only if it converges for \(\Phi_ 0\) and \(\Phi_ 1\), and the operators are uniformly bounded. Quantitative results in terms of the first modulus of continuity and orders of approximation for the functions \(\Phi_ 0\) and \(\Phi_ 1\) are presented. Furthermore, estimates are given for the order of approximation by a sequence of positive linear, uniformly bounded operators \(L_ n\) in terms of the orders of approximation for \(\Phi_ 0\), \(\Phi_ 1\), \(\Phi_ 2\), and the first modulus of continuity of f.
scientific article; zbMATH DE number 4134727