Relations between \(\Lambda\)BV and BV\((p(n) \uparrow \infty)\) classes of functions (Q812786)

The author proves embedding relations between the Waterman's class \(\Lambda\text{BV}\) and the generalized Wiener's class \(\text{BV}(p(n)\uparrow \infty)\) of functions. It is shown: Theorem 1. \(\Lambda\text{BV}\subset \text{BV}(p(n)\uparrow \infty)\) if and only if \[ \overline{\lim}_{n\rightarrow \infty} \sup_{1\leq m\leq 2^n} {m^{1/p(n)} \over \sum_{i=1}^m (1/\lambda_i)} < \infty . \] Theorem 2. Let \(\sum_{i=1}^{\infty} (1/\lambda_i) = +\infty \). Then there is a function \(f\in \text{BV}(p(n)\uparrow \infty)\cap C[0,1]\) such that \(f\not\in \Lambda\text{BV}\).