Tauberian Conditions under Which Convergence of Integrals Follows from Summability (C, 1) over R+

Given f ∈ L loc 1 (R +), we define $$s\left( t \right): = \int\limits_0^t {f\left( x \right)} {\text{ }}dx{\text{ and }}\sigma \left( t \right):\frac{1}{t}{\text{ }}\int\limits_0^t {s\left( u \right)} {\text{ }}du{\text{ for }}t >0$$ Our permanent assumption is that σ(t) → A as t → ∞, where A is a finite number. First, we consider real-valued functions, and prove that s(t) → A as t → ∞ if and only if two one-sided Tauberian conditions are satisfied. In particular, these two conditions are satisfied if s(t) is slowly decreasing (or increasing) in the sense of R. Schmidt; in particular, if f(x) obeys a Landau type one-sided Tauberian condition. Second, we extend these results for complex-valued functions by giving a two-sided Tauberian condition, being necessary and sufficient in order that σ(t) → A imply s(t) → A as t → ∞. In particular, this condition is satisfied if s(t) is slowly oscillating; in particular if f(x) obeys a Landau type two-sided Tauberian condition.