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

Abstract:

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 $\mathbb {R}$.