Tauberian theorems on ℝ⁺ and applications

Abstract

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