Tauberian theorems on ℝ⁺ and applications

Let f f be a bounded and uniformly continuous function from R R to a Banach space X X and s p ( f ) sp(f) be its Carleman spectrum. A classical Tauberian theorem states that f f is constant if and only if s p ( f ) ⊂ { 0 } sp(f) ⊂ \{ 0 \} , and f f is ω ω -periodic if and only if s p ( f ) ⊂ 2 π ω Z sp(f) ⊂ 2π /ω Z for some ω > 0 ω >0 . However, one cannot expect analogous results on R + R^+ since there is a counterexample showing that the case of R + R^+ contrasts dramatically with the case of R R . In this paper, we succeed in extending the above classical Tauberian theorem to R + R^+ and obtain an extension of the well-known Ingham theorem. We also apply our Tauberian theorems to abstract Cauchy problems and improve a result in [Russian Math. 58 (2014), pp. 1–10]. Moreover, as an application, we present an extension of a Katznelson-Tzafriri theorem in [J. Funct. Anal. 103 (1992), pp. 74–84] with weaker assumptions. In addition, it is interesting to note that several of our results and examples show that S S -asymptotically ω ω -periodic functions on R + R⁺ is just the “natural” analogue of periodic functions on R R .