Tauberian theorems on \mathbb{R}^{+} and applications.

Let f be a bounded and uniformly continuous function from \mathbb {R} to a Banach space X and \mathbf {sp}(f) be its Carleman spectrum. A classical Tauberian theorem states that f is constant if and only if \mathbf {sp}(f) \subset \{ 0 \}, and f is \omega-periodic if and only if \mathbf {sp}(f) \subset \frac {2\pi }{\omega } \mathbb {Z} for some \omega >0. However, one cannot expect analogous results on \mathbb {R}^+ since there is a counterexample showing that the case of \mathbb {R}^+ contrasts dramatically with the case of \mathbb {R}. In this paper, we succeed in extending the above classical Tauberian theorem to \mathbb {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 \mathcal {S}-asymptotically \omega-periodic functions on \mathbb {R}^{+} is just the "natural" analogue of periodic functions on \mathbb {R}. [ABSTRACT FROM AUTHOR]