Approximation of integrable, approximately continuous functions on \((0,1)^ n\) by nondecreasing functions (Q808356)

The main emphasis of this paper is on the convergence of sequences of best monotone approximations of integrable, absolutely continuous functions on \((0,1)^ n\) in the \(L_ 1\) norm. The first result concerns the best nondecreasing \(L_ 1\) approximation \(f_ 1\) to an approximately continuous function f. It is shown that if the sequence \(f^ m\) converges to \(f_ 1\) in the \(L_ 1\) norm, then \(f^ m_ 1\) also converges to \(f_ 1\) in the \(L_ 1\) norm. The second result concerns the best nondecreasing \(L_ p\) approximation \(f_ p\) to an approximately continuous function f. It is shown that \(f_ p\) converges to \(f_ 1\) a.e. and in the \(L_ 1\) norm as p converges to 1.