ISSN:
1588-273X
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1010332530381
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