Smoothing \(\Lambda\)-sequences (Q1898970)

Let \(\Lambda= (\lambda_n)\) be a nondecreasing sequence such that \(\lambda_n> 0\) for all \(n\), \(\lambda_n\nearrow \infty\) and \(\sum 1/\lambda_n= \infty\), and let \(\Gamma= (\gamma_n)\) satisfy the same conditions. Let \(\Lambda BV\) be the class of functions \(f\), defined in an interval (finite or infinite), such that \(\sum |f(b_n)- f(a_n)|/ \lambda_n< \infty\) for every sequence of nonoverlapping intervals \([a_n, b_n]\). It is shown by construction that for any \(\Lambda\) there exists a \(\Gamma\) such that \(\Gamma BV= \Lambda BV\) and \(\gamma_{n+ 1}/ \gamma_n\to 1\).