Tauberian conditions under which convergence follows from Abel summability

In this work we prove that one-sided slow oscillation of a sequence and that of its generator sequence are Tauberian conditions for the Abel summability method, using a corollary to Karamata’s Main Theorem [J. Karamata, Über die Hardy–Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes, Math. Z. 32 (1930) 319–320]. It is also shown that such conditions are Tauberian conditions for generalized Abelian summability methods.